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CANER TASKIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 2009 Z. Caner Taskm To my wife, my parents and my brother; I owe them everything I have ACKNOWLEDGMENTS I would like to express my deepest gratitude to Cole Smith for his wise, enlightening ideas, endless motivation, and patient counseling during the writing of this dissertation. He has been a great teacher, a mentor and a friend to me in the last four years, and working with him has been a privilege. I would like to thank Edwin Romeijn for introducing me to the exciting field of optimization in health care, and his invaluable contributions to this study. His guidance and support has been very helpful throughout my graduate education. I would also like to acknowledge Panos Pardalos and Douglas Dankel for taking part in my dissertation committee, and their valuable sI i..ii My sincere thanks are due to my friends in graduate school. In particular, Chase has not only been a great colleague, but also a close friend for these four years. I cannot begin to count the things that I learned from him about the culture, the 11i; I le and the language. My experience in America would not have been nearly as enjo',1'1 without him and his wife Candace. I am indebted to my parents and my brother for guiding and supporting me in all life choices I have made. Finally, I am deeply grateful to my lovely wife, Semra, for her constant encouragement, support, and understanding. She is the source of my happiness, and the secret of my success. This dissertation represents the end of my life as a graduate student. It also represents the beginning of a new stage in my life, every moment of which I am looking forward to sharing with her. TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. 4 LIST OF TABLES ....................... ............. 8 LIST OF FIGURES .................................... 9 ABSTRACT . . . . . . . . . . 10 CHAPTER 1 INTRODUCTION ................................ 13 2 STOCHASTIC EDGEPARTITION PROBLEM ....... ......... 19 2.1 Introduction and Literature Survey ........ .............. 19 2.2 Formulation and Cutting Plane Approach ...... ............ 23 2.3 A Hybrid IP/CP Approach ........ ........... .... 33 2.3.1 FirstStage Problem ................... ..... 35 2.3.2 SecondStage Problem ......................... 36 2.3.2.1 Foundations ........... ............... 37 2.3.2.2 Domain expansion .................. ..... 38 2.3.2.3 Constraint propagation ............. .. .. 39 2.3.2.4 Forward checking ................ .... .. 40 2.3.2.5 Node selection rule ............. .... .. 41 2.3.2.6 Distribution vector ordering rule . . ..... 41 2.3.3 ThirdStage Problem .................. ..... .. 42 2.3.4 Infeasibility Analysis .................. ..... .. 42 2.3.5 Enhancements for the FirstStage Problem ..... . . 43 2.3.5.1 Valid inequalities ... . . . 43 2.3.5.2 Heuristic for obtaining an initial feasible solution . 44 2.3.5.3 Processing integer solutions .... . . 45 2.4 Computational Results .................. ........ .. .. 45 3 CONSECUTIVEONES MATRIX DECOMPOSITION PROBLEM ...... 51 3.1 Introduction and Literature Survey .................. .... 51 3.2 Decomposition Algorithm .................. ........ .. 55 3.2.1 Decomposition Framework .................. .. 56 3.2.2 Master Problem Formulation and Solution Approach . ... 58 3.2.3 Subproblem Analysis and Solution Approach . . ... 61 3.2.4 Valid Inequalities for the Master Problem . . . 66 3.2.4.1 Beamontime and number of apertures inequalities . 66 3.2.4.2 Bixel subsequence inequalities . . ...... 68 3.2.5 Constructing a Feasible Matrix Decomposition . . ... 69 3.3 Computational Results and Comparisons ..... . ... 73 3.3.1 Problem Instances ............ . . .. 73 3.3.2 Implementation Issues . . ........ .... 73 3.3.3 Comparison with Langer et al. (2001) Model . . 74 3.3.4 Random Problem Instances .................. .. 75 3.3.5 Clinical Problem Instances ................ .... .. 80 4 RECTANGULAR MATRIX DECOMPOSITION PROBLEM . ... 84 4.1 Introduction and Literature Survey ................. .. 84 4.2 A MixedInteger Programming Approach .................. .. 86 4.2.1 Model Development .................. ........ .. 86 4.2.2 Valid Inequalities .................. ....... .. .. 90 4.2.2.1 Adli .. ii rectangles ................. . .. 90 4.2.2.2 Bounding box inequalities ................. .. 90 4.2.2.3 Aggregate intensity inequalities . . ...... 93 4.2.2.4 Special submatrices ................. . .. 94 4.2.2.5 Submatrix inequalities .............. .. .. 96 4.2.3 Partitioning Approach ................ ... .. 97 4.2.3.1 Separable components ............. .. .. 97 4.2.3.2 Independent regions ................ .... .. 98 4.2.3.3 Dependent regions ................ .. .. 100 4.2.3.4 Upper bound calculation ... . . ... 102 4.3 Extensions ................... . . .. 104 4.3.1 Minimize Total Treatment Time ............. .. 104 4.3.2 Optimization with BeamonTime Restrictions . . ... 105 4.4 Computational Results ............... ......... 106 5 GRAPH SEARCH PROBLEM .................. ........ .. 113 5.1 Introduction and Literature Review ................ . 113 5.2 HideandSeek Problem ............... ......... .. 115 5.2.1 Mathematical Model ............... .... .. 115 5.2.2 Solution Approach. .................... ........ .. 116 5.2.2.1 Searcher's problem ................ .. .. 117 5.2.2.2 Branchandprice algorithm ... . . 119 5.3 Pursuit Evasion Problem ............... ........ .. 119 5.3.1 Mathematical Model ............... .... .. 119 5.3.2 Solution Approach. .................... ........ 120 5.3.2.1 Searcher's problem ................ 120 5.3.2.2 Intruder's problem ................ 122 5.3.2.3 Branchcutprice algorithm. .... . ... 122 5.4 Patrol Problem ............... ............ 123 5.4.1 Problem Description ............... .... .. 123 5.4.2 Mathematical Model ............... .... .. 123 5.4.2.1 Searcher's problem ................ 124 5.4.2.2 Intruder's problem ................ 125 5.4.2.3 Branchcutprice algorithm. .... . ... 127 5.5 Branching Strategies ............... .......... 127 5.6 Computational Results .................. ........ .. 129 6 CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS . .... 134 6.1 Stochastic EdgePartition Problem ................... 134 6.2 Matrix Decomposition Problem .................. .. ... 135 6.3 Graph Search Problem . . . ....... . 138 6.4 Master Problem Reformulation in BiLevel Cutting Plane Optimization Algorithms .................. ............. .. 139 APPENDIX A AN IP MODEL FOR C1MATRIX DECOMPOSITION PROBLEM ...... 144 B COMPLEXITY OF C1PARTITION .................. ...... 147 REFERENCES .................. ................ .. .. 149 BIOGRAPHICAL SKETCH .................. ............. 158 LIST OF TABLES Table 21 Descriptions of the problem instances used for comparing algorithms . 22 Comparison of the algorithms on graphs having edge density =0.2 . 23 Comparison of the algorithms on graphs having edge density =0.3 . 24 Comparison of the algorithms on graphs having edge density = 0.4 . 25 Descriptions of the problem instances used for analyzing threestage algoi 26 ThreeStage algorithm on graphs having edge density 0.2 . . 27 ThreeStage algorithm on graphs having edge density =0.3 . . 28 ThreeStage algorithm on graphs having edge density = 0.4 . .. 31 Dimensions of clinical problem instances . ............. 32 Comparison of our base algorithm with Langer et al. (2001) model . 33 Effect of rotating the MLC head ............... . ..... 34 Computational results for our base algorithm . ..... 35 Comparison of heuristic algorithms on clinical data . . 41 Effect of valid inequalities and the partitioning strategy . . 42 Computational results on model extensions . ..... 43 Effect of maximum intensity value on solvability . ..... i page 46 . 47 . 48 . 49 thm 49 . 49 . 50 . 50 . 73 . 75 . 81 . 81 . 83 . 108 . 109 . 111 51 Average number of branchandbound nodes explored for hideandseek problem 52 Average number of searchers needed for the hideandseek problem .. ..... 53 Number of instances that are solved within time limit for the pursuit evasion p rob lem . . . . . . . . . . 54 Average number of branchandbound nodes explored for the pursuit evasion p rob lem . . . . . . . . . . 55 Average number of searchers needed for the pursuit evasion problem ....... 56 Number of instances that are solved within time limit for the patrol problem . 57 Average number of branchandbound nodes explored for the patrol problem . 58 Average number of searchers needed for the patrol problem .. .......... LIST OF FIGURES Figure page 21 (a) An instance of the deterministic edgepartition problem (b) A solution with K I = 3, 3, b = 20 .. .. ... .. .. .. ... .. .. .. .. .. ..... 19 31 (a) A multileaf collimator system (b) The projection of an aperture onto a patient 51 32 Comparison of total treatment times on random data . . 76 33 Comparison of the number of apertures on random data .......... .77 34 Comparison of beamontime values on random data . ..... 78 35 Comparison of TGI values on random data ............... .... 79 41 Example fluence map ............... ............. .. 87 42 Example start index ............... ............. .. 91 43 Example end index ............... .............. .. 91 44 Example bounding box ............... .......... .. .. 92 45 Another nondominated bounding box seeded at (6,3) .............. ..93 46 Two components of a fluence map ............... ...... 98 47 Regions of a connected component ................ ..... 98 48 Efficient frontier for number of apertures and beamontime . .... 110 51 (a) An example graph (b) Timeexpanded network for T = 2 . . ... 118 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ALGORITHMS FOR SOLVING MULTILEVEL OPTIMIZATION PROBLEMS WITH DISCRETE VARIABLES AT MULTIPLE LEVELS By Z. Caner Takmn August 2009 ('!C i: J. Cole Smith Major: Industrial and Systems Engineering In this dissertation, we investigate a class of multilevel optimization problems, in which discrete variables are present at multiple stages. Such problems arise in many practical settings, and they are notoriously difficult to optimize. Benders decomposition, which is a wellknown decomposition method for solving largescale mixedinteger programming problems, cannot be utilized for the class of problems that we consider due to the existence of discrete variables at lower levels. Cutting plane algorithms such as those proposed by Laporte and Louveaux have been designed for use in bilevel integer programming problems with binary variables at both levels. However, these are based on generic cuts, which do not utilize any problem specific structures, and hence often result in weak convergence. Our goal in this dissertation is to propose novel formulation and solution strategies for several multilevel optimization problems to solve these problems to optimality within practical computational limits. We first consider an edgepartition problem. The motivation for this study is provided by a Synchronous Optical Network (SONET) design application. In the SONET context, each edge represents a demand pair between two client nodes, and the weight of each edge represents the number of communication channels needed between the client nodes. We consider a stochastic version of the problem, in which the edge weights are not deterministic, but their underlying probability distribution is known. The problem is to design a set of SONET iiis" at minimum cost, while ensuring that the resulting network can handle the random demand with a prespecified level of service. We first model the problem as a largescale integer program, and attempt to solve it via a bilevel decomposition approach, in which both levels contain binary variables. We then propose a threelevel solution approach for the problem, which is based on a hybrid integer programming/constraint programming decomposition algorithm. We show computationally that the hybrid algorithm significantly outperforms the other approaches. Next, we focus on a matrix decomposition problem, which arises in Intensity Modulated Radiation Therapy (IMRT) treatment planning. The problem input is a matrix of intensity values that needs to be delivered to a patient, which must be decomposed into a collection of apertures and corresponding intensities. In a feasible decomposition the sum of binary shape matrices multiplied by corresponding intensity values is equal to the original intensity matrix. We consider two variants of the problem: (i) a variant in which the shape matrices used in the decomposition have to satisfy the "consecutiveones" property, and (ii) a variant in which the shape matrices have to be rectangular. For the first variant, we start by investigating an integer programming model proposed in the literature, and show how the formulation can be strengthened. We then formulate the problem as a bilevel optimization problem that has discrete variables at both stages, and ~i,. 1 a hybrid integer programming/constraint programming decomposition algorithm similar to our algorithm for the edgepartition problem. Our tests on data obtained from patients show that our algorithm is capable of solving problem instances of clinically relevant dimensions within practical computational limits. We then turn our attention to the second variant of the matrix decomposition problem. We formulate the problem as a mixedinteger program, and investigate a decomposition method for solving it. Unlike the first variant, the secondlevel problem turns out to be a linear programming problem, and therefore we are able to derive a Benders decomposition algorithm for solving this variant. Finally, we investigate a class of graph search problems. In this class of problems, an intruder is located at an unknown node on the input graph, and a group of searchers needs to be coordinated to detect the intruder within a limited amount of time. This problem arises in settings such as searchandrescue and searchandcapture operations, and patrol route design. We investigate three variants of the problem: (i) a hideandseek version, in which a stationary intruder is hiding at an unknown node, (ii) a pursuit evasion version, in which the intruder can move across the edges of the graph to avoid being detected, and (iii) a patrol problem, in which the searchers are assigned to recurring patrol routes to protect the graph from intrusion. We model each problem as a largescale integer program, and propose a branchcutprice algorithm to find the minimum number of searchers needed, and a route for each searcher. In our formulation, both the master problem and the subproblems corresponding to the searchers and the intruder contain binary variables. We model the master problem as a set covering problem and propose a solution approach that is based on dynamic column and row generation. CHAPTER 1 INTRODUCTION In most complex decisionmaking environments, there exist several types of interdependent decisions that need to be made to optimize some cost or benefit function. As a simple example, consider a production planning problem. The goal is to determine, at the very least, the types of products that are to be produced within a time period, along with the associated production quantities. There might exist individual restrictions on each type of decision, such as "the total amount of production of products A and B cannot exceed a," or "if product A is produced, so must product B." There might also exist restrictions that relate the two types of decisions, such as "if product A is produced, then the batch size must be at least 3." Modeling an optimization problem involves (i) defining a decision variable for each individual decision, (ii) formulating the restrictions on the decisions as constraints, and (iii) defining an objective function to be optimized. The field of mathematical programming seeks to optimize such models and prove the optimality of the generated solution, or prove that no feasible solution exists. Optimization problems in which some variables are restricted to take values from a discrete set are classified as discrete optimization problems. An important concern regarding building and solving discrete optimization problems is that the amount of memory and the computational effort needed to solve such problems grow exponentially with the number of discrete variables. The traditional approach, which involves making all decisions simultaneously by solving a monolithic optimization problem, quickly becomes intractable as the number of discrete variables increases. Multilevel optimization algorithms, such as Benders decomposition (Benders, 1962), have been developed as an alternative solution methodology to alleviate this difficulty. Unlike the traditional approach, these algorithms divide the decisionmaking process into several stages. For instance, in Benders decomposition a firststage master problem is solved for a subset of variables, and the values of the remaining variables are determined by a secondstage subproblem given the values of the firststage variables. In our simplistic production planning example, the master problem selects a subset of the products to be produced, by considering only the restrictions such as "if product A is produced, so must product B." Then, given the set of selected products, a subproblem seeks the production quantities considering more detailed production restrictions such as "the total amount of production of products A and B cannot exceed a," and "if a product is produced, then the batch size must be at least 3." If the subproblem is able to find a solution for the secondstage variables, then a solution for the overall problem can be obtained by combining the first and secondstage decisions. However, the subproblem can also determine that the current selection of products by the master problem does not yield a feasible solution when the detailed production constraints are considered. In this case, a constraint that eliminates the current selection of products from further consideration is added to the master problem, which is then resolved. In this manner, the optimization problem is solved via a series of "solution prop(. iI made by the master problem, and i ...i for rejection" by the subproblem. This iterative algorithm eventually converges to an optimal solution for the overall problem. In essence, multilevel optimization algorithms solve a series of small problems instead of a single large problem. Performing multiple iterations is usually justified due to the exponentially larger computational resource requirements associated with solving a larger problem. Furthermore, it is often the case that decisions for several groups of secondstage variables can be made independently given the firststage decisions. In such cases, multilevel optimization algorithms are amenable to parallel implementations. The advent of efficient parallel computing grids has allowed modern bilevel techniques to solve problems that were regarded as intractable before (Ntaimo and Sen, 2005). In some applications, solving problems in multiple stages allows effort to be conserved by avoiding the explicit solution of problems by mathematical programming, such as the evacuation network design algorithm of Andreas and Smith (2009). Multilevel optimization has recently received much attention due to the emerging importance of research in fields like stochastic programming and network interdiction. Many problems can be formulated naturally as multilevel optimization problems (\!'gdalas and Pardalos, 1996; Migdalas et al., 1997), and a wide class of optimization problems can be reformulated as multilevel optimization problems (Huang and Pardalos, 2002). Benders decomposition has been particularly successful in solving mixedinteger linear programming problems arising in a wide variety of applications. In Benders decomposition, discrete variables of the problem are kept in the master problem, and continuous variables are moved to the subproblem. In each iteration, given the values of the discrete variables by the master problem, the subproblem is solved as a linear program, and a cutting plane to be passed back to the master problem is generated using linear programming duality. However, this approach cannot be applied directly when discrete variables also appear in the second stage. The reason is that no dual information can be extracted from the subproblem if the secondstage problem contains integer variables. In this case, one can employ cutting planes such as the generalpurpose cuts of Laporte and Louveaux (1993) and combinatorial Benders inequalities (Codato and Fischetti, 2006). However, these inequalities are often very weak, and result in slow algorithmic convergence. Our main line of research is about designing efficient multilevel optimization algorithms for problems that have discrete variables at multiple stages. We first present our results on an edgepartition problem arising in a telecommunications network design context regarding Synchronous Optical Networks (SONET). The edgepartition problem considers an undirected graph with weighted edges, and simultaneously assigns nodes and edges to subgraphs such that each edge appears in exactly one subgraph, and such that no edge is assigned to a subgraph unless both of its incident nodes are also assigned to that subgraph. Additionally, there are limitations on the number of nodes and on the sum of edge weights that can be assigned to each subgraph (Goldschmidt et al., 2003). We consider a stochastic version of the edgepartition problem in C'!i ipter 2, in which we assign nodes to subgraphs in a first stage, realize a set of edge weights from among a finite set of alternatives, and then assign edges to subgraphs. We first formulate the problem as a monolithic integer programming problem, and show that this approach is not tractable due to the rapidly increasing computational requirements. We then prescribe a bilevel cutting plane approach having integer variables in both stages and examine computational difficulties associated both with the generic inequalities by Laporte and Louveaux (1993) and with our proposed cutting planes. We also prescribe a threelevel hybrid integer programming/constraint programming algorithm having discrete variables at all levels, and discuss how this hybrid approach resolves some of the difficulties associated with the bilevel cutting plane approach. ('!C lpters 3 and 4 consider a problem dealing with the efficient delivery of Intensity Modulated Radiation Therapy (IMRT) to individual patients. In particular, we consider a matrix decomposition problem that arises at the leaf sequencing stage in IMRT treatment planning. The problem input is an integer matrix of intensity values that are to be delivered to a patient from a given beam angle. To deliver this intensity profile to the patient, we must decompose the input matrix into a collection of apertures and corresponding intensities. A feasible decomposition is one in which the original desired intensity profile matrix is equal to the sum of a number of feasible binary matrices multiplied by corresponding intensity values. To most efficiently treat a patient, we wish to minimize a measure of total treatment time, which is given as a weighted sum of the number of apertures and the sum of the aperture intensities used in the decomposition. In ('!C lpter 3, we describe a version of the problem in which each aperture matrix needs to satisfy the "consecutiveones" property, which means that all matrix entries with value 1 in each row of an aperture matrix must be consecutive. Similar to C'! Ilpter 2, we prescribe a bilevel hybrid optimization algorithm in which the master problem is an integer programming problem, and we solve a subproblem for each row of the matrix by a constraint programmingbased backtracking algorithm. ('!I ipter 4 deals with another variant of the matrix decomposition problem, in which only rectangular apertures can be used in the decomposition. We develop a Benders decomposition algorithm for solving this variant. We also propose a scheme for partitioning the problem to obtain simultaneous upper and lower bounds. In ('! Ilpter 5, we study a class of graph search problems, where a group of searchers seek an intruder on a graph. Both the intruder and the searchers are located at some nodes of the graph, and the searcher can only "see" a subset of the nodes from each node. At each time period, both the intruder and the searchers can move along an edge to an .,li i:ent node, or stay at the same node. Our goal is to find the minimum number of searchers needed to locate the intruder within a given time limit. We investigate three variants of the graph search problem: (i) a hideandseek problem, in which a stationary intruder "hides" at an unknown node, (ii) a pursuit evasion problem, in which the intruder moves among the nodes to avoid being detected, and (iii) a patrol problem, in which no intruder is initially in the graph and each searcher patrols the graph to protect it from potential intrusion. We formulate these problems as a set covering problem with an exponential number of variables and constraints, and propose a branchcutprice algorithm for solving it. Both our master problem and the subproblems, which correspond to the intruder and the searchers, have discrete variables. We formulate the intruder's subproblem as a longest path problem on an auxiliary graph, and the searcher's subproblems as mixedinteger programming problems. We conclude our dissertation in ('!C ipter 6, which explores future research directions regarding multilevel optimization algorithms. We first evaluate the approaches taken in the edgepartition, matrix decomposition, and graph search problems described in C'!i ipters 25, and discuss future research topics regarding each application. We then describe our preliminary research on a master problem reformulation technique, which can be used in a variety of bilevel optimization algorithms. This reformulation technique can eliminate an exponential number of iterations by adding a quadratic number of variables to the master problem. Therefore, it has the potential of leading to significant improvements in solvability of a class of bilevel optimization problems. CHAPTER 2 STOCHASTIC EDGEPARTITION PROBLEM 2.1 Introduction and Literature Survey We begin by describing the edgepartition problem of Goldschmidt et al. (2003), which is defined on an undirected graph G(N, E). In the deterministic edgepartition problem, we create a set K of (possibly empty) subgraphs of G such that each edge is contained in exactly one subgraph, subject to certain restrictions on the composition of the subgraphs. These restrictions include the stipulations that an edge cannot be assigned to a subgraph unless both of its incident nodes belong to the subgraph, and that no more than r nodes can be assigned to any subgraph, for some r E Z+. Additionally, each edge (i,j) E E has a positive weight of and the sum of edge weights assigned to each subgraph cannot exceed some given positive number b. The objective of the problem is to minimize the sum of the number of nodes assigned to each subgraph. 2 3 2 3 3 8 3 8 10 10 14 7 6 1 4 7 6 5 12 5\ 12 (a) (b) 5 Figure 21. (a) An instance of the deterministic edgepartition problem (b) A solution with IK = 3, r 3,b =20 Figure 21 illustrates the deterministic edgepartition problem. The graph G and the corresponding edge weights are shown in Figure 2la. Figure 21b shows a feasible solution that partitions G into IKI = 3 subgraphs, where the number of nodes in each subgraph is limited by r = 3, and the total weight assigned to each subgraph is limited by b = 20. Note that the degree of node 4 is three, which implies that it must be assigned to at least two subgraphs, or else there would be at least 4 > r nodes in a subgraph. Similarly, node 5 must be assigned to at least two subgraphs. Since nodes 4 and 5 are assigned to two subgraphs, and every other node is assigned to a single subgraph, the solution represented by Figure 2lb is optimal. Goldschmidt et al. (2003) discuss the edgepartition problem (with deterministic weights) in the context of designing Synchronous Optical Network (SONET) rings. In the SONET context, each edge (i,j) E E represents a demand pair between two client nodes, and the weight ,,' represents the number of communication channels requested between nodes i and j. All telecommunication traffic is carried over a set of SONET rings, which are represented by subgraphs in the edgepartition problem. Since each demand must be carried by exactly one ring, edges must be partitioned among the rings. (Note that the term iiiig" describes only the physical SONET routing structure, and does not place any restrictions on topological properties of demand edges included on a ring. See, e.g., Goldschmidt et al. (2003) for more details.) SONET rings are permitted to carry communication between nodes only if those nodes have been connected to the ring by equipment called AddDrop Multiplexers (ADMs). There are technical limits on the number of ADMs that can be assigned to each ring (e.g., r), and on the total amount of channels (e.g., b) that can be assigned to a ring. Since ADMs are quite expensive, ring networks are preferred that employ as few ADMs as possible, which echoes the edgepartition problem's objective of minimizing the sum of nodes assigned to each subgraph. The primary contribution by Goldschmidt et al. (2003) is an approximation algorithm for a specific case of the edgepartition problem in which all ,' are equal to one. Sutter et al. (1998) propose a columngeneration algorithm for this problem, and Lee et al. (2000b) employ a branchandcut algorithm on a formulation that we adapt. For the case in which the weights on the edges can be split among multiple rings, Sherali et al. (2000) prescribe a mixedinteger programming approach augmented by the use of valid inequalities, antisymmetry constraints, and variable branching rules. Smith (2005) formulates the deterministic version of the edgepartition problem as a constraint program and examines several issues regarding symmetry and search algorithm design. Specifically, she shows how adding .. regate variables that represent the number of node copies (similar to our approach in Section 2.3) can improve performance. We consider a version of the edgepartition problem where the edge weights are uncertain, and are only realized after the nodetosubgraph assignments have been made. As we show in Section 2.2, this framework allows us to design more robust solutions than those in the literature, which are virtually all applied to deterministic data. We seek a minimumcardinality set of nodetosubgraph assignments, such that there exists an assignment of edges to subgraphs satisfying the aforementioned subgraph restrictions with a prespecified high probability. Such a probabilistic constraint is extremely hard to deal with in an optimization framework. One approach, known as scenario approximation (cf. Calafiore and Campi (2005); Luedtke and Ahmed (2008); Nemirovski and Shapiro (2005)) is to draw independent identically distributed (i.i.d.) realizations of the edge weights (called scenarios) and require the nodetosubgraph assignments to admit a feasible edgetosubgraph assignment in each scenario. It can be shown that, with a sufficiently large scenario set, a solution to this scenario approximation solution is feasible to the true probabilistically constrained problem with high confidence. In this study we develop algorithmic approaches for solving the scenario approximation corresponding to the discussed probabilistically constrained edgepartition problem. We refer to this scenario approximation as the stochastic edgepartition problem. Relatively little work has been done in SONET network design when the edge weights are uncertain. Smith et al. (2004) consider the SONET ring design problem in which edge demands can be split among multiple rings and propose a twostage integer programming algorithm. The demand splitting relaxation allows the secondstage problems to be solved as linear programs, and thus standard Benders cuts can easily be derived from the secondstage recourse problems. However, we have secondstage integer programs from which dual information cannot be readily obtained. The edgepartition problem can also be approached using stochastic integer programming theory. For problems having binary firststage variables and mixedinteger secondstage variables, such as our problem, a wellknown decomposition approach is the integer Lshaped method (Laporte and Louveaux, 1993). This method approximates the secondstage value function by linear "cuts" that are exact at the binary solution where the cut is generated, and are underestimates at other binary solutions. Typical integer programming algorithms progress by solving a sequence of intermediate linear programming (LP) problems. Using disjunctive programming techniques, it is possible to derive cuts from the solutions to these intermediate LPs that are valid underestimators of the secondstage value function at all binary firststage solutions (Sen and Higle, 2005; Sherali and Fraticelli, 2002). This avoids solving difficult integer secondstage problems to optimality in all iterations of the algorithm, providing significant computational advantage. Scenariowise decomposition methods have also been proposed (Car0e and Schultz, 1999) as an alternative to the above stagewise decomposition approaches. Here copies of the firststage variables are made corresponding to each scenario and are linked together via nonanticipativity constraints. Our proposed methodology draws on constraint programming and stochastic integer programming theory. Hybrid algorithms of this nature have recently been successfully employ, 1 to solve notoriously difficult problems. Jain and Grossmann (2001) and Boclhii 'vr and Pisaruk (2006) devise hybrid integer programming/constraint programming algorithms for solving machine scheduling problems. Thorsteinsson (2001) proposes a framework for integrating integer programming and constraint programming approaches. Hooker and Ottosson (2003) extend the Benders decomposition framework so that constraint logic programs can be used as subproblems to generate cuts that are added to a mixedinteger linear master problem. A recent work by Hooker (2007) uses logicbased Benders decomposition to solve several planning and scheduling problems. The remainder of this chapter is organized as follows. In Section 2.2, we develop a mixedinteger programming formulation for the stochastic edgepartition problem, and provide cutting planes that can be used within a twostage decomposition algorithm. In Section 2.3, we prescribe an alternative threestage algorithm to overcome the computational difficulties associated with the weakness of the proposed cutting planes. Finally, we compare the efficacy of these algorithms in Section 2.4 on a set of randomly generated test instances. 2.2 Formulation and Cutting Plane Approach Let us introduce binary decision variables xik = 1 if node i is assigned to subgraph k and 0 otherwise, Vi E N, k E K. For this formulation, we specify a value of IKI that is sufficiently large to ensure that a feasible solution exists to the problem (as discussed in Section 2.4). We denote the vector of nodetosubgraph assignments by x. Let w denote the random vector of edge weights with known distribution, and w denote a realization with components ,, .. We define binary decision variables yijk = 1 if edge (i,j) is assigned to subgraph k. Given an allowed violation probability c E (0, 1) the probabilistic edgepartition problem can be formulated as follows: Minimize xaik (21) iEN kEK subject to Y ik Xik E {0, 1} Vi c N, k E K (23) Pr {G(x,w) < b > 1 , (24) where G(x, w) =Minimize z (25) subject to Yijk 1= V(i,j) E, (26) kEK S.,, < z Vk e K (27) (i,j)EE ijk i xk, ijk < k V(i,j) E E, k e K (28) Yijk {0O, 1} V(i,j) E, kE K. (29) The objective (21) minimizes the total number of nodes assigned to subgraphs. Constraints (22) limit the number of nodes assigned to each subgraph. Constraints (26) require that the edges be partitioned among the subgraphs. Constraints (27) compute the maximum assigned weight over all subgraphs. Constraints (28) require that no edge can be assigned to a subgraph unless both of its incident nodes are assigned to that subgraph, and (23) and (29) state logical restrictions on the variables. By convention, the optimal value G(x, w) of the integer program (25)(29) is +oc if the problem is infeasible. Given a nodetosubgraph assignment vector x and edge weight vector w there exists a feasible edgetosubgraph assignment if and only if G(x, w) < b, i.e., the weight assigned to any subgraph does not exceed b. Thus the probabilistic edgepartition problem (21)(24) seeks a minimum cost nodetosubgraph assignment such that the probability that there will be a feasible edgetosubgraph assignment when the edge weights are realized is sufficiently high. To build a scenario approximation of the probabilistic edgepartition problem (21)(24), we generate an i.i.d. sample of w denoted by {wq}q), (we call each realization a scenario). The scenario approximation is then: Minimize xik (210) iEN kEK subject to ik < r Vk e K (211) iEN xik E {0, 1} Vi e N, k e K (212) G(x, wq) where the probabilistic constraint is replaced by the deterministic requirement that there must be a feasible edgetosubgraph assignment for each scenario. As mentioned before we refer to the above problem as the stochastic edgepartition problem. The following result, which follows from the general results in Luedtke and Ahmed (2008), provides justification for considering the scenario approximation. Proposition 1. Let a desired corfi,. ,..' level 6 E (0, 1) be given. If the sample size IQI Q1 > INIKI ln2 ln] (214) then i,.;, feasible solution to the stochastic edgepartition problem (210)(213) is feasible to the probabilistic edgepartition problem (21)(24) with i, ..1,,i./l./:./;/ at least 1 6. Proof. Let X denote the set of solutions satisfying the deterministic constraints (22) and (23), let X' denote the set of feasible solutions to the probabilistic edgepartition problem (satisfying (22)(24)), and let XQ denote the set of feasible solutions to the stochastic edgepartition problem corresponding to a sample Q (satisfying (211)(213)). We want to bound Q\ such that Pr{XQ C X} > 1 6. Consider a solution x E X \ X', i.e., Pr{G(x,w) < b} < 1 c. Then x E XQ if and only if G(x, wq) < b for all q e Q. Since the wq for q E Q are i.i.d. it follows that Pr{x c XQ} < ( e)I I. Now Pr{XQ X'} = Pr{3 x XQ s.t. Pr{G(x, w) < b} < 1 e} < Excx\x, Pr{x E XQ} < IX \ X'(1 c < IX(1 e)Q. Thus Pr{XQ C X'} > 1 X(1 e)IQI. To guarantee that Pr{XQ C X'} > 1 6 we need IX(1 c)I'I < 6 or equivalently Q> cIn X In6/ In a( .) The claimed bound then follows by noting that IX < 2 J11^ and ln(l/(l e)) > e. O The above result i'i' 1 that we can obtain feasible solutions to the probabilistically constrained edgepartition problem by solving the stochastic edgepartition problem with a "not too 1 i;,. number of scenarios. Key to this samplingbased approach is the ability to efficiently solve stochastic edgepartition instances having a modest number of scenarios, which is the motivation of this chapter. Next we describe an extensive form model of the stochastic edgepartition problem. Let E' be the set of edges with nonzero weights under scenario q. We define binary decision variables y 1k 1 if edge (i,j) is assigned to subgraph k in scenario q and 0 otherwise, Vq E Q, (i,j) E Eq, and k c K. The stochastic edgepartition problem can then be formulated as follows: Minimize X ik (215) iEN kEK subject to Y k = Vq e Q, (i,j) e Eq (216) kEK Y ik S". ti' k < b Vq Q, k K (218) (i,j)EEq yik < Xik, jk < Xjk Vq e Q, (i,j) e Et, k e K (219) xik E {0,1} Vi E N, k E K (220) k {0, 1} Vq e Q, (i, j) cE E, k e K. (221) Observe that if one were to solve the above extensive form problem given by (215)(221), integrality restrictions need only be imposed on the yvariables, which would in turn enforce the integrality of the xvariables at optimality. Note also that given a fixed set of xvalues, this problem decomposes into IQI separable integer programs, where the subproblem corresponding to scenario q E Q is given by: Sq(x) = Maximize 0 (222) subject to (216), (218), (219), and (221). Under the foregoing model, it is useful to define ., = min{xik, jk} as a part of the firststage decision variables, V(i,j) E E, k E K. The presence of these variables allow us to formulate stronger cutting planes than would be possible with just xvariables (see also Smith et al. (2004)). Assuming that UqEQE = E, the extensive form problem is now equivalent to: Minimize Y Yxik iEN kEK subject to YXik X Xik, < Xjk >1 V(ij) kEK xik e {0, 1} Vi e N F(v) < b Vq E F(v) Minimize maxf{ ', jk } (i,j)EEq subject to Yijk V(i,j) E Eq kEK Yijk< ., V(i,j) E, k cK yjk {0, 1} V(i, j) E, k cK. The valid inequalities (227) require that for each edge (i,j) E E, both i and j must be assigned to some common subgraph, and are useful in improving the computational efficacy of the decomposition algorithm that we propose. Note that an optimal solution exists in which = min{xik,xjk} V(i,j) E E, k E K, without enforcing integrality restrictions or lower bounds on the vvariables. (224) V(i,j) e k K EE , keK Q, where (230) (223) ( There can exist up to IKI! 1 alternative optimal solutions to this problem by simply reindexing the subgraph indices. These symmetric solutions are known to impede the performance of branchandbound algorithms (Sherali and Smith, 2001; Sherali et al., 2000). To reduce model symmetry we can rewrite the cardinality constraints (225) (or (217) for the extensive form problem) by using the following inequalities: r> x,1 > x .> > Y xiK. (234) iEN iEN iEN For a scenario q and a given vector 9, the problem (230)(233) is essentially an identical parallel machine scheduling problem to minimize makespan (P/ /Cmx) (with some assignment restrictions). In particular, there would be IKI machines and IEqI jobs, whose processing times are given by ,,'., V(i,j) E Eq. Each job must be assigned to exactly one machine, and the vvariables impose some restrictions on the assignments. The integer programming scheme developed in Smith (2004) is tailored for a similar problem in which the (weighted) number of demands that cannot be placed on one of these subgraphs is minimized (i.e., minimum weighted number of tardy jobs). This is not equivalent to solving a minimum makespan problem; however, the optimal solution of Fq(Q) is no more than b if and only if the minimum number of tardy jobs is equal to 0. If a positive lower bound to the problem of minimizing the number of tardy jobs is established, one can terminate the subproblem algorithm and conclude infeasibility. We now present a cutting plane algorithm for solving (224)(229). The scheme relaxes constraints (229) and adds cutting planes as necessary to enforce feasibility to the subproblems. Let us call the problem (224)(228) the master problem ( IP). 1. Solve MP. If MP is infeasible then STOP; the problem is infeasible. Otherwise let 9 be an optimal solution of MP. 2. For q e Q, compute Fq(9). If Fq(9) < b for all q, then STOP; the current solution is optimal. Otherwise, continue to Step 3. 3. Update MP by adding a cutting plane of the form (237) as presented in Remark 1, and return to Step 1. After a finite number of steps, the cutting plane algorithm terminates with an optimal solution or detects infeasibility. Remark 1. Suppose F4(9) > b for some scenario q and a solution vector 9 to MP. Let L be a g/1/,al lower bound on F(v), i.e., L4 < F4(v) for all v. Also 1. it,., I(v) {(ijk) : ., = 1} and O(v) = {(ijk) : ., = 0}. The integer oi//'.:,,r/,'h; cut proposed by Laporte and Louveaux (1993) for this class of problems is given by (F (q) L) < + < (F( L) L)(I(I))) L. (235) (ijk)EI(v) (ijk)EO(v) Since L4 < b for ,:;, feasible instance, and since F4(9) > b by assumption, we can 'il (Clu',,ll ii,,,.J':i,,1 to (235) by dividing both sides by (F4(9) L4) and rounding down to obtain S + (1 )>1 (236) (ijk)EO(V) (ijk)eI(V) However, the following .,:,. ,q,' l;.'; is also a valid cutting plane that dominates (236): S1 .1 > 1. (237) (ijk)EO(v) To see that (237) is valid, consider a solution v' that does not .li; fy the above .,:,. ,;,;,;;.:, i.e., vik = 0 for all (ijk) E O((). Therefore, vik < I ., for all (ijk). Then FV(v') > Fq(i) > b, and v' is not feasible. I,. ,!;,;l.:/;i (237) dominates (236) since the lefthand side of (237) is not more than that of (236), and the righthandsides are both equal to 1. Thus, (237) serves as a cutting plane that can be used in Step 3 of the above il',' .:thm. Another reformulation of our subproblem might admit stronger cutting planes than the ones of the form (237). In the parlance of machine scheduling, instead of trying to minimize the maximum makespan, we may wish to minimize the total sum of tardiness. Let Ck, Vk c K be a nonnegative variable that denotes the amount of capacity deficit in subgraph k. Then, the problem of minimizing the total capacity deficit can be formulated as problem MT (v) below: MT(v) : T(v) = Minimize Yck (238) kEK subject to y = 1 V(ij) Eq (239) kEK y < V1(i,j) E E, k K (240) Ck > q "b./kb Vk c K (241) (i,j)EEq k > 0 Vk K (242) yJ{k E 0, 1} V(i,j) E k K. (243) Clearly, Fq(v) < b if and only if Tq(v) = 0, and so we can replace master problem constraints (229) with the restrictions that Tq(v) = 0 for all scenarios q E Q. If subproblems Tq(v) are used in lieu of Fq(v), we would obtain (236) (directly, this time) from Laporte and Louveaux's integer feasibility cut. However, we can state a stronger cutting plane for a solution vector 9 having TV(r) > 0 for some scenario q, by requiring that the total amount of additional capacity that must be allocated to the collection of subgraphs is at least TV(r). This inequality is formally stated in the following proposition. Proposition 2. Suppose for some solution vector v and for some scenario q E Q, we obtain a lower bound LB4(*) > 0 for TV(r). Then the following .:,., ..,l.:/;o; is a valid cutting plane for problem MP, and is at least as strong as (237): S min{'"', LB(i)} ., > LB4(v). (244) (ijk)EO(v) Proof. Suppose by contradiction that there exists a binary vector v* such that T4(v*) = 0, but E(ijk)eO(v) i"' 6jk < LB(i). Then there exists a solution (y*,c*) to MTV(v*) having c* = 0 Vk e K. We will show that the existence of such a v* contradicts the assumption that LB4(9) is a valid lower bound on TV(9). We now build a solution (y, d) to MTV(9). First, we construct ? as follows: 1.For (i,j) E E, if y*k = 1 and ., 1, then set ijk = 1 as well. 2.For (i,j) E E4, if Yj k 1 and ., 0, then set yj = 1 for any k E K for which (ijk) e I(ir). (Note that (ijk) e 0(() since ;., 0.) 3.Set all other ijk = 0. In other words, y is constructed in two phases. In the first phase, we ensure that if edge (i,j) was assigned to subgraph k in solution y*, then (i,j) is assigned to k in y as well, unless = 0 (prohibiting this assignment). In the second phase, if yjk = 1 but = 0, then we assign (i,j) to any k such that vj k 1. Note that this assignment results in a solution feasible to (239), (240), and (243). Next, let us construct d. Observe that in the first phase of assigning edges to subgraphs based on (ijk) E I(i) for which y*jk 1 no subgraph capacities are violated since c* 0, Vk E K, and so we initialize ck = 0, Vk E K. In the second phase, we guarantee feasibility to (241) (and maintain feasibility to (242)) by increasing ck by 'ii,. Thus (y, d) is a feasible solution to MTq(r). At the end of the second phase of assignments, we have kEK Ckk (ijk)EO(,) ".i' lijk since YZkK Ck is increased by i, only when both vjk 1 and (ijk) E O(9). However, by assumption, we have that Z(ijk)EO(,) II" 'Vjk < LB4((). Since >ZkE k k Y(ijk)EO(v) "' ,ijk, we have that Y:kK Ck < LB'(1), which contradicts the fact that LB(9)) < TV(r). Therefore, all feasible solutions must obey the inequality ,, ., > LB4( ), (ijk)EO(v) from which (244) is readily derived. Finally, by dividing both sides of (244) by LB(91), we see that (244) implies (237). E Remark 2. In cutting plane implementations based on (237), once iw; scenario q is found such that the current v vector is proven to be infeasible with respect to scenario q, a cutting plane is generated and the master problem is resolved. No further scenarios are tested, since an identical cut would be generated for each infeasible scenario. However, a cutting plane implementation based on problem (238) (243) above with cutting planes (244) might '. i". from deriving multiple cuts for each infeasible scenario, since these cuts could be distinct. Remark 3. Smith et al. (2004) explore the inclusion of ',iir.,,,:'1 constraints" in the master problem, which enforce simple necessary conditions for f, ... ,;T/i to SONET problems. Denote the degree of node i E N by deg(i), and the set of nodes adjacent to i by A(i). Lee et al. (:'iiiiii) show that node i must be i',gr./ to at least subgraphs, since otherwise, more than r nodes would be assigned to some 'l'ci.',l Similarly, for scenario q E Q, the total weight associated with node i E N is given by EjeA(i) "'.. Since the total weight that can be assigned to a b;l'ai'.i, is limited by b, CA(i) is a lower bound on the number of copies of node i. We can then compute f [deg(i) EjA(i) 1  Smax r max b (245) and impose the following valid inequalities in the master problem: Y xik >i Vi N. (246) kEK Let i denote a node having the 1r/, / lower bound, so that f? > f Vi E N. Node i can be assigned ar.:l,,I'/.:l; to i1','l,,l, 1,... ,C, and we fix xil = 2 = x = 1 accordingly. Note that the symmetrybr'.rl. .:, constraints (234) need to be adjusted so that they are enforced separately for I.'l'g.'li' 1,... ,Cf, and f+ 1,..., IK1. Sherali et al. (2000) show comp,llI.:..a,,ll;, that such a variablef i:,:.j scheme improves ...1.tl /ii, of problem instances. Smith et al. (2004) note that a node i cannot be assigned to a ;1'.g.Jli k in an optimal solution unless an adjacent node is also assigned to the same ;'1li.g,', Therefore, we also include the following constraints in MP: Xik< j xjk Vi e N, k e K. (247) jEA(i) Smith (2005) describes valid inequalities that can be derived by ,n,,l;, ...:,i the './''.*'.* it of the pgil'r/, First, consider an edge (i,j) E E such that ,i = j 1. Let A(i,j) A(i) U A(j) {i,j} denote the set of distinct nodes that are adjacent to i or j. If A(i,j) > r 1, then i or j must be assigned to at least two 'l';'g,.'l, Similarly, we , I,,,: W(i, j) = EkA(i,j)('q + t 't,) + ', and note that if Wq(i,j) > b for some q E Q, then we cannot f/. .il;;, assign nodes i and j to a single .,l'q',.l, If A(i,j) > r 1 or Wq(i, j) > b, then we state the following valid .:,, ,;,,;.l/;;, S Xik + xjk > 3. (248) kEK kEK Second, for each edge (i,j) E E, suppose deg(i) > r, deg(j) < r, and IA(i,j)l > 2r 3. Smith (2005) shows that nodes i and j collectively need to be assigned to at least four '1,''1qrl'' which we state as: 5 Xik + xk > 4. (249) kEK kEK 2.3 A Hybrid IP/CP Approach The cutting plane algorithms presented in Section 2.2 are preferable to solving stochastic edgepartition instances by the extensive form problem given by (215)(221), as we show in Section 2.4. However, the twostage cutting plane algorithms still suffer from several computational difficulties. First, the master problem, MP, contains IN IK binary variables, IEIIKI continuous variables, and O(IEIIKI) constraints, which results in large integer programs. Second, the linear programming relaxation of MP is quite weak for i rn 'i: problem instances. Furthermore, the lower bound improves slowly as cuts of the type (237) or (244) are added to MP in each iteration. The main reason for this slow convergence is the existence of symmetry in MP. Inequalities (234) reduce, but do not completely eliminate, symmetric solutions in MP. Therefore, when a solution of MP is found to be infeasible to a subproblem, MP often simply switches to a symmetric solution having the same objective function value. On the other hand, stronger antisymmetry constraints tend to make MP very difficult to solve. In this section we develop a new decomposition framework to remedy these difficulties. We combat symmetry due to reshuffling of subgraphs by representing subgraphs as configurations. A configuration c is identified by a subgraph node set N1 (we allow N = 0) and a positive integer ac, which gives the number of subgraphs having node set N1. A solution is represented by a configuration multiset C whose elements are pairs (N,, ac). We eliminate symmetry by ensuring that no isomorphic configuration multisets (i.e., those that are identical after reindexing configuration indices) are encountered in our search. A configuration multiset C satisfies the following necessary feasibility conditions. Fl: Ecc ac= IKI (partitions E into IKI subgraphs) F2: IN, < r, V c C (no subgraph contains more than r nodes) F3: V(i,j) E E, 3c E C such that i E N1, j E N1 (for each edge (i,j), there is at least one subgraph to which (i,j) can be assigned) A multiset C that satisfies Fl, F2, and F3 represents a feasible solution if all edges can be partitioned on the set of subgraphs corresponding to C without violating the weight restrictions for any scenario. Note that the number of distinct configurations in C, which we denote by  C, is dynamically determined in our algorithm. We now provide an overview of our threestage hybrid algorithm. 1. The firststage problem determines (via optimal solution of a mixedinteger program) the number of times we assign each node to the configurations in C. For instance, in the example given in Figure 2la, we could specify that we must use two copies of nodes 4 and 5, and one copy of the other nodes. 2. In the second stage, we seek a multiset C that uses exactly the number of node assignments specified in the first phase and satisfies Fl, F2, and F3. In the example mentioned above, a multiset C having configurations {1, 2, 4}, {3, 4, 5}, and {5, 6} (each with multiplicity one) could be generated based on the firststage solution. 3. Finally, in the third stage, we determine whether C is feasible. If C is feasible then we stop with an optimal solution. Else, we return to the second stage, and generate a different multiset meeting the stated criteria. If no such multiset exists, a cut is added to the firststage problem, which is then resolved. For the example given above, the multiset yields a feasible solution (see Figure 2lb). 2.3.1 FirstStage Problem For all i E N, let zi be an integer variable that represents the number of copies of node i to be used in forming configurations. We v that an INIdimensional vector z induces a multiset C if C contains exactly zi copies of node i, Vi E N. The firststage problem can succinctly be written as: Minimize > zi (250) iEN subject to z induces a feasible multiset (251) S< i < IKI Vi N (252) zi integer, (253) where i is a lower bound on the number of copies required for node i, as given in (245). To formulate the firststage problem as an integer program, we rewrite (251) as an exponential set of linear inequalities by considering the zvectors that violate it. We first need to introduce auxiliary binary variables tik, Vi E N, k = Li,..., K so that tik 1 if zi = k. Then, given a vector 2 that does not induce a feasible multiset, we note that no z such that zi < zi, Vi E N, induces a feasible multiset. Hence, at least one component of z must be increased, and so IKI tik > 1 (254) iEN k= i+1 is a valid inequality. Our firststage problem can now be expressed as the following integer program: Minimize > zi (255) iEN IKI subject to i > ktik Vi e N (256) k =4 IKI tik 1 Vi N (257) k ~= 5 tik > 1 V2 eZ (258) iEN k= i+1 tik binary Vi N, k ..., K, (259) where Z is the set of all zvectors that do not induce a feasible multiset. (The zvariables are in fact unnecessary in this formulation, but we keep them for ease of exposition.) In our algorithm we relax constraints (258) in the firststage problem, and add them in a cutting plane fashion. In every iteration we solve the firststage problem to find 2, and solve the second and thirdstage problems to seek a feasible multiset induced by 2. If a feasible multiset is found, then 2 induces an optimal solution and we stop. Otherwise, we add a cut of type (258) and resolve the firststage problem. 2.3.2 SecondStage Problem Our secondstage problem seeks a multiset induced by 2 that satisfies Fl, F2, and F3, using a constraint programming search. Given a set of constraints, a set of variables, and the domain of each variable (i.e., the set of values that each variable can take), constraint programming seeks a value assignment to each variable that satisfies all constraints. Constraints are propagated to reduce variable domains, which in turn trigger new constraint propagations. When no more domain reductions are possible, the algorithm searches for a solution by fixing a variable to a value in its domain, then recursively propagating constraints and reducing variable domains. If the domain of a variable becomes empty during constraint propagation, then the algorithm backtracks. We refer the reader to Smith (1995), Lustig and Puget (2001), and Rossi et al. (2006) for a thorough discussion of constraint programming techniques. 2.3.2.1 Foundations In our secondstage algorithm, a solution corresponds to a multiset C induced by 2 that meets conditions Fl, F2, and F3. In a solution each node i has a corresponding C dimensional distribution vector i, which represents the number of copies of node i to be allocated to each existing configuration in C. Note that < cannot exceed ac, and that ecc pc = i. The domain of a node i E N is the set of possible p'vectors that i can take. We ic that a node i is processed if we have selected its distribution vector P3. A partial multiset is constructed by processing a subset of the nodes in N. For instance, consider a fivenode graph, and let the zvector obtained by the firststage problem be 2 = (2, 3, 1, 4, 3). Suppose that nodes 1, 2, and 3 have been processed, and the following partial multiset with CI = 3 has been obtained: * N = 0, ac = 5, * N= {1,2}, a2= 2, and SN3 {2, 3}, a3 1. Suppose that we process node 4 by choosing its distribution vector as 4 = (2, 1,1). Adding node 4 to two of the five copies of N1 creates a new configuration N2 whose node set consists only of node 4 (with multiplicity two) and reduces the multiplicity of N1 by two. After similarly adding one copy of node 4 to N2 and one copy of node 4 to N3, we obtain the following partial multiset with IC' = 5: * N1 0, a1 = 3 (reduced ca), * NI {4}, a' = 2 (generated from configuration 1 by adding node 4 to N1), * N = {1,2}, a2= 1 (reduced a2), * N2 {1, 2, 4}, a' = 1 (generated from configuration 2 by adding node 4 to N2), and * N3 {2, 3, 4}, a3 = 1 (added node 4 to N3). In general, when we process node i by choosing a distribution vector /', we update the partial multiset C as follows. For each configuration c E C if 3 = 0, then no changes are made to c (since no copies of node i are added to c). If 3 = ac, then we update configuration c by setting Nc = N, U {i}. Finally, if 0 < Pf < a,, then we create a new configuration c' having N, = N, U {i}, = P, and update configuration c by setting aOc = c O3. Remark 4. Recall that the configurations in a partial multiset C can be ordered in C! ,lii,,I ii.,,, Our l1'j.,,:hm avoids this symmetry by generating only one such ordering after processing a node. Furthermore, the conr fii ,il..'i multisets that we compute by processing node i according to the u'vectors in its domain must be pairwise nonisomor phic, since the f'values in the domain of node i are distinct. Hence, we never encounter isomorphic configuration multisets in the secondstage search. 2.3.2.2 Domain expansion Processing a node modifies the current partial multiset, and therefore distribution vectors of the remaining unprocessed nodes need to be updated. Domains of nodes are reduced by constraint propagation as we describe in the next section, but must also be expanded as new configurations are generated. We describe the initialization and expansion of node domains below. In the beginning of the second stage we initialize our multiset C with a single configuration having N1 = 0 and ac = K. Each node can only be added to the lone configuration, and so the domain for node i is initially the single onedimensional vector fi = (z). Our algorithm next processes some node i E N and updates the existing set of configurations: N1 = 0, a1 = IKI zi and N2 = {i, a2 = i. Next, the domains of all unprocessed nodes are updated to reflect the changes in C. For each unprocessed node j, we enumerate all possible vv of partitioning zj copies into node sets N1 and N2. This logic is repeated at all future steps as well. For instance, in the example given above, suppose that 35 = (2, 0, 1) was the only vector in the domain of node 5 before processing node 4. Since processing node 4 modifies the first configuration by reducing ac and generates a new configuration (N', a'), we expand the domain of node 5 by enumerating all possible vv of assigning 3 = 2 copies of node 5 to configurations (N1, al) and (NI, a'c). On the other hand, since 35 does not assign node 5 to the second configuration, the distribution vectors in the expanded domain do not add node 5 to (N2, a2) or (2N, 2). Finally, since processing node 4 does not generate any new configurations from the third configuration, all distribution vectors in the expanded domain of node 5 assign a single copy of node 5 to (N3, a3). After processing node 4 and updating the configurations as described above, the domain of node 5 is expanded to: {(2,0,0,0, 1), (1, 1,0,0, 1), (0,2,0,0, 1)}. 2.3.2.3 Constraint propagation The constraints we impose in the secondstage problem limit the number of nodes in each configuration (F2) and require that each edge has both its end points in at least one configuration (F3). Condition Fl (requiring IKI total configurations) is implicitly satisfied. We apply constraint propagation algorithms to remove distribution vectors inconsistent with F2 or F3 from the expanded node domains. Let i E N be the last processed node, and let Ci C C represent the subset of configurations to which node i has been added. We only need to execute constraint propagation for configurations c E CQ, since these are the only newly modified configurations. To enforce F2, the propagation algorithm identifies all configurations to which r nodes have been assigned. For each such configuration c, we remove all distribution vectors 3P having P3 > 0 from the domains of all unprocessed nodes j e N. To enforce F3, the propagation algorithm iterates over the domains of the unprocessed nodes j .lIi i.:ent to i, and removes all distribution vectors that do not add at least one copy of j to any configuration in C. Otherwise, the configurations containing node i would be disjoint from those containing node j, which violates F3. 2.3.2.4 Forward checking After all constraints are propagated, we first check whether the domain of any unprocessed node is empty; if so, then we backtrack. Else, we further analyze the current partial multiset before resuming the search with the next unprocessed node. This step identifies whether the current partial multiset can eventually yield a feasible multiset as early as possible to avoid performing unnecessary backtracking steps (van Beek, 2006). We call one such test implied node assignment i,.il,.: Suppose that we identify a processed node i such that zi 1, and the configuration c to which i has been assigned. By condition F3 it follows that all unprocessed nodes j .,.i i,:ent to i must also be assigned to configuration c. We use this analysis to augment partial configurations with implied node assignments, and then check whether any augmented configuration contains more than r nodes, and hence violates F2. We also perform an implied edge i/:I..:w, l, i,.rl, i.:j by finding all edges that can only be assigned to a single configuration. For each (i, j) E E, if both nodes i and j have been processed, then we check whether both i and j are in a single configuration c for which ac 1. In this case edge (i,j) can only be assigned to configuration c. On the other hand if (without loss of generality) node i has been processed but node j has not yet been processed, and Zi 1, then edge (i,j) can only be assigned to the configuration to which i has been assigned. After finding all implied edge assignments, we check whether F3 is violated for any scenario. Finally, we consider a singleton node wi,';l,..:> in which we ensure that each node is .,1i ,i:ent to at least one other node in each configuration. For each processed node i, and for all configurations c E Ci, we seek a node j .Ii] i,:ent to i so that either j E Nc (if j also has been processed), or f3 > 0 for some distribution vector in the domain of j (if j has not been processed). If no such j can be found for a configuration c E Ci, then the current partial solution cannot lead to an optimal solution; node i can ultimately be removed from configuration c without affecting feasibility conditions, leading to a reduction in the objective function value. 2.3.2.5 Node selection rule The order in which variables are processed can significantly affect the performance of constraint programming algorithms (Lustig and Puget, 2001; Smith, 1995). Especially for infeasible secondstage problem instances, processing the 1'l I.' i,, i, c" nodes first can quickly lead to the detection of infeasibility and can result in significant savings in computational time. We employ a dynamic node selection rule in which the order of nodes considered can vary in different sections of the search tree. In accordance with the "failfirst" principle widely used in constraint programming algorithms (Haralick and Elliott, 1980; van Beek, 2006), our node selection rule first picks an unprocessed node that 1. has the fewest number of distribution vectors in its domain, 2. has the fewest number of copies to be partitioned, and 3. has the largest number of unprocessed .,.i i:ent nodes, breaking ties in the given order. In this manner, we can quickly enumerate all possible distribution vectors of a few key nodes, allowing constraint propagation to quickly reduce the size of the remaining search space. 2.3.2.6 Distribution vector ordering rule Once the next node to be processed has been identified, all distribution vectors in its domain need to be tried onebyone to see if any of them leads to a feasible multiset. For an infeasible secondstage problem instance, the order in which these vectors are instantiated does not matter, because all vectors must be enumerated before infeasibility can be concluded. However, for feasible problem instances it is important to find a vector that leads to a feasible multiset as soon as possible to curtail our search. Our ordering rule attempts to sort the distribution vectors in nonincreasing order of the likelihood that the vector leads to a feasible multiset. We calculate the feasibility likelihood score of a distribution vector /P in the domain of an unprocessed node i with respect to a partial multiset C as: FL(i, C, ') = 3 {j N : (ij) e E}. (260) cEC FL(i, C,/3 ) measures the total number of .,li i,:ent node pairs (i, j) that would be added across all configurations if fO is selected to be the distribution vector for node i. Our vector ordering rule sorts vectors in the domain of the chosen node in nondecreasing order of their FLscores. By allowing for a higher degree of flexibility in assigning edges, we increase the likelihood that a feasible partition of edges to subgraphs can be found. 2.3.3 ThirdStage Problem Given a solution of the secondstage problem that consists of a configuration multiset C satisfying Fl, F2, and F3, the thirdstage problem must verify whether C is feasible. We first generate the set of subgraphs from the multiset C {cl, c2,..., clcl} by assigning the nodes in Nc, to the first ac, subgraphs, then assigning the nodes in N12 to the next ac2 subgraphs, and so on. Since we have enforced cecc a= K this transformation creates exactly IKI subgraphs, some of which can be empty. Then we iterate over all subgraphs and set ., = 1 if nodes i and j are in subgraph k, and ., = 0 otherwise. We then use formulation (238)(243) to solve the thirdstage problem. Note that this transformation reintroduces symmetry into the thirdstage problem. However, the solution of the thirdstage problems does not constitute a bottleneck in the algorithm, and symmetrybreaking constraints appended to the transformed subproblem will not impact the computational efficacy of the overall algorithm. 2.3.4 Infeasibility Analysis If a zvector is found not to induce a feasible multiset, we add a constraint to the firststage problem so that the same zvector is not generated in subsequent iterations. Constraints (258) state that the number of copies of some node must be increased, but they do not contain any information about which nodes need to be added. We observe that the progress of our secondstage algorithm can be analyzed to identify a 11 ''I" in I 1c" subset of nodes whose corresponding zvalues cause infeasibility regardless of other variable values. Given a vector 2 for which no feasible multiset exists, if a node i E N has not been processed, or has not been identified as the reason of infeasibility in any step of the backtracking algorithm, then 2 will not induce a feasible multiset for any value of zi. Let P C N denote the set of nodes that have been processed, or whose domains have become empty due to constraint propagation in the secondstage algorithm, possibly during different backtracking steps. The following is a valid inequality: IKI S ik t> 1. (261) iEP k =i+1 Constraints (261) clearly dominate (258) for any P C N, and get stronger as P decreases. Based on this observation, we update our node selection rule by giving preference to selecting nodes that have already been added to P. Our revised node selection rule first picks a node that 0. has been added to P in a previous backtracking step, 1. has the fewest number of distribution vectors in its domain, 2. has the fewest number of copies to be partitioned, and 3. has the largest number of unprocessed .,.1] i:ent nodes, again breaking ties in the stated order. 2.3.5 Enhancements for the FirstStage Problem Our computational studies revealed that the firststage integer programming model solution represents the bottleneck operation of our algorithm. To decrease the computational time spent by the firststage problem, we investigate several strategies. 2.3.5.1 Valid inequalities The valid inequalities that we discuss in Remark 3 can be adapted to the firststage problem to eliminate the zvectors that violate the corresponding necessary feasibility conditions. In particular, constraints (246) translate to simple lower bounds (252) on the zvariables. Constraints (248), which are written for node pairs that satisfy the conditions discussed in Remark 3, can be written as: z, + zj > 3. (262) Similarly, each constraint of type (249) can be equivalently represented as following: z, + zj > 4. (263) Smith (2005) discusses an additional valid inequality, which cannot be represented using the xvariables in our twostage algorithm, but can be written in terms of the z and tvariables in the firststage problem of our hybrid algorithm. For nodes i E N and j E N, if (i,j) E, deg(i) < r 1,deg(j) < r 1, IA(i,j) > r 1, and there exists a common neighbor k E N so that k c A(i), k c A(j), deg(k) > r, and if i,j, k have more than 2r 4 distinct neighbors in total, then zi = 1, zy = 1 implies zk > 3. This condition can be written as: zk > 1 + 2(tl + tji), (264) which reduces to Zk > 3 for zi j = 1, and is redundant otherwise. 2.3.5.2 Heuristic for obtaining an initial feasible solution The existence of a good initial feasible solution can help improve the performance of the firststage problem because it provides a good upper bound, and allows the solver to apply strategies such as reduced cost fixing. We first solve the firststage model enhanced with valid inequalities (262)(264) to obtain an initial solution z, and execute the second and thirdstage algorithms to seek a feasible multiset. If one is found, we terminate with an optimal solution. Otherwise, we investigate the set of processed nodes P C N, and pick a node c E P having the fewest number of copies (breaking ties by picking a node having the largest degree). We then set z, = z, + 1 and reinvoke the second and thirdstage algorithms. This algorithm eventually finds a feasible multiset or concludes that the entire problem is infeasible after generating the solution zi = KI, Vi E N. We also generate a cut of type (261) for each 2 generated before a feasible multiset is found, which we add to the firststage problem to improve the lower bound. 2.3.5.3 Processing integer solutions We can interrupt the branchandbound solution process of the firststage problem each time the solver finds an integer solution 2, and check whether 2 induces a feasible multiset by solving the second and thirdstage problems. If a feasible multiset exists, we accept 2 as the new incumbent and resume solving the firststage problem. Otherwise, we reject 2, generate a constraint of type (261), and again resume the solution process. The same idea is also applicable to the master problem (lI'P) of the twostage algorithm discussed in Section 2.2. In our tests, this approach turned out to be more effective than solving the firststage problem to optimality in each iteration, adding a cut, and resolving it. The reason is that the problem is solved using a single branchandbound tree, which we tighten by adding cuts as necessary on integral nodes, instead of repeatedly generating a branchandbound tree in each iteration. It also allows us to obtain good feasible solutions for problem instances that are too difficult to solve to optimality. We note that this approach requires a minor modification to the secondstage algorithm. All constraint propagation (Section 2.3.2.3) and forward checking rules (Section 2.3.2.4) except for singleton node analysis are based on necessary conditions for feasibility of configurations, and therefore they are valid for any integral 2. However, singleton node analysis is based on an optimality condition and hence can only be used if 2 is a candidate optimal solution to the firststage problem. 2.4 Computational Results We implemented the algorithms discussed in the previous sections using CPLEX 11.1 running on a Windows XP PC with a 3.4 GHz CPU and 2 GB RAM. Our base set of test problem instances consists of 225 randomly generated problem instances for which the expected edge density of the graph (measured as i l) takes values 0.2, 0.3, and NJ.xQNJ 1)1 e aus0.,03 n 0.4, the number of nodes ranges from 5 to 15, and the number of scenarios is between 1 (corresponding to the deterministic edgepartition problem) and 100. There is no practical limit on the number of subgraphs (IKI), but a limit needs to be specified to model the problem (see Goldschmidt et al. (2003); Sherali et al. (2000); Smith (2005)). C'! .. ig IKI too small may make the problem infeasible, and large values of IKI increase difficulty of the problem. In our tests, we chose IKI sufficiently large to yield a feasible edge partition in each problem instance. In generating instances we first picked a random subset of edges to have a positive weight, and then we assigned a weight uniformly distributed between 1 and 10 to each edge in each scenario. We generated five problem instances for each problem size, which is determined by the expected edge density, the number of nodes, and the number of scenarios. The data set names and details used in our experiments are given in Table 21. Table 21. Descriptions of the problem instances used for comparing algorithms Name INI IKI IQI r b Name INI IKI IQI r b 51 5 5 1 4 20 121 12 10 1 5 50 530 5 5 30 4 20 1230 12 10 30 5 50 5100 5 5 100 4 20 12100 12 10 100 5 50 81 8 7 1 4 35 151 15 10 1 8 70 830 8 7 30 4 35 1530 15 10 30 8 70 8100 8 7 100 4 35 15100 15 10 100 8 70 101 10 8 1 5 40 1030 10 8 30 5 40 10100 10 8 100 5 40 We used the default options of CPLEX for solving the extensive form problems. Preliminary computational experience on our twostage algorithm indicated that the best implementation includes the valid inequalities (227), (246)(249), and the symmetrybreaking constraints (234), and uses the model given by (238)(243) for the subproblem, which is the formulation that minimizes the total tardiness. In our base setting for the threestage algorithm, we used our heuristic to find an initial feasible solution, generated valid inequalities (262)(264), and (similar to the twostage algorithm) we used formulation (238)(243) for the thirdstage problem. We used callback functions of CPLEX to generate a single branchandbound tree for both twostage and threestage algorithms as discussed in Section 2.3.5. We imposed a halfhour (1800 seconds) time limit past which we halted the execution of an algorithm in all our experiments. Our first experiment compares the performance of the extensive form, twostage, and threestage algorithms. Table 22 summarizes the results of these three algorithms on low density graphs having expected edge density 0.2. For each problem size, we report the following statistics calculated over five random instances: (i) the number of problems solved to optimality ("Solvh 'i ), (ii) the average optimality gap obtained at the root node ("Root C p1 ), (iii) the average final optimality gap for instances that could not be solved within the allowed time limit ("Final C(; ), (iv) the average amount of time spent by each algorithm on the instances that were solved to optimality ("Time"). Out of the 75 instances in this data set, CPLEX could solve the extensive form to optimality for 61 instances, while both twostage and threestage algorithms solved all 75 instances to optimality within a few seconds. The results reveal that the performance of the extensive form formulation deteriorates rapidly as the number of scenarios increases, but the effect of the number of scenarios is mitigated for the twostage and threestage algorithms. We observe that the average optimality gap obtained by the threestage algorithm at the root node is 1..!' which is significantly less than the initial gaps obtained using other approaches. Table 22. Comparison of the algorithms on graphs having edge density = 0.2 Extensive Form TwoStage ThreeStage Root Final Root Final Root Final Name Solved Gap Gap Time Solved Gap Gap Time Solved Gap Gap Time 51 5 0.00% 0.1 5 5.00% 0.1 5 0.00% 0.1 530 5 18.33% 6.6 5 4.00% 0.2 5 0.00% 0.1 5100 5 12.38% 5.4 5 11.00% 0.6 5 0.00% 0.3 81 5 25.90% 0.4 5 6.67% 0.1 5 0.00% 0.1 830 5 12.89% 4.0 5 3.64% 0.2 5 0.00% 0.1 8100 5 37.61% 223.1 5 14.84% 1.2 5 0.00% 0.3 101 5 19.58% 0.5 5 17.80% 0.4 5 0.00% 0.1 1030 5 57.01% 147.3 5 10.71% 0.8 5 0.00% 0.2 10100 4 30.35% 7.14% 684.1 5 13.94% 2.0 5 0.00% 0.4 121 5 47.25% 8.1 5 24.66% 2.2 5 0.00% 0.1 1230 4 55.09% 25.00% 507.3 5 17.99% 4.3 5 3.08% 0.3 12100 2 62.21% 24.88% 713.1 5 36.28% 4.7 5 2.11% 0.8 151 5 31.85% 33.0 5 64.38% 16.4 5 4.56% 0.2 1530 1 65.29% 21.65% 864.6 5 39.13% 27.1 5 7.29% 0.6 15100 0 57.33% 28.47% 5 24.49% 20.4 5 4.86% 1.2 Tables 23 and 24 compare the three approaches on denser graphs having edge density 0.3 (medium density) and 0.4 (high density), respectively. We observe that performances of all three algorithms deteriorate as the edge density increases, which is not surprising due to the nature of the edgepartition problem. The number of instances Table 23. Comparison of the algorithms on graphs having edge density = 0.3 Extensive Form TwoStage ThreeStage Root Final Root Final Root Final Name Solved Gap Gap Time Solved Gap Gap Time Solved Gap Gap Time 51 5 0.00% 0.1 5 2.86% 0.1 5 0.00% 0.1 530 5 25.76% 10.4 5 6.15% 0.5 5 0.00% 0.1 5100 5 10.00% 3.1 5 10.77% 0.4 5 0.00% 0.2 81 5 31.30% 0.5 5 11.20% 0.1 5 0.00% 0.1 830 5 42.57% 18.0 5 7.48% 0.4 5 0.00% 0.2 8100 4 39.37% 7.14% 110.0 5 16.19% 1.3 5 1.43% 0.3 101 5 32.42% 3.7 5 16.27% 0.6 5 1.18% 0.1 1030 4 51.33% 21.05% 953.0 5 40.82% 8.2 5 5.83% 0.3 10100 2 61.24% 29.05% 382.6 5 35.43% 302.7 5 8.89% 0.5 121 5 53.85% 312.0 5 39.49% 16.8 5 4.65% 0.2 1230 0 63.41% 27.06% 5 46.98% 120.5 5 9.31% 0.8 12100 0 84.24% 65.50% 4 42.78% 4.35% 89.0 5 11.99% 1.4 151 4 46.88% 11.54% 460.4 5 72.86% 250.5 5 12.93% 0.9 1530 0 66.01% 42.41% 2 72.20% 16.02% 30.4 5 13.51% 3.5 15100 0 80.76% 74.48% 0 53.05% 13.21% 5 16.31% 4.1 that can be solved by the extensive form decreases from 61 for low density graphs to 49 for medium density graphs, and finally to 36 for highdensity graphs. The twostage algorithm also exhibits a similar behavior; it can solve 75, 66, and 61 instances for low, medium, and highdensity graphs, respectively. On the other hand, the threestage algorithm is able to solve almost all instances, failing to solve two instances in the highdensity 1530 and 15100 data sets to optimality within the allowed time limit. Table 24 clearly shows that the threestage algorithm dominates the other approaches, and the twostage algorithm provides better results than directly solving the extensive formulation. Our analysis of optimal solutions obtained for the problem instances shown in Tables 2224 showed that the average objective function value for the deterministic (singlescenario) problem instances is 14.8. This value is smaller than the average objective function value for 30scenario and 100scenario instances (15.52 and 15.6, respectively). We also observe that several subgraphs can be empty in an optimal solution. Table 24. Comparison of the algorithms on graphs having edge density = 0.4 Extensive Form TwoStage ThreeStage Root Final Root Final Root Final Name Solved Gap Gap Time Solved Gap Gap Time Solved Gap Gap Time 51 5 5.00% 0.1 5 0.00% 0.1 5 0.00% 0.1 530 5 24.67% 2.6 5 19.79% 0.3 5 0.00% 0.1 5100 5 12.38% 5.6 5 8.31% 0.6 5 0.00% 0.2 81 5 41.32% 2.0 5 3.33% 0.1 5 0.00% 0.1 830 5 48.89% 140.9 5 17.68% 1.1 5 1.43% 0.1 8100 3 47.23% 22.50% 113.0 5 21.08% 8.7 5 2.50% 0.4 101 5 45.08% 48.3 5 32.36% 3.5 5 2.16% 0.1 1030 0 61.52% 20.64% 5 56.55% 39.5 5 8.45% 0.4 10100 0 64.47% 50.91% 3 54.82% 7.50% 151.7 5 12.73% 1.5 121 1 67.13% 14.30% 33.2 5 40.60% 327.3 5 7.86% 0.5 1230 0 88.61% 46.74% 5 42.93% 160.9 5 3.16% 0.8 12100 0 84.37% 68.24% 5 51.54% 583.7 5 13.91% 1.7 151 2 60.11% 11.21% 369.6 3 53.01% 5.56% 410.0 5 11.57% 0.9 1530 0 85.29% 65.29% 0 66.72% 22.66% 3 18.03% 4.74% 120.2 15100 0 96.00% 86.92% 0 62.62% 24.58% 3 19.98% 6.45% 173.8 . Descriptions of the problem instances  IKI r b 5 4 20 7 4 35 8 5 40 10 5 50 10 8 70 10 8 100 10 10 120 10 10 140 used for analyzing threestage algorithm Our next experiment analyzes the performance of our threestage algorithm for larger instances. For this experiment, we generated additional random problem instances using the parameter settings given in Table 25. Similar to our previous experiments, we Table 26. ThreeStage algorithm on graphs having edge density = 0.2 6 = 0.05 = 0.01 Root Final Heuristic Root Final Heuristic Name Q Solved Gap Gap Time Gap Q Solved Gap Gap Time Gap 5 407 5 0.00% 0.8 2.86% 2194 5 0.00% 3.5 0.00% 8 837 5 0.00% 2.2 1.54% 4343 5 0.00% 12.7 3.33% 10 1169 5 10.88% 5.6 5.09% 6006 5 1.33% 19.6 1.33% 12 1724 5 1.11% 13.2 4.19% 8779 5 3.00% 58.5 3.33% 15 2140 5 7.24% 22.1 2.74% 10858 5 12.41% 170.9 4.37% 17 2417 5 10.19% 41.0 4.78% 12245 5 9.82% 211.1 8.01% 20 2833 5 16.55% 79.8 7.01% 14324 5 12.40% 403.7 4.51% 22 3110 5 14.77% 128.2 6.49% 15710 5 15.78% 699.9 6.46% generated problem instances for which the expected edge density of the graph takes values 0.2, 0.3, and 0.4. For each data set, we calculated the number of scenarios corresponding to e, 6 = 0.05 and e, 6 = 0.01 using Proposition 1. Hence, inequality (214) ensures that we can be 95'. (9''., respectively) certain that all demands can be satisfied 95' (9I'' respectively) of the time. We generated five random instances for each data set, Table Name 5 8 10 12 15 17 20 22 Table 27. ThreeStage algorithm on graphs having edge density = 0.3 c, = 0.05 = 0.01 Root Final Heuristic Root Final Heuristic Name Q Solved Gap Gap Time Gap Q Solved Gap Gap Time Gap 5 407 5 0.00% 0.8 2.86% 2194 5 0.00% 3.5 0.00% 8 837 5 0.00% 2.6 2.86% 4343 5 3.33% 13.0 2.86% 10 1169 5 6.58% 7.5 8.99% 6006 5 11.86% 27.4 4.80% 12 1724 5 8.61% 16.1 4.51% 8779 5 8.22% 71.7 4.31% 15 2140 5 15.45% 45.1 3.05% 10858 4 18.53% 3.45% 176.4 4.25% 17 2417 5 13.63% 42.4 3.43% 12245 5 9.93% 189.3 2.68% 20 2833 5 17.47% 362.5 3.24% 14324 5 18.13% 639.1 3.32% 22 3110 4 16.18% 4.76% 159.3 5.82% 15710 5 15.86% 738.5 3.45% resulting in 240 instances in total. In addition to the columns given in Table 22, Tables 26, 27, and 28 show the relative gap between the quality of the solution found by our initial heuristic (Section 2.3.5.2) and the best lower bound obtained ("Heuristic C; Ip ). Our algorithm can solve 206 instances out of 240 to optimality, and provides an average Table 28. ThreeStage algorithm on graphs having edge density = 0.4 c, = 0.05 e, = 0.01 Root Final Heuristic Root Final Heuristic Name Q Solved Gap Gap Time Gap Q Solved Gap Gap Time Gap 5 407 5 0.00% 0.8 2.22% 2194 5 0.00% 3.2 0.00% 8 837 5 7.71% 2.7 2.43% 4343 5 7.25% 17.3 5.33% 10 1169 5 16.38% 9.4 5.71% 6006 5 15.84% 52.1 9.73% 12 1724 5 16.71% 63.2 5.41% 8779 5 14.13% 118.4 5.45% 15 2417 4 16.24% 2.86% 338.8 4.07% 12245 2 16.55% 4.71% 549.8 6.86% 17 2140 1 23.46% 8.46% 993.7 9.23% 10858 1 24.77% 11.44% 1515.5 13.07% 20 2833 0 18.83% 9.46% 9.46% 14324 0 20.01% 11.30% 11.81% 22 3110 0 18.36% 11.05% 11.47% 15710 0 17.83% 9.90% 10.29% optimality gap of 9.21 for the 34 instances that it cannot solve to optimality. The maximum optimality gap obtained for the entire data set is 21.22''. The results also ,i., I that our heuristic for finding an initial feasible solution is quite effective: the average optimality gap for our heuristic is 4.97'. and the maximum optimality gap is 22.72''. Since these calculations are based on the lower bounds obtained for the problem instances that could not be solved to optimality, our reported gaps possibly overestimate the true gap between heuristic and optimal objective values. CHAPTER 3 CONSECUTIVEONES MATRIX DECOMPOSITION PROBLEM 3.1 Introduction and Literature Survey Cancer is one of the leading causes of death throughout the world. In the last century, external beam radiation therapy has emerged as a very important and powerful modality for treating many forms of cancer, either in primary form or in conjunction with other treatment modalities such as surgery, chemotherapy, or medication. In the United States .1 I,iv, approximately twothirds of all newly diagnosed cancer patients receive radiation therapy for treatment. Since the radiation beams employ, ,1 in radiation therapy damages all cells traversed by the beams, both in targeted areas in the patient that contain cancerous cells as well as any cells in I. i111!:' organs and tissues, the treatment must be carefully designed. This can partially be achieved by delivering radiation from several different directions, also called beam orientations. Therefore, patients receiving radiation therapy are typically treated on a clinical radiationdelivery device that can rotate around the patient. The most common device is called a linear accelerator and is typically equipped with a socalled multileaf collimator (_llC) system which can be used to judiciously shape the beams by forming apertures, thereby providing a high degree of control over the dose distribution that is received by a patient (see Figure 31 1 ). This Multileaf Photon collimator It therapy system beam Left and right leaves form aperture, creating an I .. irregularly shaped beam '. (a) (b) Figure 31. (a) A multileaf collimator system (b) The projection of an aperture onto a patient technique has been named :i/. .i ih modulated radiation therapy (IMRT). Since the mid 1990's, largescale optimization of the fluence applied from a number of beam orientations around a patient has been used to design treatments from MLCequipped linear accelerators. A typical approach to IMRT treatment planning is to first select the number and orientations of the beams to use as well as an intensity profile or fluence map for each of these beams, where the fluence map takes the form of a matrix of intensities. This problem has been studied extensively and can be solved satisfactorily, in particular when (as is common in clinical practice) the beam orientations are selected manually by the physician or clinician based on their insight and expertise regarding treatment planning. For optimization approaches to the fluence map optimization problem with fixed beam orientations we refer to the review paper by Shepard et al. (1999). More recently, Romeijn et al. (2006) proposed new convex programming models, and Hamacher and Kiifer (2002) and Kiifer et al. (2003) considered a multicriteria approach to the problem. Lee et al. (2000a, 2003) studied mixedinteger programming approaches to the extension of the fluence map optimization problem that also optimizes the number and orientations of the beams to be used. However, to enable delivery of the optimal fluence maps by the MLC system, they need to be decomposed into a collection of deliverable apertures. (For examples of integrated approaches to fluence map optimization, also referred to as aperture modulation, we refer to Shepard et al. (2002), PreciadoWalters et al. (2004), and Romeijn et al. (2005).) The vast i i ii iy of MLC systems contain a collection of leaves that can be moved in parallel, thereby blocking part of the radiation beam. This architecture implies that we can view each beam as a matrix of beamlets or bixels (the smallest deliverable square beam that can be created by the MLC), so that each aperture can be represented by a collection of rows (or, by rotating the MLC head, columns) of bixels, each of which 1 Varian Medical Systems; http://www.varian.com/orad/prd056.html. should be convex. In other words, each fluence map should be decomposed into either constantintensity rowconvex apertures or constantintensity columnconvex apertures. Due to the time required for setup and verification, clinical practice prohibits using both types of apertures for a given fluence map, so that without loss of generality we focus on rowconvex apertures only. Note that while some manufacturers of MLC systems impose additional constraints on the apertures, we assume that all rowconvex apertures are deliverable. As an example, consider the fluence map given by the following 2 x 3 matrix of bixel intensities (see Baatar et al. (2005)): 3 6 4 2 1 5 If we represent an aperture by a binary matrix in which an element is equal to one if and only if the associated bixel is exposed (i.e., not blocked by either the left or right leaf of the MLC system), rowconvexity corresponds to the property that, in each row of the corresponding matrix, the elements that are equal to one are consecutive (often referred to as the consecutiveones 1y, '. i/;i). Now observe that this fluence map can be decomposed into three apertures with corresponding intensities: 3 6 4 1 0 0 1 1 0 0 1 1 1x +2x +4x 2 1 5 0 1 1 1 0 0 0 0 1 Since, in general, there are many v  of decomposing a given fluence map into rowconvex apertures, it is desirable to select the decomposition that can be delivered most efficiently. The two main efficiency criteria that pl li a role are the total beamon time, i.e., the total amount of time that the patient is being irradiated, and the total setup time, i.e., the total amount of time that is spent shaping the apertures. The former metric is proportional to the sum of intensities used in the decomposition, while the latter is approximately proportional to the number of matrices used in the decomposition. Although closely related, these two efficiency criteria are not equivalent. The example given above shows the unique decomposition using only three apertures and with a beamontime of 7. However, the minimum beamontime for this fluence map is 6, which can be realized by four apertures using the following decomposition: 3 6 4 1 1 00 1 0 111 0 11 1x +lx +2x +2x 2 1 5 1 0 0 1 1 1 0 0 1 0 0 1 The problem of decomposing a fluence map while minimizing beamontime is polynomially solvable and has been widely studied, leading to several solution approaches for this problem. Bortfeld et al. (1994) proposed the sweep method, which Almi and Hamacher (2005) (who derived an equivalent method) showed to indeed yield an optimal solution; other exact algorithms were proposed by Kamath et al. (2003), and Siochi (1999). In addition, Baatar et al. (2005), Boland et al. (2004), Kalinowski (2005a), Kamath et al. (2004a,b,c,d), Lenzen (2000), and Siochi (1999) studied the problem of minimizing beamontime under additional hardware constraints, while Kalinowski (2005b) studied the benefits of allowing rotation of the MLC head. Although the time required by the MLC system to transition between apertures formally depends on the apertures themselves, the fact that these times are similar and the presence of significant (apertureindependent) verification and recording overhead times justifies the use of the total number of setups (or, equivalently, the total number of apertures) to measure the total setup time. In addition, delivering IMRT with a small number of apertures provides the additional benefits of less wearandtear on the collimators (less stopping and starting) and a less errorprone delivery as IMRT delivery errors are known to be proportional to the number of apertures (see Stell et al. (2004)). The problem of decomposing a fluence map into the minimum number of rowconvex apertures has been shown to be strongly NPhard (see Baatar et al. (2005)), leading to the development of a large number of heuristics for solving this problem. Notable examples are the heuristics proposed by Baatar et al. (2005) (who also identify some polynomially solvable special cases), Agazaryan and Solberg (2003), Dai and Zhu (2001), Que (1999), Que et al. (2004), Siochi (1999, 2004, 2007), Van Santvoort and Heijmen (1996), Xia and Verhey (1998). In addition, Engel (2005), Kalinowski (2005a), and Lim and Choi (2007) developed heuristics to minimize the number of apertures while constraining the total beamontime to be minimal. Finally, Langer et al. (2001) developed a mixedinteger programming formulation of the problem, while Kalinowski (2004) proposed an exact dynamic programming approach for the related problem of minimizing the number of apertures that yields the minimum beamontime. Baatar et al. (2007) described integer programming and constraint programming models for the same problem, and Ernst et al. (2009) proposed a constraint programming approach for minimizing the number of apertures. However, computational studies reported in Baatar et al. (2007); Ernst et al. (2009); Kalinowski (2004); Langer et al. (2001) reveal that these approaches can only be used to efficiently solve small problem instances to optimality. Our primary contribution is that we develop the first algorithm capable of solving clinical problem instances to optimality (or to provably nearoptimality) within clinically acceptable computational time limits. In this chapter, our focus is on the problem of finding a decomposition of a fluence map into rowconvex apertures that minimizes total treatment time, as measured by the sum of the total setup time and beamontime. In Section 3.2 we develop our decompositionbased solution approach. In Section 3.3 we discuss the application of our algorithm on a collection of clinical and randomly generated test data, and compare its performance with alternative exact and heuristic techniques. 3.2 Decomposition Algorithm Throughout this chapter, we denote the fluence map to be delivered by a matrix B E 1' where the element at row i and column j, (i,j), corresponds to a bixel with required intensity bi. Let w, be the time required by the MLC to form an aperture and w2 denote the time required for the delivery of one unit of fluence. We refer to the problem of minimizing the total treatment time, i.e., the sum of the aperture transition times and the total delivery time, as the optimal leaf sequencing problem. We start this section by describing a decomposition framework for the optimal leaf sequencing problem in Section 3.2.1 and use this to formulate our master problem in Section 3.2.2. We introduce our subproblem in Section 3.2.3, prove its complexity, and provide a combinatorial search algorithm for its solution. We then enhance the empirical performance of our decomposition algorithm by introducing classes of valid inequalities to the master problem in Section 3.2.4, and finally describe an algorithm for constructing a feasible solution with medically desired properties in Section 3.2.5. 3.2.1 Decomposition Framework To establish motivation for our approach, observe that if the objective is to minimize beamontime, the optimal leaf sequencing problem is decomposable by the rows of the fluence map. In particular, if the beamontime is minimized for each bixel row, the maximum of the corresponding beamontime values is equal to the minimum beamontime for the overall fluence map (see, e.g., Ehrgott et al. (2008)). However, this approach is not directly applicable when the objective is to minimize the total treatment time. Even though the optimal leaf sequencing problem is not directly decomposable by rows, the fact that leaves corresponding to different rows can be positioned independently can still be exploited. Denote a particular positioning of left and right leaves for a row as a leaf position; an aperture is composed of a leaf position for each row of B. Our main observation is that given a collection of intensities, which can be used in apertures that collectively cover the fluence map, the rows are independent of one another. That is, we can determine the leaf positions to be used for covering each row independently, and then form apertures for covering the entire fluence map by combining individual leaf positions for each row that are assigned to the same intensity. We define an allowable I':/. i,/l multiset to be a collection of (potentially nonunique) intensity values, each of which can be assigned to a single aperture in our solution. We .iv that an allowable intensity multiset is compatible with a row if there exists a feasible decomposition of the row into leaf positions using a subset of that allowable intensity multiset. If an allowable intensity multiset is compatible with all rows, then it corresponds to a feasible decomposition of the fluence map and we call it a feasible I'/. ui.i, multiset. As an example, consider the fluence map given by the following 3 x 3 matrix: 148 B= 3 8 5 (31) 453 4 5 3 Consider the allowable intensity multiset {1, 3, 5}. Assigning each of these values to at most one leaf position, the first row can be decomposed as [1 4 8] x [1 1 0] + 3 x [0 1 1] + 5 x [0 0 1], (32) so that the allowable intensity multiset is compatible with the first row. Similarly, the second row can be decomposed as [3 8 5] 3 x [1 1 0] + 5 x [0 1 1]. (33) However, the first bixel in the third row must be covered by two leaf positions assigned to intensities 1 and 3, and the second bixel must be covered by a single leaf position assigned to intensity 5. Therefore, all allowable intensities must be used to cover the first two bixels, and the third bixel with required intensity 3 cannot be covered. Hence, the allowable intensity multiset is not compatible with the third row. Alternatively, consider an allowable intensity multiset that contains the values 1, 3, and 4 for the same fluence map. The rows can be decomposed as [1 4 8] x [1 1 1] + 3 x [0 1 1] + 4 x [0 0 1] , [3 8 5] 1 x [0 1 1] + 3 x [1 1 0] + 4 x [0 1 1] and (34) [4 5 3] x [0 1 0] + 3 x [0 0 1] + 4 x [1 1 0]. Since the allowable intensity multiset is compatible with all rows, it is a feasible intensity multiset having three leaf positions and a beamontime of 8. Furthermore, observe that the intensity requirements of the bixels in the first row strictly increase from left to right, implying that a leaf position must start at each bixel. Thus, any feasible decomposition of the first row uses at least three leaf positions, which yields a lower bound on the number of apertures. Also, the largest element of B is 8, which yields a lower bound on the beamontime. Since the given decomposition achieves the lower bounds on both objectives, we have an optimal solution to the optimal leaf sequencing problem. 3.2.2 Master Problem Formulation and Solution Approach We represent an allowable intensity multiset by an integer vector x = (x,..., XL), where L = maxi= 1... m;j=1,...,nb bij is the maximum intensity value in the fluence map, and where xr is the number of times that intensity value E occurs in the allowable intensity multiset. It is easy to see that, assuming all allowable intensity values are used, the number of apertures and the beamontime are, respectively, equal to L L xi and EUx. (35) e=1 e= 1 The master problem can therefore succinctly be written as L L minimize wi E + w2 (36) e=1 e= 1 subject to x is compatible with row i V i 1,...,m (37) r integer V 1,... ,L. (38) Clearly, our model contains the problem of minimizing the number of apertures as a special case by setting wl = 1 and w2 = 0. Moreover, if we wish to minimize the number of apertures required while limiting the beamontime to no more than T, we simply add the following constraint to the model: L fx, where of course T cannot be less than the minimum achievable beamontime z (which can be found in polynomial time using the algorithms mentioned in Section 3.1). To formulate our master problem as an integer programming problem, we introduce binary variables ye,, V 1,..., L, r = 1,..., Re, where ye = 1 if and only if xa = r, and Re is an upper bound on the number of apertures having intensity used in an optimal solution. (We can compute Re by computing an initial upper bound on the optimal objective function value via any of the heuristics mentioned in Section 3.1, and then setting Rf to the largest value such that wAlR + W2Re is no more than this bound.) Using these decision variables, we can reformulate the master problem (M!lP) as follows: L L minimize w, xy + w2 x (310) e=1 e= 1 subject to RE Y.ry Vxf 1,...,L (311) r=i RE Ye < 1 V f 1,...,L (312) r=i x is compatible with row i V i 1,..., m (313) X' integer V 1,...,L (314) yNr binary V = 1,...,L, r 1,...,R. (315) We next formulate (313) as a set of linear inequalities by deriving valid inequalities that cut off precisely those vectors x that violate (313). To this end, consider a particular allowable intensity multiset represented by i that is incompatible with at least one row. It is then clear that we should only consider vectors x that are different from 5 in at least one component. We can achieve this by imposing the following constraint: L Re yi > 1. (316) =1 r=l 1 Since all integer solutions except for i satisfy (316), it is indeed a valid inequality. Constraint (316) can be tightened by observing that if the solution i is incompatible with row i, then any solution x such that xL < Vx, V 1,..., L, is also incompatible with row i. Therefore, we require that x contain at least one component that is larger than its corresponding component in i, which yields the stronger valid inequality L Re y E r>l (317) =1 r= e+l Constraint (317) can, in turn, be tightened further by explicitly considering the rows for which x is incompatible. Let Li = maxj, 1,.., bij be the maximum intensity in the fluence map for row i. By the same argument as above, if the current solution i is incompatible with row i, then any solution x such that xL < VX, V = 1,..., Li, is also incompatible with row i, since no leaf positions with intensity greater than Li can be used in decomposing row i. Therefore, we require that x is larger than i in at least one component 1,... Li: Li Re SZ y&r > 1 V rows i incompatible with k. (318) =1 r= e+1 Since (318) is stronger than (316) or (317), we use the latter inequalities in our model. Note also that (318) stated for row i1 dominates a cut generated for row i2 if Li, < Li,. Thus, we consider the bixel rows in nondecreasing order of their Livalues, halt when an infeasible row is detected, and add a single inequality of the form (318). This sequence also tends to minimize subproblem execution time, since rows having a small maximum intensity are easier to solve by the nature of the backtracking algorithm discussed in Section 3.2.3. Since the collection (318) contains an exponential number of valid inequalities, we add them only as needed in a cutting plane fashion. In particular, this means that we relax (318), solve the relaxation of (l IP) and generate an xsolution representing a candidate allowable intensity multiset. We then solve a subproblem for each bixel row to determine if the allowable intensity multiset is incompatible with that row. If not, we have found an optimal solution to (\! I). Otherwise, we add a constraint of the form (318) to (\! P) that cuts off that solution. 3.2.3 Subproblem Analysis and Solution Approach In this section, we consider the subproblem of checking whether a given intensity multiset x is compatible with a particular bixel row. For convenience and wherever the interpretation is clear from the context, we suppress the index i of the bixel row and denote a typical row of the fluence map B by b = (bi,..., b,). We represent a feasible decomposition as a collection of ndimensional binary vectors v&r (e = 1,... L; r = 1,... r). The values of ve, that equal 1 correspond to the (consecutive) exposed bixels in the rth aperture having intensity For example, the decomposition in equation (32) corresponds to vi = (1, 0), v31 = (0, 1, 1), V51 (0, 0, 1), and ve = 0 for other r. (Note that this decomposition would be feasible as long as xl, X3, Xs > 1.) The subproblem can then formally be presented as follows: CIPARTITION INSTANCE: An ndimensional vector of nonnegative integers b and an integer vector x= (Xi,... ,XL). QUESTION: Do there exist ndimensional binary vectors v&r (e 1, ..., L; r 1,..., xr) that satisfy the consecutiveones property such that C L1 1 1V r = b? Proposition 3. CIPARTITION is strongly NPcomplete. Proof. See Appendix B. In principle, the CIPARTITION problem can be formulated and solved as an integer program. However, we have developed a computationally more effective backtracking algorithm that focuses on partitioning intensity requirements individually for each bixel. An integer vector p = (pJ,... ,p ) provides a bixel decomposition of bixel j E {1,..., n} in row b if and only if bj = I1 p". We then attempt to form a collection of leaf positions that realizes the individual bixel partitions. We call such a collection of leaf positions a leaf decomposition of b. To more effectively conduct our subproblem searches, we describe a property that holds in some leaf decomposition (if one exists) that satisfies the given collection of bixel decompositions. Lemma 1. Consider candidate bixel decompositions for bixels j and j + 1, for some j E {1,..., n 1}, and suppose that these have a common decomposition ;/,l.. './i value f, i.e., ,p jp > 0. Then, if a leaf decomposition exists, one exists in which a leaf position having ",'1. i, ii exposes both bixels j and j + 1. Proof. Assume that there exists a leaf decomposition V in which bixels j and j + 1 are exposed by two separate leaf positions, vi and v2, respectively, each having intensity f. Now consider the leaf position V3 = V1+v2 having intensity f. Then V' = {v3}UV\{v, v2} is also a leaf decomposition that realizes the given bixel decomposition. E We next derive a necessary condition that any feasible bixel decomposition must satisfy so that the corresponding set of leaf positions is compatible with a given allowable intensity multiset x. Similar to the idea behind Lemma 1, if p > p+l, then p p+l leaf positions having intensity f must expose bixel j but not j + 1. Lemma 2 formalizes this idea. Lemma 2. Let x represent an allowable ';/,.I. ,'li multiset, and pJ7' denote candidate bixel decompositions for bixels j,, Vr 1, ..., n' such that 1 < ji < .. < jn, < n. The following set of conditions must be /.:/7. in I ,; feasible solution. n/ Z max{Op"p + p" Proof. If p& > p", at least p" p leaf positions having intensity must expose bixel Ji but not j,. Also, at least pj"' leaf positions having intensity must expose bixel j,'. Since all leaf positions listed above are necessarily disjoint, the lemma holds. E We next describe our backtracking algorithm. In this algorithm, we first enumerate all possible vi of decomposing the bixel intensities in b using a subset of the allowable intensity multiset given by x. We denote the set of all candidate bixel decompositions for bixel j by Pj, where for each p E Uj I'j, we must have pe < xz, V = 1,..., L. The backtracking algorithm for solving the subproblem is stated formally in Algorithm 1. We begin by enumerating each possible element of Pj, V j = 1,..., n. We denote the set of processed bixels by F (for which a candidate Il, bixel decomposition has been established), and the set of unprocessed bixels by R. In each iteration, we check to see if the set of candidate bixel decompositions Pj for any j E R is empty. If so, the current active bixel decompositions do not yield a feasible solution, and the algorithm backtracks. Otherwise, we consider an unprocessed bixel j E 7, and choose an untried bixel decomposition p' E CPy to be active for bixel j. Next, we move j from R to F, creating updated sets R' and F', and invoke Lemma 2 to update the set of bixel decompositions for the bixels in R'. Specifically, for each j E R' and pJ E 'j, we calculate the number of leaf positions that would be required due to selecting pJ as the active bixel decomposition for bixel j, in addition to those already selected for bixels in F'. We eliminate pJ if a condition of type (319) is violated. We then recursively call the procedure to continue with a new bixel j' E R'. We stop either when we find a feasible bixel decomposition for all bixels, or when we exhaust all bixel decompositions without finding a feasible solution. In the former case, a leaf decomposition that realizes the bixel decompositions for bixels j e {1,..., n} can be found by invoking Algorithm 3, which is based on the repeated application of Lemma 1. To see that Algorithm 3 recovers a feasible leaf decomposition, note that Algorithms 1 and 2 provide bixel decompositions that satisfy Lemma 2, and in particular, the condition max{0,pF 1 + Algorithm 3 recovers a feasible leaf decomposition if, in the outer whileloop corresponding to each 1,..., L, the counter r is never incremented more than xr times. Note that r is incremented each time the inner whileloop terminates, which occurs either when J > n (a total of p7 times), or when p = 0 (p1 pj times) for = 2,..., n. The total number of times that r is incremented in the outer whileloop for 1,..., L is thus the lefthandside of (320), which is no more than xe, as required. If we exhaust all bixel decompositions without finding a feasible solution, we conclude that the current allowable intensity multiset is incompatible with the current row. Algorithm 1 C1PARTITION(b, x) Input: b {ndimensional vector representing bixel intensity requirements} Input: x {Ldimensional vector representing an allowable intensity multiset} {This algorithm finds whether there exists a CIPARTITION of b compatible with x} S< 0 {F is the set of processed bixels} R < {1,..., n} {7R is the set of unprocessed bixels} for all j E {1,...,n} do Pj  Enumerate all bixel decompositions compatible with x for bixel j < {P ,.... ,P } return C1PARTITIONRECURSIVE(b, x, F, 7, P) Since Algorithm 1 is a backtracking algorithm, and therefore in the worst case investigates all possible bixel decompositions, it is of exponential time complexity (as expected, due to Proposition 3). However, the empirical running time of the algorithm can be reduced using the following observations: (i) If two .,.li i.:ent bixels in a row have the same required intensity value, there must exist an optimal solution in which they are exposed by the same leaf positions. This result can be proven in a similar way as Lemma 1, and is therefore omitted for brevity. This observation implies that we can preprocess the data by merging Algorithm 2 C1PARTITIONRECURSIVE(b, x, F, 7, 7P) if 7R 0 then return true {all bixels have been processed, P represents a feasible solution} else if 3j ER : Pj = 0 then return false {there is no remaining way of decomposing bixel j} else j argminj,'Pj {j is a bixel having the smallest number of bixel decompositions} for all p e c P do p7' ' p, 'P < {pJ} {p is now the active decomposition for bixel j} F' Fu { j}, R' R\ {j} for all j R' do  Pj\ {all elements eliminated by Lemma 2, given the active decompositions p3 for jE C '} if C1PARTITIONRECURSIVE(b, x, F', R', 7') then return true {a feasible solution that uses pi to decompose bixel j is found} return false {all bixel decompositions of bixel j have been exhausted} Algorithm 3 RECOVERLEAFDECOMPOSITION(b, x, P) Require: Pj {pJ} Vj e {1,..., n} {all bixels have been processed} Output: vr ( = 1,..., L; r = 1,... ,xf) {ve, is an ndimensional binary vector that represents a leaf position} for all c {1,..., L}, r {1,..., x} do vr < 0 for all f {1,..., L} do r < 1, j < 1 while j < n do if pi > 0 then j < j {a new leaf position must start at bixel j} while j < n and p' > 0 do {expand the new leaf position as much as possible} +  1  pi 1,J J+1 r r+ 1 else j  j + 1 {all leaf positions that start at bixel j have been recovered} all .,ili i, ent bixels in a bixel row having the same intensity requirement, thereby reducing the dimensionality of the problem instance. (ii) In choosing the next bixel to be processed, we pick a bixel j E R having the smallest number of remaining candidate bixel decompositions. In this manner, we can quickly enumerate all possible bixel decompositions for a few key bixels and eliminate a significant portion of bixel decompositions for the remaining bixels without wasting effort by unnecessary backtracking steps. (iii) In choosing the next candidate bixel decomposition pJ E Pj for a chosen bixel j E 7?, we select an untried bixel decomposition having the fewest number of intensity values. Since each intensity value used in decomposing a bixel needs to be assigned to a different aperture, this rule favors a bixel decomposition using the fewest number of apertures to decompose the chosen bixel. Therefore, it tends to retain the availability of more elements of the allowable intensity multiset (and hence apertures) for the remaining bixels, making it easier to find a feasible solution (if one exists). 3.2.4 Valid Inequalities for the Master Problem The initial optimal solution to the relaxation of ( lP) in which none of the inequalities (318) have yet been added to the model will set all variables equal to zero, which is clearly incompatible with all rows. In this section, we derive some characteristics of all feasible solutions and use these to define valid inequalities for (\! I). In this way, we attempt to improve the convergence rate of the decomposition algorithm by eliminating some clearly infeasible solutions before the initial execution of the master problem. 3.2.4.1 Beamontime and number of apertures inequalities Our first observation uses and generalizes the fact that the beamontime, number of apertures, and total treatment time required for the decomposition of any single row into leaf positions provide lower bounds on the minimum beamontime, number of apertures, and total treatment time, respectively, needed to deliver the entire fluence map. More generally, consider any collection of nonnegative objective weights w' and w' in place of w1 and w2, and let Ti(w', w') be the minimum value of the objective with respect to these weights over all decompositions for row i only. Then the following are valid inequalities for (\!P): L L w' x+w' x > T, w,w ) V i 1,...,m. (321) e=1 e=1 We formulate an integer programming model to determine T (w w%) for a given row i. First, denote the set of possible leaf positions for that row by /C, and define ndimensional binary vectors vk for k E /C (where 1C = O(n2)), such that = 1 if and only if bixel j is exposed by leaf position k. In addition to decision variables xe as in ( \ P), define binary decision variables zke, V k E /C, f 1,..., Lk such that zk = 1 if and only if leaf position k is used with intensity (where Lk = min,: .v=1 b is an upper bound on the intensity of leaf position k.) Then T (w', w') is the optimal objective function value of the following optimization problem, (SR): L L minimize wu4 Y x' + w' x' (322) e=1 e= 1 subject to Svk i =z b Vj = 1,...,n (323) Lk 5 Zke < 1 V k C (324) e= 1 zkU V f 1,...,L (325) kEIC:Lk>_ zke {0,1} V k cIC, = 1,...,Lk (326) xe > 0 and integer V 1,...,L. (327) Constraints (323) ensure that each bixel receives exactly its required amount of dose while constraints (324) guarantee that each leaf position is either not used or is assigned to a single intensity value. Finally, constraints (325) relate the x and zvariables. A practical difficulty in implementing the valid inequalities of the form (321) is that we must determine appropriate values for the weights w' and w'. However, Baatar (2005) shows that, when decomposing a single bixel row, there exists a set of leaf positions that simultaneously minimizes both beamontime and the number of apertures. If we let Ni Ti(l, 0) represent the minimum number of apertures for row i, and i = Ti(0, 1) represent the minimum beamontime for row i, this implies that Ti(w, w') = wNi+ + w'i, so that we can replace (321) by L L w xe+w' x>wwNi + ,,'_ V 1, ...,m. (328) 1 1 It is easy to see that we can capture all of these valid inequalities by restricting ourselves to the coefficient pairs (w, w) = (1, 0) and (0, 1) only: L x > max {NJ} (329) =1 L x, > max {Js}. (330) =1 We can generalize this idea as follows. Let R() denote the set of rows for which the maximum intensity requirement is bounded by L for some L E {1,..., L}, i.e., R() = {i {1,... ,m} : Li < }. Since intensity values greater than L cannot be used in decomposing the rows in R(), a similar approach to the one above can be used to derive the following family of valid inequalities xe > max NJ} V 1,..., L (331) 1iER(L) e 1 > x > max {zJ} V 1, ..., L. (332) 1iER(L) = 1 Finally, note that the values of Ni and zi can be found by solving (SR) with w' = 1, w 1 or by using the method of Kalinowski (2004), since there exists a solution that minimizes both beamontime and the number of apertures (Baatar, 2005). 3.2.4.2 Bixel subsequence inequalities Recall that (3 16) (3 18) represent necessary conditions for feasibility of an allowable intensity multiset with respect to a particular row. It is possible to develop stronger necessary conditions if we examine subsequences of a row, i.e., a subset of the required intensity values in a row that preserves their order in the fluence map. First, Lemma 3 shows that, if a given allowable intensity multiset is incompatible with a subsequence s of row i, then it also must be incompatible with row i. Lemma 3. Consider an allowable <;I/. <,;iu multiset x, an ndimensional vector b that represents the :<,l. ,i1/; requirements of the bixels in some row of B, and an n'dimensional vector s = (bl,... ba,) where 1 < jl < i < jn < n. If x is not compatible with s, then it is also not compatible with b. Proof. We prove the equivalent statement that if x is compatible with b, then it is also compatible with s. Assume that x is compatible with b. By definition, there exists a bixel decomposition for each bixel j 1,..., n so that the resulting set of leaf positions is compatible with x. The bixel decompositions corresponding to only the bixels in s are also compatible with x, since the order of the bixels in s is the same as that in b. O Note that we can invoke Lemma 3 to associate a subproblem with each of the 0(2') subsequences of a bixel row b. Each of these subproblems can then be used to generate cutting planes of the form (318), as well as valid inequalities of the form (331) and (332). However, since the strength of (318), (331) and (332) depend on the largest intensity value in a bixel row, we form subsequences of each bixel row by, for L 1,... L, considering only those bixels having required intensity less than or equal to L. The valid inequalities generated by the O(min(n, L)) subsequences generated in this fashion imply all 0(2") valid inequalities associated with all possible subsequences. 3.2.5 Constructing a Feasible Matrix Decomposition Our algorithm finds an optimal allowable intensity multiset and a bixel decomposition for each bixel row. To construct a corresponding matrix decomposition, we need to apply Algorithm 3 to find a leaf decomposition for each row. We can then generate aperture matrices by arbitrarily combining leaf positions using the same intensity values in different rows. We have found empirically that this simple approach yields a feasible matrix decomposition very quickly. Since any pair of leaf positions assigned to the same intensity value in different rows can be combined, there are up to (nH ,(X!) aperture matrices that can be constructed from a given feasible leaf decomposition for each row. Even though each such choice represents an alternative optimal solution to the optimal leaf sequencing problem, some matrix decompositions may clinically be preferable to others based on their structural properties. Perhaps the most challenging structural consideration pertains to the socalled "tongueandgiuui, effect observed in MLCs. We refer the reader to the works of Deng et al. (2001) and Que et al. (2004) for technical details of the tongueandgroove effect in dynamic MLC dose delivery. For the purposes of this study, it is sufficient to understand that leaves in .,.i ,i:ent rows often interlock with a tongue on the bottom of one row sliding along a groove in the top of another row. Tongueandgroove underdosage occurs since a leaf's tongue blocks dosage intended for cells beneath it. Therefore, it is desirable to limit such underdosages. To measure the amount of tongueandgroove effect in a treatment plan, Que et al. (2004) note that it is generally not desirable to deliver one aperture in which some bixel (i,j) is blocked by a leaf while bixel (i + 1,j) is not blocked, if another aperture is being delivered where (i,j) is not blocked by a leaf while (i + 1,j) is blocked. Based on this observation, Que et al. (2004) derive the following tongueai,:l,roove index (TGI). Suppose a treatment plan consists of K apertures described by binary values vjk, where v =k 0 if cell (i,j) is blocked by a leaf in aperture k and vk = 1 otherwise, for each i = 1,..., j = 1,..., n, k = ,..., K. Let Ik be the intensity delivered in aperture k = 1,..., K. Then the TGI of a matrix decomposition is defined as: m1 n K1 K min{Ik, I ik (t V+1,k) t )U i+1, i 1 j=1 k=1 =k+1 + (1 Vk) (kvj 1 +) (333) i v (t We thus can calculate the TGI component induced by rows 1 and 2 (of all aperture pairs), then rows 2 and 3, and so on, down to rows m 1 and m. This observation allows us to focus on pairs of rows instead of pairs of entire aperture matrices while reducing TGI, allowing us to design an efficient algorithm for TGI reduction given a set of bixel decompositions for each row. Given a pair of .,l1i ient rows, we attempt to match individual leaf positions in the two rows to minimize the TGI induced by the .,li ,i:ent row pair. To limit computational overhead in this phase of our algorithm, we reduce TGI indirectly by the following scheme. Let us denote a leaf position for row i by a binary nvector v', where vj = 1 if the leaf position exposes bixel j in row i. We measure the overlap between two leaf positions having the same intensity value in consecutive rows by counting the number of columns that both leaf positions expose simultaneously. Formally, we define the overlap between leaf positions vi and vi+1 as 0(vi, vi+1) = E, vjv1'. Our approach is to heuristically minimize TGI by maximizing the total overlap between all leaf position pairs, which can efficiently be solved as an assignment problem. The efficiency of the assignment problems can be further improved by noting that the problem decomposes over the intensity values Se {1,..., L}, since only leaf positions having the same intensity value can be combined. Therefore, we can generate a matrix decomposition by finding a leaf decomposition for each row, and then matching leaf positions in .,li i,:ent rows having the same intensity value by solving an assignment problem so that the total overlap is maximized. The TGI minimization step described in the previous paragraph can be improved as follows. Typically, multiple bixel decompositions exist for each row that are compatible with a given feasible intensity multiset. Algorithm 2 can be modified in a straightforward manner so that it finds all leaf decompositions of a row, instead of stopping once the first feasible bixel decomposition for all bixels is found. Since different bixel decompositions for a bixel row correspond to different leaf decompositions, considering alternative bixel decompositions can lead to a matrix decomposition having a smaller TGI. Given alternative leaf decompositions for each row, the problem of minimizing TGI can be formulated as a shortest path problem as follows. We create al 1,. I network in which each l? r corresponds to a bixel row i E {1,..., m}, and node Nid represents the dth leaf decomposition of row i. We add a directed arc from each node Nid to all nodes N(1i+l)d, for all i = ,... m 1. The cost of the arc from node Nid to N(i+l)dl is given by the TGI value resulting from the assignment solution corresponding to the candidate leaf decompositions represented by d for row i, and d' for row i + 1. Finally, we add a start node S and a finish node F. We create zerocost arcs from S to all nodes in the first 1~.lr, and from all nodes in the last l?r to F. A shortest SF path in this graph represents a matrix decomposition having a minimum TGI from among the provided options. Since the graph is .,. i, the shortest path problem can be solved in O(A) time, where A is the set of all arcs. Remark 5. The shortest path approach to i,,.,:.:i,, .t:, TGI can be difficult to solve !.':. 1./ when bixel rows have a ,,',, number of alternative leaf decompositions, since an arc joins each pair of nodes corresponding to adjacent bixel rows. To y., ;/.:ill,/ overcome this difficulty, we limit the number of bixel decompositions found by Algorithm 2 by terminating once 250 feasible bixel decompositions have been .,J. ,.,1/; Next, note that a straightforward '. ;/. /.:'. shortest path implementation processes 7.';,.. one at a time, and does not generate a feasible SF path before processing the last 17;, ,. Since being able to I'... :fy a time limit is a desired feature in a practical setting, we use a i/;,l, .:i .r'i' .:hm for solving the shortest path problem. Our i1'.,i,.:hm starts by processing 7.;1,.. onebyone, i.,l.':,.,j node labels as usual. If a shortest path is not found when a given initial time limit expires, our il'.>rithm switches to a depthfirstsearch (DFS) procedure, which we terminate after a given final time limit. We start DFS from an unprocessed node Nid having a smallest label, select a minimumcost arc (Nid, N(i+l)d,) exiting that node, and update the label of N(i+l)di if we have found a new shortest SN(i+l)d, path. Else, the i1,.' ,:thm backtracks and seeks another arc from Nid. We then return the shortest SF path found by this procedure when the final time limit is reached. 3.3 Computational Results and Comparisons 3.3.1 Problem Instances In our experiments we have used two classes of problem instances. Our base set of test problem instances consists of 25 clinical problem instances that were obtained from treatment plans for five headandneck cancer patients treated using five beam angles each. Table 31 reports the problem characteristics for these problem instances in terms of the matrix dimensions m and n. The maximum intensity value is L = 20 for all these instances. In addition, to allow comparison of our results with published results on other approaches to the problem, we generated 100 random problem instances of dimensions 20 x 20 having maximum intensity value L = 10. However, since these problem instances are generally too large to be solvable by the integer programming model from Langer et al. (2001) and its modification described in Appendix A, we also randomly generated eight instances ( i. I .:;. ", ..., "test6x7b") to demonstrate the computational limitations of the latter approaches. Unless otherwise specified, we used wl = 7 and w2 = 1 as the objective weights for the number of apertures and beamontime, respectively. Table 31. Dimensions of clinical problem instances Name m n Name m n Name m n Name m n Name m n clbl 15 14 c2bl 18 20 c3bl 22 17 c4bl 19 22 c5bl 15 16 clb2 11 15 c2b2 17 19 c3b2 15 19 c4b2 13 24 c5b2 13 17 clb3 15 15 c2b3 18 18 c3b3 20 17 c4b3 18 23 c5b3 14 16 clb4 15 15 c2b4 18 18 c3b4 19 17 c4b4 17 23 c5b4 14 16 clb5 11 15 c2b5 17 18 c3b5 15 19 c4b5 12 24 c5b5 12 17 3.3.2 Implementation Issues We have implemented our decomposition algorithm using CPLEX 11.0 running on a Windows XP PC with a 3.4 GHz CPU and 2 GB RAM. We use callback functions of CPLEX to generate a single branchandbound tree in which we solve the subproblems corresponding to each integer solution found in the tree, and add cuts to tighten the master problem as necessary. This implementation turned out to be consistently faster than one which resolves the master problem each time a cutting plane is added to the model. Furthermore, in our base algorithm, we use the subsequence inequalities (331) and (332) described in Section 3.2.4.2. We also use Engel's heuristic (Engel, 2005), which executes in well under one CPU second for each instance and generates a solution having minimum beamontime, to (i) obtain an initial upper bound and (ii) compute the upper bounds Re (f 1,..., L). 3.3.3 Comparison with Langer et al. (2001) Model Our first experiment compares our base algorithm that minimizes the total treatment time to that of Langer et al. (2001) and to the modification of their model as described in Appendix A. We choose randomly generated test instances of various dimensions to identify the problem sizes that can be solved by each algorithm, as well as four of the smallest clinical instances to compare the effectiveness of the algorithms on clinical instances. We imposed a onehour time limit past which we halted the execution of an algorithm. For these experiments we disabled the use of Engel's heuristic as an initial heuristic to test the ability of these models to efficiently find goodquality upper bounds. Table 32 summarizes the results of these three algorithms in terms of the execution time, the best upper and lower bounds found within the time limit, and the optimality gap (calculated as the difference between the upper and lower bound as a percentage of the upper bound). Our decomposition algorithm can solve all 15 instances in this data set within a few seconds, whereas only six instances can be solved to optimality within an hour by either integer programming formulation. We conclude that, even though the integer programming formulation given in (Langer et al., 2001) can solve small instances to optimality, it cannot be used to solve clinical problem instances to optimality within practical computation time limits. Table 32. Comparison of our base algorithm with Langer et al. (2001) model Two stage Langer Modified Langer Name m n L CPU Optimal CPU UB LB Gap CPU UB LB Gap test3x3 3 3 8 0.1 29 1.6 29 29.00 0.0% 0.9 29 29.00 0.0% test3x4 3 4 8 0.1 37 5.2 37 37.00 0.0% 1.6 37 37.00 0.0% test4x4 4 4 8 0.1 36 30.4 36 36.00 0.0% 10.7 36 36.00 0.0% test5x5a 5 5 10 0.2 45 2069.6 45 45.00 0.0% 86.4 45 45.00 0.0% test5x5b 5 5 15 0.2 50 198.2 50 50.00 0.0% 92.6 50 50.00 0.0% test5x6a 5 6 10 0.2 55 3600 61 33.53 45.0% 3600 55 40.95 25.5% test5x6b 5 6 18 0.4 71 3600 84 51.58 38.6% 3600 77 58.67 23.8% test6x6a 6 6 13 0.3 55 3600 55 45.63 17.0% 3600 55 48.00 12.7% test6x6b 6 6 13 0.3 52 3600 57 43.82 23.1% 3600 57 50.00 12.3% test6x7a 7 6 10 0.2 45 690.0 45 45.00 0.0% 435.1 45 45.00 0.0% test6x7b 6 7 15 0.4 74 3600 94 35.69 62.0% 3600 80 47.88 40.1% clbl 15 14 20 1.3 111 3600 336 48.58 85.5% 3600 273 42.00 84.6% clb2 11 15 20 0.8 104 3600 280 38.26 86.3% 3600 132 39.55 70.0% clb5 11 15 20 3.1 104 3600 280 46.20 83.5% 3600 140 49.29 64.8% c5b4 14 16 20 2.5 124 3600 360 34.00 90.6% 3600 360 39.11 89.1% 3.3.4 Random Problem Instances For our next experiment, we first solved each of the 20 x 20 random problem instances in our data set to optimality for the problems of (i) minimizing total treatment time ("Total Time"), (ii) minimizing the number of apertures while constraining the beamontime to be minimal ("Lexicographic"), and (iii) minimizing the number of apertures ("# Ap. i I i. ). We also implemented three heuristic algorithms proposed by Siochi (2007), Engel (2005), and Xia and Verhey (1998), which we executed on the same data set. (The results we present from Siochi (2007) refer to the Variable Depth Recursion (VDR) algorithm without tongueandgroove constraints, using the parameters recommended in the paper. We discuss the effect of including tongueandgroove considerations in the algorithm below.) Figure 32 summarizes the total treatment times associated with the solutions generated by the six algorithms we tested. Each algorithm is represented by a curve that depicts quality of the solutions obtained by the corresponding algorithm. For each value T of total treatment time on the horizontal axis, each curve plots the number of problem instances for which the corresponding algorithm was able to find a solution having total treatment time no more than T. For instance, Figure 32 shows that Siochi's heuristic found a solution with a total treatment time of at most 175 time units in 5' of the problem instances, while an optimal solution (represented by "Total Time") has C) 0 I 50 LL0 40 30 20 / p XiaVerhey / SiochiVDR Engel 10 # Apertures SLexicographic STotal Time 0 150 200 250 300 Total Treatment Time Figure 32. Comparison of total treatment times on random data the same quality level in 97'. of the problem instances. We observe that all three exact algorithms find solutions having similar treatment times. Solution qualities generated by the Engel and Siochi heuristics are similar, with the Siochi heuristic being slightly better. A comparison of the heuristic solutions with optimal solutions reveals that average optimality gaps for Siochi, Engel and XiaVerhey heuristics are 10.1 12.0' and 51.5' , respectively. Figure 33 compares the algorithms with respect to the number of apertures used in their respective solutions. We note that our algorithm that minimizes total treatment time ("Total Time") finds a solution that also minimizes the number of apertures for most problem instances. As expected, lexicographic minimization of the two objective functions C) = 50 L0 40 30 20 XiaVerhey / SiochiVDR /  Engel 10 // # Apertures Lexicographic e  Total Time 0 14 16 18 20 22 24 26 28 30 32 Number of Apertures Figure 33. Comparison of the number of apertures on random data results in an increased number of apertures. For this objective the "# Ap. i Ii i  algorithm finds optimal solutions. Average optimality gaps for the heuristics of Siochi, Engel, and Xia\Vrl v are 15.1,'. 18.9' and 62.;:', respectively. We analyze the beamontime values of the solutions generated by each algorithm in Figure 34. Since both Engel's heuristic and our "Lexicographic" algorithm find optimal solutions having minimum beamontime, their curves overlap. We observe that the Siochi heuristic and our "Total Time" algorithm tend to generate solutions having small beamontime values, but the solutions generated by our "# Ap. iIi algorithm, and by the XiaVerhey heuristic have higher beamontime values. We calculated the average optimality gaps for the latter two algorithms as 12.1,'. and 32.1 respectively. C 0 50 LL 40 30 20 XiaVerhey  SiochiVDR  Engel 10 / # Apertures Lexicographic STotal Time 0 r r r T I  45 50 55 60 65 70 75 80 85 BeamonTime Figure 34. Comparison of beamontime values on random data The final measure of solution quality that we consider is TGI, which is a measure of the tongueandgroove effect given by (3 33). Figure 35 reveals that the solutions obtained by all three variants of our decomposition algorithm have significantly lower TGI values than the heuristic procedures. This result implies that, even though our TGIreduction algorithm described in Section 3.2.5 does not guarantee a minimum TGI, it is highly effective in finding solutions with TGI values superior to the other heuristic approaches. To estimate optimality gaps for the heuristics we compare heuristic solutions with the solutions generated by our "Lexicographic" algorithm, which provides the best TGI among all methods mentioned above. We note that average gaps for Siochi, Engel and XiaVerhey heuristics are 162.1 164. !' and 205.!' respectively. We also note 90 80 70 60 0 50 40 30 20 XiaVerhey  SiochiVDR 7Engel 10 # Apertures Lexicographic STotal Time 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 TongueandGroove Index (TGI) Figure 35. Comparison of TGI values on random data that these heuristics do not attempt to minimize TGI, and it might be possible to modify them to obtain solutions with lower TGI values. It is interesting to note that a variant of Siochi's algorithm (Siochi, 2007) is capable of completely eliminating TGI at the expense of creating additional apertures. This variant is reported to increase the number of apertures by 10('. to 3 :' i relative to the variant that does not remove TGI (Siochi, 2007). Finally, the Engel and XiaVerhey heuristics took less than one second of CPU time in all instances we tested. The average CPU time for Siochi's heuristic, "Total Time" algorithm, "# Ap. ii wi. algorithm, and "Lexicographic" algorithm were 31.5, 963.1, 414.8, and 421.4 seconds, respectively. We note that all variants of our twostage algorithm showed a "heavytail" behavior, where about 1I' '. of the problem instances were solved to optimality in less than the average solution time. For instance, using the "# Ap. 11 i  algorithm, we were able to solve 40 instances within one minute, 58 within two minutes, 81 within 414.8 seconds (the average solution time for this algorithm), 90 within 15 minutes, and all but three instances were solved within an hour. The remaining three instances were solved within three hours. 3.3.5 Clinical Problem Instances Recall from Section 3.1 that in clinical practice, we can deliver each fluence map using a decomposition into either rowconvex or columnconvex apertures, where the latter requires rotation of the MLC head. Our final set of experiments compares the algorithms on clinical problem instances in our data set, allowing for MLC head rotation. We first show the results of applying our decomposition algorithm to decompose each of the 25 clinical fluence maps into rowconvex apertures, and columnconvex apertures, where the latter is achieved by applying our algorithm to the transpose of each fluence map. Table 33 reports the performance of our algorithm when the objective function is set to minimize total treatment time, and dip'1 v the number of apertures ("nAper"), beamontime ("BOT"), total treatment time ("Time"), tongueandgroove index ("TGI"), and CPU time used ("CPU") for the algorithm. Our algorithm finds an optimal solution to several instances within a few seconds while four instances take more than 10 minutes of CPU time to be solved to optimality. Comparing the solutions obtained for rowconvex and columnconvex decompositions, we observe that rotating the MLC head is most beneficial (in terms of treatment time) for instances in which the number of rows is much smaller than the number of columns. These benefits are most apparent on instances c4b2 and c4b5, where rotating the MLC head can result in more than 50'. reduction in total treatment time. We also note that several problem instances require much less computational time to solve for a columnconvex decomposition compared to a rowconvex decomposition. Table 33. Effect of rotating the MLC head RowConvex ColumnConvex Name nAper BOT Time TGI CPU nAper BOT Time TGI CPU clbl 10 41 111 102 1.1 11 38 115 50 5.5 clb2 10 34 104 80 0.8 8 23 79 14 0.7 clb3 11 31 108 97 11.4 9 28 91 59 1.0 clb4 11 33 110 74 37.0 11 37 114 146 7.0 clb5 10 34 104 133 4.3 8 32 88 49 1.2 c2bl 14 34 132 134 26.5 12 30 114 187 11.5 c2b2 13 41 132 159 20.1 11 33 110 192 8.0 c2b3 13 49 140 245 14.7 11 28 105 151 3.1 c2b4 14 51 149 316 87.3 12 34 118 148 8.3 c2b5 13 41 132 217 395.6 10 27 97 120 2.0 c3bl 13 41 132 323 310.0 14 40 138 254 23.0 c3b2 14 46 144 320 4759.8 8 23 79 86 1.1 c3b3 13 49 140 533 10373.9 12 40 124 360 18.6 c3b4 12 44 128 481 524.9 12 40 124 327 428.2 c3b5 13 34 125 133 3.3 9 27 90 75 2.6 c4bl 16 40 152 216 34.9 12 46 130 244 10.6 c4b2 16 69 181 450 20901.0 9 27 90 149 15.8 c4b3 14 41 139 130 44.7 10 32 102 129 3.3 c4b4 14 44 142 246 164.3 10 27 97 163 8.0 c4b5 17 76 195 470 14511.4 9 24 87 48 4.0 c5bl 10 26 96 68 0.5 10 35 105 41 0.5 c5b2 12 41 125 59 14.3 8 25 81 27 0.6 c5b3 10 34 104 155 3.1 9 23 86 42 1.0 c5b4 12 40 124 105 2.2 10 32 102 87 4.3 c5b5 12 46 130 151 51.9 8 31 87 17 0.8 Table 34. Computational results for our base algorithm Total Time # Apertures Lexicographic Name nAper BOT TGI CPU nAper BOT TGI CPU nAper BOT TGI CPU clbl 10 41 102 4.7 10 41 102 2.3 11 38 50 4.8 clb2 8 23 14 1.1 8 23 14 1.1 8 23 14 1.1 clb3 9 28 59 3.0 9 28 59 4.5 9 28 59 4.5 clb4 11 33 74 41.2 11 37 128 27.1 11 33 74 12.2 clb5 8 32 49 2.1 8 34 56 1.3 9 26 9 1.9 c2bl 12 30 187 15.6 12 30 187 14.9 12 30 187 14.4 c2b2 11 33 192 10.8 11 38 161 6.9 11 33 146 7.8 c2b3 11 28 149 8.9 11 28 113 9.9 11 28 197 10.8 c2b4 12 34 148 16.8 12 34 148 16.8 12 34 148 17.1 c2b5 10 27 120 6.1 10 31 155 6.2 10 27 120 6.2 c3bl 13 41 323 315.0 12 51 521 62.1 13 41 325 31.4 c3b2 8 23 86 4.4 8 26 87 4.5 8 23 62 5.6 c3b3 12 40 360 27.4 12 40 360 894.7 12 40 365 20.1 c3b4 12 40 327 442.2 12 46 284 548.8 13 38 928 55.1 c3b5 9 27 75 5.6 9 27 75 5.4 9 27 75 5.7 c4bl 12 46 244 16.8 12 46 227 10.6 12 46 227 11.3 c4b2 9 27 149 45.5 9 32 150 56.2 9 27 135 35.0 c4b3 10 32 129 15.7 10 34 108 14.9 10 32 129 15.6 c4b4 10 27 163 32.0 10 28 112 32.6 11 26 72 29.9 c4b5 9 24 48 27.8 9 24 48 27.7 9 24 48 27.0 c5bl 10 26 68 1.2 10 26 68 1.2 10 26 68 1.2 c5b2 8 25 27 1.1 8 25 27 1.0 9 23 8 1.1 c5b3 9 23 42 3.6 9 24 45 3.2 9 23 83 3.1 c5b4 10 32 87 5.8 10 41 101 2.7 10 32 87 2.8 c5b5 8 31 17 1.4 8 33 16 1.2 8 31 71 1.1 Motivated by this observation, we modify our algorithm to directly solve for the best orientation by using obtained upper and lower bounds to quickly prove whether rotating the MLC head is beneficial. Assume that we have lower and upper bounds for the rowconvex and columnconvex problems, and suppose that the lower bound of the rowconvex problem is greater than the upper bound of the columnconvex problem. In this case, we can conclude that an optimal solution minimizing total treatment time for the given fluence map must be a columnconvex decomposition. We use this argument to solve one of the problems, and then use the bound information to avoid having to solve the other one to optimality. We pick the first problem to solve by selecting one having the least initial lower bound, breaking ties if applicable by choosing the problem for which n < m, since the subproblems tend to solve faster for smaller values of n. Table 34 shows the nAper, BOT, TGI, and CPU metrics obtained from our algorithm enhanced with the above bounding scheme, corresponding to the "Total Time," "# Apertures," and "Lexicographic" objectives. Observe that all 25 instances, under any metric, terminate in under 15 minutes of CPU time with a solution that is optimal with respect to the corresponding objective, and all instances are solved to optimality within a minute using the "Lexicographic" algorithm. Recall that the "BOT" column in "Lexicographic" reports the minimum achievable beamontime, and the "nAper" column under the objective "# Ap. 'i I,. reports the minimum number of apertures needed to decompose each instance. Perhaps surprisingly, in comparing these values with the results of "Total Time," we observe that there exists a solution that minimizes both the number of shapes and the beamontime simultaneously in 19 of the 25 instances. Finally, we analyze performance of the three heuristics on clinical data, where we execute each heuristic on each problem instance and its transpose (corresponding to rowconvex and columnconvex decompositions), and pick the solution yielding the smallest treatment time. Table 35 shows the number of apertures, beamontime, TGI Table 35. Comparison of heuristic algorithms on clinical data Siochi Engel XiaVerhey Name nAper BOT TGI CPU nAper BOT TGI CPU nAper BOT TGI CPU clbl 11 38 245 14.0 12 38 261 < 1 13 40 219 < 1 clb2 8 23 109 3.0 8 23 127 < 1 10 32 133 < 1 clb3 9 28 213 4.0 10 28 192 < 1 12 34 198 < 1 clb4 12 34 306 9.5 11 37 398 < 1 14 42 355 < 1 clb5 9 26 103 3.6 9 26 175 < 1 12 35 124 < 1 c2bl 12 30 652 11.2 12 30 738 < 1 15 45 635 < 1 c2b2 12 33 395 17.8 12 33 464 < 1 15 45 460 < 1 c2b3 12 28 625 34.8 12 28 429 < 1 15 43 459 < 1 c2b4 12 34 628 43.2 12 34 723 < 1 18 56 417 < 1 c2b5 11 27 463 15.3 11 27 465 < 1 14 41 375 < 1 c3bl 14 43 828 36.3 15 40 1054 < 1 17 55 765 < 1 c3b2 9 23 143 11.1 9 23 127 < 1 12 36 289 < 1 c3b3 14 40 1316 40.8 14 40 869 < 1 19 60 1038 < 1 c3b4 13 48 678 33.3 14 38 765 < 1 17 55 553 < 1 c3b5 9 28 263 7.0 9 27 325 < 1 13 45 261 < 1 c4bl 13 46 617 29.4 14 46 625 < 1 18 62 531 < 1 c4b2 10 29 295 73.8 10 27 466 < 1 14 44 350 < 1 c4b3 11 32 339 19.4 11 32 365 < 1 14 48 428 < 1 c4b4 11 26 489 13.5 11 26 540 < 1 15 46 424 < 1 c4b5 9 24 236 89.8 9 24 328 < 1 15 44 328 < 1 c5bl 11 26 188 4.6 12 26 176 < 1 12 38 185 < 1 c5b2 9 23 129 6.9 9 23 100 < 1 10 33 145 < 1 c5b3 9 26 201 5.1 10 23 293 < 1 12 32 189 < 1 c5b4 11 32 218 11.2 11 32 322 < 1 13 46 243 < 1 c5b5 8 32 217 7.2 9 31 211 < 1 11 35 138 < 1 metrics for each solution as well as the CPU time spent by each heuristic. Comparison with the "Total Time" columns in Table 34 reveals that even though the heuristics consistently generated highquality solutions, the Siochi and Engel heuristics were able to find an optimal solution in only five problem instances, and XiaVerhey heuristic could not find an optimal solution to any instance. CHAPTER 4 RECTANGULAR MATRIX DECOMPOSITION PROBLEM 4.1 Introduction and Literature Survey Over the past decade, Intensity Modulated Radiation Therapy (IMRT) has developed into the most successful externalbeam radiation therapy delivery technique for many forms of cancer. This is due to its ability to deliver highly complex dose distributions to cancer patients that enable the eradication of cancerous cells while limiting damage to nearby L. il' 1!: organs and tissues. Patients treated with IMRT therefore often experience a higher chance of cure, suffer from fewer side effects of the treatment, or both. In this chapter, we study an optimization problem that is related to the efficient clinical implementation of IMRT using a simpler technology than currently used, which, if successful, will reduce the cost as well as the complexity of delivering IMRT and thereby make such superior treatments accessible to significantly more patients worldwide. Externalbeam radiation therapy is delivered from multiple angles by a device that can rotate around a patient. The use of multiple (typically 39) angles is one of the tools that allow for the treatment of deepseated tumors while limiting the radiation dose to surrounding functioning organs. Conventional conformal radiation therapy then further uses blocks and wedges to shape the beams (see, e.g., Lim (2002) and Lim et al. (2004, 2007)). IMRT is a more powerful therapy that instead modulates beam intensity. The most common technique for achieving this modulation is to dynamically shape beams with the help of a multileaf collimator (! IC) system. Such systems can dynamically form many complex apertures by independently moving leaf pairs that block part of the radiation beam. Unfortunately, MLC systems are very costly and technologically advanced, and are therefore difficult and expensive to operate and maintain. Moreover, MLC systems are currently only available for use with a socalled linear accelerator that generates highenergy photon beams for treatment. However, the use of radioactive 60Co (Cobalt) sources for radiation therapy is still ubiquitous in many parts of the world and is poised to experience a revival in the United States and Europe through the RenaissanceTM device that is under development by ViewRay, Inc. based in Cleveland, Ohio. Without a MLC, IMRT delivery may be achieved through the use of compensators: highdensity blocks that control the intensity profile of a radiation beam. Such blocks are custommade for each individual patient, which makes compensatorbased IMRT not only labor and storage space intensive, but it also makes the actual treatment very timeconsuming due to the fact that therapists must enter the treatment room to place each individual compensator. In addition, compensators have several undesirable properties that make it difficult to perform accurate dose calculations, thereby reducing the advantages of IMRT (see, e.g., Earl et al. (2007)). Recently, researchers have begun to explore the clinical feasibility of delivering IMRT using conventional jaws that are already integrated into radiation delivery devices and can create apertures that are rectangular in shape (see, e.g., Earl et al. (2007), Kim et al. (2007), and Men et al. (2007)). Successful application of this much simpler delivery technique depends critically on the ability to eff;. : /ibi deliver highquality treatment plans. We therefore develop and test new optimization approaches to minimize the treatment time required for a particular treatment plan using rectangular apertures only. Solving a socalled fluence map optimization problem yields an optimal IMRT treatment plan that resolves different, and conflicting, clinical measures of treatment plan quality related to tumor control and side effects (see, e.g., Shepard et al. (1999) for a review; Lee et al. (2000a, 2003) for mixedinteger programming approaches; Romeijn et al. (2006) for convex programming models; and Hamacher and Kiifer (2002) and Kiifer et al. (2003) for a multicriteria approach). A treatment plan then consists of a collection of nonnegative intensity matrices, often referred to as fluence maps, one corresponding to each beam angle. To limit treatment time, each of these matrices is then expressed as a multiple of an integral fluence map in which the maximum element is on the order of 1020. To allow delivery of the treatment plan, each of these fluence maps should be decomposed into a number of apertures and corresponding intensities, where the collection of apertures that may be used depends on the delivery equipment. For MLC delivery this problem is called the leaf sequencing problem and is very widely studied; for examples, we refer to Al!mi and Hamacher (2005), Boland et al. (2004), Kamath et al. (2003), Engel (2005), Kalinowski (2005a), and Taskm et al. (2009b). (Note that integrated approaches to fluence map optimization, also referred to as aperture modulation, have been proposed as well; we refer to, e.g., PreciadoWalters et al. (2004), Romeijn et al. (2005), and Men et al. (2007).) The problem that we study is the decomposition of an integral fluence map into rectangular apertures and corresponding intensities. While Dai and Hu (1999) proposed a straightforward heuristic for a variant of this decomposition problem, we develop the first computationally viable optimization approach to this problem. In Section 4.2 we consider the core problem of decomposing an (integral) fluence map while minimizing the number of rectangular apertures. In Section 4.3 we then extend our models to the problems of (i) minimizing total treatment time (as measured by the sum of the required aperture setup times and the beamontime, i.e., the actual time that radiation is being delivered); and (ii) minimizing the number of apertures subject to beamontime being minimal. Finally, Section 4.4 discusses our computational results on a collection of clinical fluence maps. 4.2 A MixedInteger Programming Approach We begin in Section 4.2.1 by formally describing the optimization model under investigation and modeling it with a mixedinteger programming formulation. We next describe several classes of valid inequalities in Section 4.2.2. Finally, we discuss methods for partitioning the input matrix in Section 4.2.3, which leads to effective lower and upper bounding techniques. 4.2.1 Model Development In this section, we discuss an integer programming approach to decomposing a fluence map into a minimum number of rectangular apertures and corresponding intensities. We denote the fluence map to be delivered by a matrix B E 1' where the element at row i and column j, (i,j), corresponds to a bixel with required intensity bi. We call a bixel having an intensity requirement of zero a zerobixel. We also define a nonzerobixel analogously. Figure 41 shows an example fluence map, which we use throughout this chapter. 2 3 0 8 2 4 2 2 1 0 5 1 2 1 3 0 0 5 0 0 3 5 0 2 8 6 0 3 0 8 14 10 9 0 3 5 8 20 7 1 0 4 5 9 5 4 0 0 3 Figure 41. Example fluence map Let R be the set of all O(n2r2) possible rectangular apertures (i.e., submatrices of B having contiguous rows and columns) that can be used to decompose B, excluding those that contain a zerobixel. For each rectangle r E R we define a continuous variable x, that represents the intensity assigned to rectangle r, and a binary variable y, that equals 1 if rectangle r is used in decomposing B (i.e., if xr > 0), and equals 0 otherwise. Let Cr be the set of bixels that is exposed by rectangle r. We define Mr = min(ij)ec,{bij} to be the minimum intensity requirement among the bixels covered by rectangle r. Furthermore, we denote the set of rectangles that cover bixel (i,j) by R(i,j). Given these definitions, we can formulate the problem as follows: IPR: Minimize Syr (41) rER subject to: Y z xr bij Vi 1,...,m, j 1,...,n (42) rER(i,j) Xr xr > 0, yr binary Vr e R. (44) The objective function (41) minimizes the number of rectangles used in the decomposition. Constraints (42) guarantee that each bixel receives exactly the required dose. Constraints (43) enforce the condition that x, cannot be positive unless yr = 1. Finally, (44) states bounds and logical restrictions on the variables. Note that the objective (41) guarantees that y, = 0 when x, = 0 in any optimal solution of IPR. Formulation IPR contains two variables and a constraint for each rectangle, resulting in a largescale mixedinteger program for problem instances of clinically relevant sizes. Furthermore, the Mrterms in constraints (43) lead to a weak linear programming relaxation; with no valid inequalities or branching yet performed on the problem, we have that y, = x,/Mr at optimality to the linear programming relaxation of IPR. An alternative formulation that does not require Mrterms employs a decomposition method. Recall that we investigated the problem of decomposing an integer matrix into "consecutiveones" matrices in Chapter 3, where in each decomposed matrix all nonzero values take the same value and appear consecutively on each row. Our computational results showed that solvability of the problem is significantly improved by applying a bilevel optimization algorithm. A similar approach for the problem we consider in this chapter would formulate a master problem as: MP: Minimize Y yr (45) rER subject to: y corresponds to a feasible decomposition (46) yr binary Vr E R, (47) where we address the form of (46) in the sequel. Given a vector y, we can check whether constraint (46) is satisfied by solving the following linear program: SP(f): Minimize 0 (48) subject to: x xr bij Vi 1,...,m, j 1,...,n (49) rER(i,j) x, < 31 i^ Vr E R x, > 0 Vr e R. (411) Associating variables ac with (49), and /3 with (410), we obtain the dual formulation: DSP(y): Maximize bijaij + If (4 12) i=1 j=1 rER subject to: Y c + f3 > 0 Vr ER (413) (ij)ecC, yij unrestricted Vi 1,...,m, j 1,...,n (414) 0,< 0 Vr R. (415) Our Benders decomposition strategy first solves MP, which yields 9. If SP(S) is feasible, then 9 corresponds to a feasible decomposition and is optimal. Else, DSP(f) is unbounded (since the trivial allzero solution guarantees its feasibility). Let (&, 3) be an extreme dual ray of DSP(9) such that 1 1 bijij + TrER 1i 1 > 0. Then, all yvectors that are feasible with respect to (46) must satisfy Z bZ j + (M,,r)yr < 0. (416) i=1 j=1 rER We add (416) in a cutting plane fashion as necessary. Remark 6. Even though the number of r. in,.J, that can be used in ,';,1.:1.:;/.,':.i; the input matrix B is O(n2m2), we observe that optimal solutions 'I/,'./ .l'/; use only a small percentage of the total number of r. /, ,..,jl, This observation suggests that another way to overcome the dimensional ,. m,;pl. :~ ii associated with solving IPR is to apply a column generation approach. In this approach, we start with a feasible set of columns and rows corresponding to a subset of r,. I,gl, and generate additional columns and rows as necessary within a branchcutprice (BCP) il,'., :thm. Even though this approach requires the solution of much smaller linear p.ji 'i':r,",:,J' relaxations, several features of the branchandcut ili,rithm such as preprocessing and automatic cutting plane (410) generation are not applicable. As a result, our implementation of the BCP approach was not ,,i*,*l/,/'L, *,', ll; competitive with the other il' ', :thms we presented, and further details are therefore omitted. 4.2.2 Valid Inequalities In this section we discuss several valid inequalities and optimality conditions for our problem. All inequalities that we describe in this section are applicable to both the integer programming formulation and the master problem of the Benders decomposition approach we described in Section 4.2.1. 4.2.2.1 Adjacent rectangles We call two nonoverlapping rectangles rl and r2 adjacent if either of the following conditions is satisfied: (a) rl and r2 cover an identical range of columns, with rl having bottom row i and r2 having top row i + 1, or (b) rl and r2 cover an identical range of rows, with ri having rightmost column j and r2 having leftmost column j + 1. We observe that there exists an optimal solution in which no two .,i.i i:ent rectangles are used in the decomposition. To see this, assume that .,.i i:ent rectangles rl and r2 have intensities x,1 and x,2, respectively, where x,, < x,2 without loss of generality. In this case, an alternative optimal solution can be constructed by extending ri into r2. Specifically, let r' be the rectangle for which C,, = C,, U C2. An alternative optimal solution that does not contain any .,.i i,:ent rectangles uses r2 having intensity x,2 x,,, and r' having intensity xr1. This dominance criterion can be written as: y~r + y,2 < 1 V .,l.i ,.ent rectangles r, r2, (4 17) which states that no pair of .,I.i ient rectangles can be selected in an optimal solution. 4.2.2.2 Bounding box inequalities We first observe that intensity requirements of .,Ii i.ent bixels can be used to derive certain necessary conditions that any feasible decomposition of a matrix needs to satisfy. We z that a rectangle starts at bixel (i, j) if the upperleft corner of the rectangle is located at (i,j). Consider the bixel (5, 3) marked with dark gray in Figure 42. Since b43 = 2, the total intensity delivered to (5, 3) by all rectangles that start in rows i 1,..., 4 cannot exceed 2. However, b53 = 14 > 2, and hence at least one rectangle that starts in row 5 is required to cover bixel (5, 3). Similarly, b53 > b52 implies that at least one rectangle that starts in column 3 is required to cover the same bixel. These results can be strengthened by considering both (4, 3) and (5, 2) simultaneously. Since b53 > b43 + b52, we conclude that at least one rectangle that starts at bixel (5, 3) is required in any feasible decomposition of the fluence map. In general, a rectangle must start at (i,j) if bij > b(_i)j + bi(j_) is satisfied. Figure 43 illustrates a similar idea, where we 2 3 0 8 2 4 2 2 1 0 5 1 2 1 3 0 0 5 0 0 3 5 0 2 8 6 0 3 0 8 10 9 0 3 5 8 20 7 1 0 4 5 9 5 4 0 0 3 Figure 42. Example start index compare the intensity requirement of bixel (6, 4) with the bixel below it, and the one on its right. Using arguments similar to the ones regarding starting indices, we conclude that a rectangle must end (i.e., have a lowerright corner) at (6, 4) since b64 > b74 + b65. 2 3 0 8 2 4 2 2 1 0 5 1 2 1 3 0 0 5 0 0 3 5 0 2 8 6 0 3 0 8 14 10 9 0 3 5 8 20 1 0 4 5 9 5 4 0 0 3 Figure 43. Example end index Starting and ending index conditions can be generalized further as follows. Assume that there exist integers u E [0, i 1], d E [i + 1, m + 1], 1 E [0,j 1], and re [j + 1, n + 1] so that bij > bi + bj + bi, + bdj, where we define bio = boj = bm+l,j = bi,+l = 0 for i E {0,... ,m + 1},j E {0,..., n + 1}. In this case, we i that (1,u, r,d) is a bounding box for bixel (i,j). Figure 44 illustrates a bounding box for bixel (6, 3) (marked in dark gray), which corresponds to (1, u, r, d) = (2,4,5, 7). The four bixels that represent the borders of a bounding box are marked in light gray. We note that any rectangle that 2 3 0 8 2 4 2 2 1 0 5 1 2 1 3 0 0 5 0 0 3 5 0 2 8 6 0 3 0 8 14 10 9 0 3 5 8 7 1 0 4 5 9 5 4 0 0 3 Figure 44. Example bounding box contains bixel (6,3), and does not start inside the bounding box (at (5,3) or (6,3)) or end inside the bounding box (at (6,3) or (6,4)), has to contain at least one of the four bixels on the border. Therefore, the sum of intensities of those rectangles is bounded by the total required intensity of the bixels in light gray. Since the intensity of the dark gray bixel cannot be satisfied by those rectangles alone, it follows that at least one rectangle contained within the bounding box must be used to cover bixel (6, 3). Let BByj represent the interior of a bounding box for bixel (i, j), i.e., given (1, u, r, d) all bixels at the intersection of rows u + 1,..., d 1 and columns 1 + 1,... ,r 1. We denote the set of rectangles in R(i,j) that are contained within BBBy by R(BBBy). In this case, the following inequality is valid: yr > 1. (418) rER(BBij) Note that (0, 0, n+ l, m+ 1), which corresponds to the input matrix, is a bounding box for any bixel. Therefore there can be multiple bounding boxes associated with each bixel. Let BBij and BBj be two bounding boxes for bixel (i,j). We that BBBy dominates BBI if R(BBij) c R(BB'B). Since the inequality (418) that corresponds to a dominated bounding box is implied by the inequality that is associated with the corresponding dominating bounding box, we are only interested in generating nondominated bounding boxes. Figure 45 di pl'1 another nondominated bounding box for the bixel considered in Figure 44. 2 3 0 8 2 4 2 2 1 0 5 1 2 1 3 0 0 5 0 0 3 5 0 2 8 6 0 3 0 8 14 10 9 0 3 5 8 7 1 0 4 5 9 5 4 0 0 3 Figure 45. Another nondominated bounding box seeded at (6,3) To generate nondominated bounding boxes, we first make the following observation. A nondominated bounding box for bixel (i, j) is minimal in the sense that none of its edges can be shifted closer to (i,j) without violating the bounding box intensity property. We use this observation to design an algorithm that finds several nondominated bounding boxes associated with a given bixel. In our algorithm, we start at a bixel (i,j), and first move in a vertical or horizontal direction until we encounter a bixel (i',j') having bi/j, < bij. We mark (i',j') as an edge of the bounding box, reduce bij by bi//, and return to (i,j). We then move in the remaining directions onebyone, updating bij after each step, to find the remaining edges of the bounding box. We repeat the same procedure for all 4! permutations of the directions, and obtain a nondominated bounding box in each iteration. Finally, we eliminate duplicates to obtain a set of nondominated bounding boxes, and we generate a constraint of type (418) for each bounding box. 4.2.2.3 Aggregate intensity inequalities We derive a simple class of valid inequalities by observing that the total intensity that can be delivered to each bixel needs to be greater than or equal to its required intensity. Formally, > Mryr, > bi Vi 1,... m, ,j 1,... n. (419) rER(i,j) We note that inequalities (419) are implied by (42) and (43) in IPR. However, (419) can be used to tighten the master problem of the Benders decomposition approach discussed in Section 4.2.1. Furthermore, various tightening procedures can be applied to (419) for use in either the direct solution of IPR or in the Benders master problem. In our implementation, we apply a C'!Orv; 1 Gomory rounding procedure (see, e.g., Nemhauser and Wolsey (1988)) in which we divide both sides of the inequality by the smallest Mr coefficient on the lefthandside (unless bij is divisible by that number), and round up coefficients on both sides of the inequality. If bij is divisible by the smallest Mrcoefficient on the lefthandside of (419), then the rounding procedure yields an inequality implied by (419), and hence we do not generate it. 4.2.2.4 Special submatrices An alternative strategy to the one described in Section 4.2.2.3 divides both sides of (419) by bij 1, provided that bij > 2, and then rounds up all coefficients and the righthandside. Noting that all coefficients on the lefthandside are bounded from above by bij, this process yields: yr + 2 > 2 Vi 1,.,m, j 1,.,n. (420) rER(i,j): rER(i,j): Mr Equations (420) imply that bixel (i,j) can either be covered by a single rectangle having a maximum intensity of bij, or otherwise needs to be covered by at least two rectangles. The idea behind (420) can be extended to other special cases. For instance, consider the following lemma. Lemma 4. Consider ':1; 1 x 2 or 2 x 1 submatrix of B in which both elements equal a common nonzero value, q. D. fi;,. A1 as the set of r,. la,.jl, that cover il. /11 one of the two bixels, and have a maximum ':i,. I,.l'i of q. Let A' be the set of all r,. In,.l/l. that cover, i. i//:l one of the two elements, and have a maximum ':,/ :,il/i less than q. D, Fi, Ay and Af i",i., i.'..li for r.. ,l.i, that cover both elements. The following .,, 8.;,/.:/u is valid: 4 y + 2 y, + 2 y, + y, > 4. (421) rEA2 rEA2 rEA1 rEA1 Proof. Consider any feasible solution, and let vector v denote how many rectangles exist in the solution belonging to Ay, Af, A1, and A<, respectively. We claim (without proof, for brevity) that the following vectors Vl,..., v6 are minimal, in the sense that v > vi for at least one i = 1,..., 6, for every feasible v: v = (1,0, 0, 0), 2 = (0,1,0, 2),V3 (0, 2,0, 0), V4 (0, 0, 1, 2),V5 = (0,0,2, 0), V6 (0, 0, 0, 4). Note that each solution represented by vi satisfies (421), and thus all v corresponding to a feasible solution must also satisfy (421). E Similarly, consider submatrices of the form Lr quR or its transpose, where we assume 0 < qL < qR without loss of generality. We define AL and A< to be the sets of rectangles that cover qL, but not qR, with maximum intensity qL, and less than qL, respectively. Let AR and A< be defined for rectangles that cover qR but not qL, with a maximum intensity greater than or equal to (qR qL) and less than (qR qL), respectively. We define Ay and A< as before, with a maximum intensity of qL, and less than qL, respectively. A similar analysis as in proof of Lemma 4 reveals that the following inequality is valid: 2 y + 2 yr + 2 yr + yr + yr + r > 4. (422) rEAL rEAr rEA2 rEA< rEA< rEA2 The last special case that we consider is a nonzero submatrix of the form: q q q q We define Ai to be the sets of rectangles having maximum intensity equal to q, and covering exactly i elements of the 2 x 2 submatrix, for i 1, 2, and 4. Similarly, define A< to be the sets of rectangles having maximum intensity less than q, and covering exactly i elements of the submatrix. Given these definitions, we obtain: 8 y + 4 y y + 4 y + 2 + 2 yr + 2 > 8. (423) rEAT rEA A^ EAA reA2 reA rEAA 4.2.2.5 Submatrix inequalities It is possible to generate valid inequalities using arguments similar to the ones discussed in Section 4.2.2.4 for other submatrices as well. However, this process is very tedious, and there is a large number of possible submatrix combinations. In this section we describe a similar set of inequalities, which are weaker than those described in the previous section, but are easier to generate. We first observe that the formulation IPR can be solved quickly for small input matrices. Let S denote a submatrix of the input matrix, and R(S) represent the set of rectangles that cover at least one bixel in S. Let LB(S) be a lower bound on the number of rectangles required to decompose S. Since LB(S) constitutes a lower bound on the total number of rectangles required, the following inequality is valid for any submatrix S: Syr > LB(S)]. (424) reR(S) We can obtain LB(S) by formulating an auxiliary integer programming problem of type IPR for S, and setting a limit on the maximum solution time. 4.2.3 Partitioning Approach In this section, we propose a partitioning approach for our problem. We first propose an algorithm for detecting completely separable regions of the input matrix, which can be solved independently. Next, we explore methods for partitioning the large components, to obtain simultaneous upper and lower bounds, which we use to improve the solvability of our formulation. 4.2.3.1 Separable components Our observations on clinical data sets sI., 1 that input matrices can usually be decomposed into several small components, and one or two large components. The small components can usually be solved to optimality by formulation IPR enhanced with the valid inequalities discussed in Section 4.2.2. We observe on clinical data that several regions of the input matrix are completely surrounded by zerobixels. Since no rectangle can cover a zerobixel, each of these regions can be solved independently. A connected subset of the input matrix obeys the property that a rectilinear path exists between any two nonzerobixels of the subset, such that each bixel in the path is also a nonzerobixel that belongs to the subset. We call a connected set of nonzerobixels a component of the input matrix if it is .,i1i ient to zerobixels across all of its boundaries (i.e., if the subset is not contained within a larger connected subset). To identify the components of the input matrix, we generate a graph G in which each nonzerobixel has a corresponding node. We add an arc between a pair of nodes if and only if the corresponding bixels are .,11] went in the input matrix. We then identify connected components on G by running a standard depthfirstsearch algorithm. Each connected component on G corresponds to a component of the input matrix, which can be solved independently of other components. Figure 46 depicts the components of the fluence map given in Figure 41. Figure 46. Two components of a fluence map 4.2.3.2 Independent regions After finding separable components of the input matrix, we attempt to further partition each component into smaller regions. We iv that distinct regions of a component are independent if no rectangle intersects two bixels belonging to different regions without also intersecting a zerobixel. In Figure 47, the regions with light and dark gray background are independent. If we solve IPR separately over all independent regions, the sum of rectangles required to decompose each independent region yields a lower bound on the objective function for the corresponding component. 23 30 S 14 10 9 061 5 8 0 7 1 0 5 9 5 4 0 0 Figure 47. Regions of a connected component In general, there are multiple viva of partitioning a component into independent regions, with each yielding possibly different lower bounds. The problem of finding a partition that yields the best lower bound can be thought of as a "dual" of finding the minimum number of rectangles to decompose a component. To solve this dual problem, we need to balance two conflicting criteria: The number of bixels assigned to each independent region needs to be small enough so that each region can be solved quickly. 2 3 0 8 2 4 2 2 1 0 5 1 2 1 3 0 0 5 0 0 3 5 0 2 8 6 0 3 0 8 14 10 9 0 3 5 8 207 1 0 4 5 9 5 4 0 0 3 595411003 * The number of bixels not assigned to any independent regions needs to be as small as possible to obtain a good lower bound. We use a heuristic procedure to partition a component into independent regions, which employs an auxiliary objective of maximizing the number of component bixels covered by an independent region. Each bixel (i,j) is called "committed" if it either belongs to an independent region, or if (i, j) is contained within some rectangle in R that also covers bixels in an independent region (and hence, (i,j) cannot belong to another independent region). All other bixels are called "uncommitted." We select our independent regions one at a time, until no more uncommitted bixels remain. The procedure's details are described as follows. Initialization. Labels all nonzerobixels as "uncommitted." Step 1. Each candidate independent region (or just "candidate") is seeded from a rectangle r E R such that rectangle r contains only uncommitted bixels, and such that the number of bixels in the rectangle is no more than some limit L. For each such rectangle r, define to be the (initial) candidate region. Step 2. For each candidate r, if f, covers exactly L bixels, then go to Step 4. Else, continue to Step 3. Step 3. For each candidate r, determine if there exists an uncommitted bixel (i,j) .,i i,:ent to r (i.e., a bixel (i,j) } r such that either (i 1,j), (i + 1,j), (i,j 1), or (i,j + 1) belongs to e,), such that for every r' E R(i,j), all bixels in r' either belong to r,, or would already become committed due to the selection of 4r as an independent region. That is, adding (i,j) to 4, would not increase the number of bixels committed by selecting fr as a new independent region. If such a bixel exists, then add (i,j) to r, and return to Step 2. Else, continue to Step 4. Step 4. For each candidate r,, compute Kr = the number of bixels in ,, and K, = the number of uncommitted bixels (i,j) such that some rectangle in R includes both (i,j) and a bixel in 4,. If any candidates exist such that ,K = 0, then choose e, to be any such candidate. Else, choose to be any candidate that maximizes K /0. Go to Step 5. Step 5. Create an independent region corresponding to f*. For each bixel (i,j) that can be covered by a rectangle in R intersecting at least one bixel in e", change the status of (i,j) to "committed." (This includes all bixels in e4 itself.) If all bixels are committed, terminate the procedure; else, return to Step 1. In our algorithm for solving a component, we execute the foregoing heuristic to find a set of independent regions. We formulate IPR for each region, with a limit on the maximum solution time. We then use the lower bound obtained for each region to generate an inequality of type (424). (It is often prudent to skip this step if only one region is computed for a component.) 4.2.3.3 Dependent regions In this section, we attempt to improve the lower bound obtained using independent regions by focusing on those bixels not included in the union of independent regions. We define a dependent region to be a connected set of bixels in a component that does not overlap with any of the independent regions in that component. In our example, the region with black background in Figure 47 is a dependent region. Let D represent the set of bixels in a dependent region, and let R(D) represent the set of rectangles that cover only a subset of the bixels in D. To improve our lower bound, we wish to compute the minimum number of rectangles required to cover D; however, we wish to avoid doublecounting those rectangles used to cover bixels in independent regions. Accordingly, we seek the minimum number of rectangles in R(D), perhaps in concert with rectangles outside R(D), required to cover the bixels in D. Using the x and yvariables as before, we formulate the following variation of IPR to find the minimum number of rectangles in R(D) required to partition DPR: Minimize > y, (425) rER(D) subject to: Xr bij V(i,j) E D (426) rER(ij) S< V ,..., n, j) D (4 27) rER(i,j) x, <3 1, Vr E R(D) (428) x, > 0 Vr e R, yr binary Vr e R(D) (429) Objective (425) minimizes the number of rectangles in R(D) used in the solution. Constraints (426) ensure that the bixels in D get partitioned exactly, where (427) limit the intensity delivered to the remaining bixels. Constraints (428) relate the x and y variables as done in IPR, and finally (429) define variable types. As before, we set a time limit for the solution of DPR, and obtain a lower bound on the objective function value, which we denote by LB(D). Given this value, the following inequality is valid: Syr > [LB(D)]. (430) rER(D) In our example, the optimal value of DPR for the black (dependent) region is 1 since the intensity requirement of bixel (1, 4) cannot be satisfied completely by rectangles that cover bixels in the gray (independent) regions (in fact, this result can also be seen due to the bounding box constraint implying that one rectangle representing the singleton bixel (1,4) must appear in any feasible solution). We note that the rectangles in R(D), by definition, do not intersect any other (dependent or independent) regions. Therefore, the lower bounds obtained for all regions can be summed to obtain a lower bound on the minimum number of rectangles required to decompose a component. 4.2.3.4 Upper bound calculation In this section, we discuss how a related approach leads to a heuristic algorithm to obtain a feasible decomposition of a component. We first note that a feasible decomposition of a component can be obtained by combining feasible solutions obtained for individual regions within a component. Feasible solutions for independent regions are readily available from the integer programming problems solved for obtaining lower bounds on those regions, as discussed in Section 4.2.3.2. Feasible solutions for dependent regions can be extracted from solutions of the formulation given by DPR. However, since DPR minimizes the number of rectangles that are contained within a dependent region, and not necessarily the total number of rectangles required to decompose a dependent region, the solutions obtained from DPR potentially use an unnecessarily large number of rectangles not contained in R(D). A better way of obtaining feasible solutions for dependent regions is to formulate the problem IPR for each dependent region. Since IPR explicitly minimizes the total number of rectangles required, we expect this approach to result in feasible solutions of higher quality. However, this approach does not consider the fact that some of the rectangles that are already used for decomposing independent regions can be extended into dependent regions without increasing the total number of rectangles. To permit the use of rectangles that intersect independent and dependent regions, we require a revised integer programming formulation. In our approach, we solve the integer programming formulations for decomposing the independent regions first, and store the best feasible solutions found within the allowed time limit. Let x, represent the intensity assigned to rectangle r for decomposing independent regions. Next, we generate a feasible solution for each dependent region, one at a time, as follows. We first find the set of rectangles that can be extended into the current dependent region, and determine how those rectangles can be extended. Let E(D, r) represent the set of rectangles in R that extend rectangle r into dependent region D. We also define the parameter I(r)e equal to one if bixel (i,j) E D is covered by extension e of rectangle r, and zero otherwise. Let ze be a binary variable that equals 1 if and only if extension e of rectangle r is used in the solution. We define the x and y variables as before, and formulate the following problem: EPR: Minimize > Y, (431) rER(D) subject to: x, b, (xI(r) ,) z V(i,j) e D (432) rER(i,j) rER eEE(D,r) SZr < 1 Vr R (433) eEE(D,r) x, < 3 Vr E RZ(D) (434) x, > 0, yr binary Vr e R(D) (435) zr, binary Vr e R, e E (D, r). (436) We generate a feasible solution by combining three types of rectangles: (i) rectangles used to decompose independent regions that are not extended by EPR; (ii) rectangles obtained by extending rectangles from independent regions into dependent regions by EPR; and (iii) rectangles in R(D) used by EPR. Note that the optimal value of EPR for the dependent region given in Figure 47 is 1. This can be seen by observing that the rectangle(s) that cover bixel (3, 4) can be extended up to fully satisfy the intensity requirement of bixel (2, 4) without any penalty on the objective function of EPR formulated for the dependent region. Therefore, a single rectangle contained in the dependent region solves EPR optimally. Since the optimal value of DPR for the dependent region is also 1, our partition solves the problem of finding the minimum number of rectangles to optimality. 4.3 Extensions In this section, we briefly discuss how our model can be adjusted to tackle the problems of minimizing total treatment time, and lexicographically minimizing beamontime and number of apertures. 4.3.1 Minimize Total Treatment Time The total time spent delivering a given fluence map is composed of (i) time required to move the jaws to form the next rectangular aperture (setup time), and (ii) time during which radiation is delivered (beamontime). Even though the setup time required for switching from one rectangular aperture to the next one depends on the jaw settings corresponding to these apertures, and hence is sequencedependent, we make the common assumption that total setup time is proportional to the total number of apertures used. With this assumption, our model can easily be adjusted to explicitly minimize the total treatment time by changing the objective function of IPR to Minimize w Y r + .r, (437) rER rER where w is a parameter that represents the average setup time per aperture relative to the time required to deliver a unit of intensity. The Benders decomposition procedure discussed in Section 4.2.1 also needs to be adjusted accordingly. We first add a continuous variable t to MP, which "predicts" the minimum beamontime that can be obtained by the set of rectangles chosen by MP. The updated master problem can be written as follows. MPTT: Minimize w y 2 + t (438) rER subject to: y corresponds to a feasible decomposition (439) t > minimum beamontime corresponding to y (440) yr binary Vr E R. (441) Given a vector 9, we can find the minimum beamontime for the corresponding decomposition, if one exists, by solving: SPTT(f): Minimize x, (442) rER subject to: x= bij Vi 1,...,m, j 1,...,n (443) rER(i,j) xr < i, Vr R (444) xr > 0 Vr e R. (445) Note that SPTT is obtained by simply changing the objective function of SP. If SPTT(y) is infeasible, then we add a Benders feasibility cut of type (416) as before, and resolve MPTT. Otherwise, let the value of t in MPTT be t, and the optimal objective function value of SPTT be t*. If t = t*, then (y, t) is an optimal solution of MPTT that minimizes the total treatment time. However, if t > t*, then we add the following Benders optimality cut mn n t ^bipiij + ),y, (446) i=1 j=1 rER where &yi and /3 are optimal dual multipliers associated with constraints (443) and (444), respectively. 4.3.2 Optimization with BeamonTime Restrictions Another related problem that we consider is finding the minimum number of rectangles that yields the minimum beamontime. Note that the minimum beamontime required to decompose a fluence map can be found (in polynomial time) by solving SPTT, which is a linear program, for the vector yr 1, Vr E R. Let T* denote the optimal objective function value of SPTT(1), where 1 is the vector of all 1's. Given this value, it is sufficient to add x, < T* (447) rER to minimize the number of rectangles while limiting beamontime to T*. The modifications required for the Benders decomposition algorithm are also straightforward. To enforce the minimum beamontime restriction, we add (447) to SP, which checks whether a given set of rectangles can decompose the fluence map. The updated feasibility cut is given by > 1 &(M + ,M)r)y, + TtO < 0, (448) i= j=1 rER where 0 is the dual variable associated with (447) in SP. Finally, we need to check whether the solution generated by our heuristic discussed in Section 4.2.3.4 satisfies constraint (447); if so, then it can be used as an initial upper bound. 4.4 Computational Results We have implemented our algorithms using CPLEX 11 running on a Windows XP PC with a 3.4 GHz CPU and 2 GB RAM. Our base set of test problem instances consists of 25 clinical problem instances ("clbl", ..., "c5b5"). These instances were obtained from treatment plans for five patients treated using five beam angles each. We report problem characteristics in terms of the number of rows m, the number of columns n, and the maximum intensity value L. We imposed a time limit of 1800 seconds (30 minutes) in all of our tests. For problem instances that were not solved to optimality within the imposed time limit, we report the best upper and lower bounds obtained, where we round lower bounds up for the cases in which the objective function is guaranteed to have an integral value. Our preliminary computational tests showed that the naive implementation of our Benders decomposition approach, in which we add a cut and resolve the master problem in each iteration, was not computationally competitive with solving the explicit integer programming formulation. This is due to the fact that repetitively solving the master problem, which is an integer programming problem, is computationally very expensive. We instead used callback functions of CPLEX to generate a single branchandbound tree in which we solve SP(9) (or (SPTT(9)) corresponding to each integer solution found in the branchandbound tree, and add cuts to tighten the master problem as necessary. While this approach produced better results than the naive implementation, it still yielded inferior bounds than those obtained from the explicit formulation. Therefore, we omit further Bendersbased computational results. Our first experiment quantifies the effects of the valid inequalities discussed in Section 4.2.2, and the partitioning approach discussed in Section 4.2.3 on solution quality and execution time. In Table 41, the set of columns labeled "Default CPLEX" shows the results we obtained by solving the formulation IPR on each problem instance using default CPLEX options. The "+ Valid inequ I!i i. columns represent the IPR formulation enhanced with the .,i1] i:ent rectangle inequalities (417), bounding box inequalities (418), strengthened .i,regate intensity inequalities (419) and (420), and 1 x 2 submatrix inequalities (421) and (422). (Additional computational results showed that the 2 x 2 submatrix inequalities (423) and the arbitrary submatrix inequalities (424) did not improve the solvability of the model.) The set of columns labeled "+ Partitions" shows the results we obtained by partitioning the problem into separable components (Section 4.2.3.1), further partitioning each component into independent and dependent regions (Sections 4.2.3.2 and 4.2.3.3), and using our upper bounding heuristic (Section 4.2.3.4) in addition to the valid inequalities used for the tests in the previous set of columns. We refer to the latter settings as our base il/,.>rithm in the remaining computational tests. Each set of columns in Table 41 di pl1iv the time spent for each problem instance ("CPU"), and upper bound ("UB"), lower bound ("LB"), and optimality gap ("GAP") obtained. We also report the average and maximum gaps over all problem instances. We observe that none of the problem instances were solved to optimality using the default CPLEX options, whereas clb2 and c5b2 were solved to optimality after adding the valid inequalities of Section 4.2.2. An additional instance (c5b5) was solved using the partitioning strategy described in Section 4.2.3. We note that even though our approach Table 41. Effect of valid inequalities and the partitioning strategy Default CPLEX + Valid inequalities + Partitions Name m n L CPU UB LB Gap CPU UB LB Gap CPU UB LB Gap clbl 15 14 20 1800 66 60 0.09 1800 63 62 0.02 1800 64 62 0.03 clb2 11 15 20 1800 48 47 0.02 138.2 48 48 0 1009.9 48 48 0 clb3 15 15 20 1800 57 54 0.05 1800 57 54 0.05 1800 58 54 0.07 clb4 15 15 20 1800 61 52 0.15 1800 61 53 0.13 1800 59 55 0.07 clb5 11 15 20 1800 47 45 0.04 1800 46 45 0.02 1800 47 45 0.04 c2bl 18 20 20 1800 114 79 0.31 1800 119 85 0.29 1800 103 87 0.16 c2b2 17 19 20 1800 95 69 0.27 1800 96 81 0.16 1800 94 82 0.13 c2b3 18 18 20 1800 98 73 0.26 1800 103 77 0.25 1800 94 77 0.18 c2b4 18 18 20 1800 114 80 0.3 1800 115 84 0.27 1800 105 88 0.16 c2b5 17 18 20 1800 94 64 0.32 1800 98 72 0.27 1800 91 72 0.21 c3bl 22 17 20 1800 121 69 0.43 1800 134 79 0.41 1800 119 79 0.34 c3b2 15 19 20 1800 73 46 0.37 1800 71 52 0.27 1800 70 52 0.26 c3b3 20 17 20 1800 119 69 0.42 1800 119 75 0.37 1800 107 77 0.28 c3b4 19 17 20 1800 103 69 0.33 1800 106 73 0.31 1800 99 78 0.21 c3b5 15 19 20 1800 73 55 0.25 1800 71 58 0.18 1800 73 58 0.21 c4bl 19 22 20 1800 106 79 0.25 1800 107 89 0.17 1800 109 89 0.18 c4b2 13 24 20 1800 88 54 0.39 1800 99 58 0.41 1800 91 58 0.36 c4b3 18 23 20 1800 95 71 0.25 1800 99 75 0.24 1800 93 77 0.17 c4b4 17 23 20 1800 103 78 0.24 1800 102 81 0.21 1800 98 83 0.15 c4b5 18 24 20 1800 93 62 0.33 1800 93 66 0.29 1800 87 67 0.23 c5bl 15 16 20 1800 66 64 0.03 1800 66 65 0.02 1800 66 65 0.02 c5b2 13 17 20 1800 58 57 0.02 102.1 58 58 0 213.6 58 58 0 c5b3 14 16 20 1800 63 54 0.14 1800 68 56 0.18 1800 65 57 0.12 c5b4 14 16 20 1800 63 57 0.1 1800 64 59 0.08 1800 62 59 0.05 c5b5 12 17 20 1800 53 47 0.11 1800 51 48 0.06 36.2 49 49 0 was not able to provide provably optimal solutions for most instances, it significantly improved both lower and upper bounds for several instances. Our next experiment tests our base algorithm under the extensions discussed in Section 4.3. The set of columns labeled as "Total Time" in Table 42 presents the extension in which the objective function is defined as a linear combination of the beamontime and the number of rectangles. The actual value of w depends on the particular treatment delivery equipment used in the clinic, where values of w in the range 110 are typical (see, e.g., Dai and Hu (1999), and Takm net al. (2009b)). In our experiments, we therefore used w = 7 as a representative value. The next set of columns ("Lexicographic") is dedicated to the extension in which we first minimize beamontime, T*, and then find the minimum number of rectangles that yields the minimum beamontime. The column "BOT" represents the value of T*, and "Total Time" represents the total treatment time associated with the solution found, where we again use w = 7 as the average setup time per rectangle. We observe that our algorithm could solve more problem instances to optimality for both extensions compared to the Table 42. Computational results on model extensions Total Time Lexicographic Name m n L CPU UB LB Gap CPU UB LB Gap BOT Total Time clbl 15 14 20 255.9 621 621 0 36.5 66 66 0 176 638 clb2 11 15 20 330.3 459 459 0 132.6 50 50 0 121 471 clb3 15 15 20 1800 548 542.72 0.01 130.4 62 62 0 147 581 clb4 15 15 20 1800 557 542.49 0.03 186.9 62 62 0 136 570 clb5 11 15 20 1800 451 443.63 0.02 30.9 53 53 0 115 486 c2bl 18 20 20 1800 962 814.24 0.15 1800 107 104 0.03 194 943 c2b2 17 19 20 1800 883 797.74 0.1 1800 96 92 0.04 207 879 c2b3 18 18 20 1800 918 797.6 0.13 1800 96 88 0.08 237 909 c2b4 18 18 20 1800 1028 889.36 0.13 1800 111 106 0.05 258 1035 c2b5 17 18 20 1800 890 721.13 0.19 1800 92 83 0.1 207 851 c3bl 22 17 20 1800 1161 858.9 0.26 1800 116 103 0.11 266 1078 c3b2 15 19 20 1800 668 533.24 0.2 1800 70 64 0.09 151 641 c3b3 20 17 20 1800 1066 847.09 0.21 1800 111 95 0.14 278 1055 c3b4 19 17 20 1800 1023 857.91 0.16 1800 103 95 0.08 287 1008 c3b5 15 19 20 1800 722 610.01 0.16 204.4 76 76 0 182 714 c4bl 19 22 20 1800 1044 918.57 0.12 1800 108 105 0.03 275 1031 c4b2 13 24 20 1800 895 656.15 0.27 1800 95 76 0.2 232 897 c4b3 18 23 20 1800 858 743.62 0.13 1800 92 89 0.03 189 833 c4b4 17 23 20 1800 943 834.32 0.12 1800 101 96 0.05 235 942 c4b5 18 24 20 1800 913 740.19 0.19 1800 86 77 0.1 260 862 c5bl 15 16 20 271.4 626 626 0 5.5 71 71 0 158 655 c5b2 13 17 20 33.4 597 597 0 19.9 63 63 0 156 597 c5b3 14 16 20 1800 623 597.96 0.04 869.2 68 68 0 180 656 c5b4 14 16 20 1800 584 571.15 0.02 192.4 66 66 0 145 607 c5b5 12 17 20 90.4 503 503 0 37.2 57 57 0 147 546 problem of finding the minimum number of rectangles. To understand why this is the case, we first note that the difficulty of the matrix decomposition problem varies greatly based on the objective function used. On one hand, minimizing the number of rectangles is strongly NPhard, even for fluence maps having a single row (see Baatar et al. (2005)). On the other hand, minimizing the beamontime is a polynomially solvable problem (see Section 4.3). Therefore, we expect that the problem should become easier as the weight of the beamontime term in the objective function increases. The reason the lexicographic minimization problem is easier to solve than the other two variations is because the additional beamontime constraint considerably shrinks the feasible solution space. Another way of looking at the problem of balancing the number of apertures and the beamontime is to view the problem as a multicriteria optimization problem. In this setting, we are interested in constructing the Pareto efficient frontier of solutions with the property that neither of the two criteria can be improved without deteriorating the other. Note that the lexicographic approach that we considered above determines a particular Pareto optimal solution to the multicriteria problem. To generate other nondominated solutions for the multicriteria version of the problem, we sequentially impose different upper bounds on the number of apertures allowed, , 7, and find the corresponding minimum beamontime for these values of 7. As an example, we considered the problem instance c5b5. For this instance, we note that the minimum number of apertures is 49 (see Table 41) with a corresponding beamontime of 160, while the minimum beamontime for this problem instance is 147 (see Table 42) which requires 57 apertures. Figure 48 then depicts (i) the nondominated solutions; (ii) the Pareto efficient frontier for values of 7 E [49, 57], and (iii) the (boundary of the) convex hull of the Pareto set. The solutions on the latter are the optimal solutions to the problem of minimizing total treatment time that can be obtained with different values of w. 162 160 ,.\ [  158 156 \ 0154 E 152 150 o 15 2 148 ~  146 48 49 50 51 52 53 54 55 56 57 58 Number of apertures Figure 48. Efficient frontier for number of apertures and beamontime Our final experiment analyzes the effect of the maximum intensity value L. Usually fluence maps are obtained by solving a nonlinear optimization problem for each beam angle to determine an intensity profile for each beam angle, which is represented by a Table 43. Effect of maximum intensity value on solvability L 5 L 10 L 15 Name m n CPU UB LB Gap CPU UB LB Gap CPU UB LB Gap clbl 15 14 4.4 305 305 0 22.9 441 441 0 1800 539 536.99 0 clb2 11 15 1.4 238 238 0 4.4 320 320 0 498.5 394 394 0 clb3 15 15 9.6 287 287 0 228.6 377 377 0 1800 495 487.67 0.01 clb4 15 15 5.8 269 269 0 1800 393 377.42 0.04 1800 513 493.13 0.04 clb5 11 15 2.4 216 216 0 28.9 326 326 0 316.7 411 411 0 c2bl 18 20 31.3 440 440 0 1800 648 635.1 0.02 1800 826 732.68 0.11 c2b2 17 19 51.5 448 448 0 1800 625 599.84 0.04 1800 759 690.67 0.09 c2b3 18 18 144.4 428 428 0 1800 645 615.64 0.05 1800 800 704.61 0.12 c2b4 18 18 1593.1 487 487 0 1800 755 678.49 0.1 1800 919 796.13 0.13 c2b5 17 18 197.9 429 429 0 1800 606 538.62 0.11 1800 728 621.9 0.15 c3bl 22 17 1359.4 480 480 0 1800 747 662.47 0.11 1800 1080 779.16 0.28 c3b2 15 19 1800 280 274.97 0.02 1800 414 376.01 0.09 1800 532 461.6 0.13 c3b3 20 17 1800 461 446.77 0.03 1800 731 641.2 0.12 1800 908 755.25 0.17 c3b4 19 17 1800 463 456.42 0.01 1800 713 634.26 0.11 1800 900 762.02 0.15 c3b5 15 19 1800 332 325.87 0.02 1800 481 466.6 0.03 1800 582 532.2 0.09 c4bl 19 22 39.9 529 529 0 1800 758 719.53 0.05 1800 899 827.73 0.08 c4b2 13 24 1800 422 408.14 0.03 1800 595 503.92 0.15 1800 764 582.69 0.24 c4b3 18 23 126.3 409 409 0 1800 579 564.5 0.03 1800 695 666.23 0.04 c4b4 17 23 321.4 444 444 0 1800 662 649.1 0.02 1800 815 742.81 0.09 c4b5 18 24 1194.7 414 414 0 1800 636 573.02 0.1 1800 794 662.74 0.17 c5bl 15 16 5.9 342 342 0 5.6 442 442 0 28.9 584 584 0 c5b2 13 17 4.3 289 289 0 5.5 439 439 0 49.3 516 516 0 c5b3 14 16 1574.4 294 294 0 1800 453 441.44 0.03 1800 566 538.22 0.05 c5b4 14 16 3.6 239 239 0 1800 473 468.49 0.01 1800 534 524.31 0.02 c5b5 12 17 2.1 252 252 0 9.3 354 354 0 63.9 441 441 0 matrix of real numbers. Later, these matrices are rounded to integer matrices to limit the delivery time. To analyze the tradeoff between roundoff errors and treatment time, we started from clinical treatment plans, and applied rounding with different levels of granularity. Specifically, we generated problem instances from the same fluence maps with L {5, 10, 15, 20}, and used our algorithm to find the minimum total treatment time required as a measure of delivery efficiency. Table 43 shows the results of our experiments. We observe that our algorithm produces smaller optimality gaps as L decreases, which is not surprising since IPR becomes tighter as the Mrcoefficients (which are bounded by L) decrease. Furthermore, delivery efficiency is also higher for small values of L. The average treatment time (calculated over the lower bounds) for all problem instances increases from 366.05 for L = 5 to 513.79 for L = 10, 609.79 for L = 15, and 684.96 for L = 20, which is calculated using the set of columns labeled "Total Time" in Table 42. Our results show that the choice of granularity chosen for rounding has a significant effect on the treatment time. For each individual patient, the risks associated with the deterioration in treatment plan quality due to the rounding of intensities needs to be weighed against the disadvantages of a longer treatment time by the physician or clinician. CHAPTER 5 GRAPH SEARCH PROBLEM 5.1 Introduction and Literature Review In this chapter we consider several variants of a search problem on graphs, which can be seen as a game between an intruder and a group of searchers. Consider an undirected graph G = (N, E). The intruder and searchers occupy some nodes of the graph. At each time period, a pl liv r (intruder or searcher) located at node i e N can move along an edge to an .,I.i ient node, or stay at their current position. A searcher located at node i E N can detect an intruder if it is located at some node in S(i) C N. Searchers can be deploy, .1 at any node, but have no information about the location of the intruder. Therefore, they have to systematically search the graph to be able to detect the intruder, even if the intruder has perfect information about the searchers' plan, and utilizes this knowledge to evade detection for as long as possible. The problem is to find the minimum number of searchers needed and a routing plan for each searcher that guarantees detection of the intruder within a given time limit. The graph search problem was initially defined by Parsons (1978) in the context of seeking a person lost in a cave. The cave is represented as a graph, where tunnels of the cave correspond to edges of the graph. Searchers have to sweep edges of the graph to locate the missing person, who is assumed to be wandering unpredictably or is purposefully trying to evade searchers. The search number s(G) of a graph G is defined to be the minimum number of searchers needed so that the missing person can be found even if he could move infinitely fast along any path not occupied by searchers (Parsons, 1978). Computing s(G) is NPhard for general graphs (Bienstock and Seymour, 1991; LaPaugh, 1993; Megiddo et al., 1988), but it can be computed in linear time for trees (Alspach, 2004; Megiddo et al., 1988; Peng et al., 2000). The search number of a graph has been shown to be related to other important parameters such as treewidth, pathwidth, and vertex separation (Dendris et al., 1997; Ellis et al., 1994; Seymour and Thomas, 1993). Several variants of the graph search problem have been investigated in the literature. In decontamination problems, edges or nodes of a graph are infected by a contaminant such as a computer virus or a chemical agent, which spreads across the graph (Flocchini et al., 2008; LaPaugh, 1993; Penuel and Smith, 2009). In rendezvous problems different p1 i, rs, who are not aware of the location of others, try to meet at a common node as quickly as possible (Alpern, 1995; Alpern and Gal, 2003; Kikuta and Ruckle, 2007). Hideandseek problems consider an intruder that "hides" in a stationary location, while the searchers try to locate the intruder in minimum time (Alpern, 2008; Jotshi and Batta, 2008). Such problems also arise in searchandrescue settings (Benkoski et al., 1991). Pursuit evasion (or "copsandrobber") games model an intruder that tries to avoid being captured by searchers (Aigner and Fromme, 1984; Alspach et al., 2008; Hahn, 2007; Isler and Karnad, 2008). In some applications nodes of a graph need to be patrolled for protection or supervision (C'!, i. I!. yre et al., 2004; Sak et al., 2008). In particular, one interesting application coordinates automated software searchers so that they patrol the Internet to find web sites that exploit browser vulnerabilities (Wang et al., 2005). We refer the reader to Alpern and Gal (2003); Alspach (2004); Fomin and Thilikos (2008) for detailed surveys of the literature on search problems and applications in various practical settings. Most of the previous research on graph search problems has focused on theoretical aspects of the problems (e.g. C'!, i,!yre et al. (2004); Dendris et al. (1997); Ellis et al. (1994); Goldstein and Reingold (1995); Seymour and Thomas (1993)) or designing algorithms for solving the problems on special graph structures (e.g. Alpern (2008); Flocchini et al. (2008); Kikuta and Ruckle (2007); Peng et al. (2000)). Our contribution is an exact optimization algorithm for solving several variants of the search problem on general graphs (see also Penuel and Smith (2009) for a decontamination problem in which the intruder location has been determined). In particular, we consider three specific graph search problems: (i) a hideandseek problem, (ii) a pursuit evasion problem, and (iii) a patrol problem. We model these problems as largescale integer programs, and propose a branchcutprice algorithm that is capable of solving all three problems with some modifications. A variant of the branchandbound algorithm, which adds cutting planes to linear programming relaxations to tighten dual bounds is called branchandcut, and is emploiv '1 in most commercial solvers for solving integer programs ( r'chand et al., 2002; Nemhauser and Wolsey, 1988; Wolsey, 1998). An effective method for solving integer programs having a large number of variables is branchandprice, which is based on dynamic column generation (Barnhart et al., 1998). Branchcutprice is essentially an algorithm that combines dynamic column generation with dynamic row generation (Jiinger and Thienel, 2000). The remainder of this chapter is organized as follows. In Section 5.2 we describe a hideandseek problem and propose a column generation algorithm for solving its linear programming relaxation. Similarly, Sections 5.3 and 5.4 analyze the pursuit evasion and patrol problems, respectively. We describe some branching rules that can be used in all three algorithms to obtain an optimal solution to these problems in Section 5.5. Finally, we give computational results in Section 5.6. 5.2 HideandSeek Problem We start our discussion with a hideandseek problem, in which a group of searchers seek to locate a stationary intruder within T time steps. While this problem is relatively easy to model and analyze, it serves as the basis for our more complex search algorithms. 5.2.1 Mathematical Model We denote the set of nodes .,.i i,'ent to node i e N by A(i). Since we allow the intruder and searchers to stay at their current location, we assume that i E A(i), Vi E N. Recall that a walk on a graph G = (N, E) is a sequence i1,..., ir of nodes such that for all 1 < k < r 1, ik+l E A(ik) (Almi I et al., 1993). We define the length of a walk as the number of traversed edges on the walk. Let P(T) denote the set of all possible walks of length at most T that can be taken by a searcher. We assume that each node i e N can be observed from some node, i.e., there exists a j E N such that j E S(i). Let dpi be a parameter whose value is 1 if a searcher following walk p can detect an intruder located at node i, and 0 otherwise. Let Ap be a binary variable that equals 1 if a searcher is assigned to follow walk p, and 0 otherwise. Given these definitions, our hideandseek problem can be formulated as the following set covering problem. HS: minimize Ap (5 1) pEP(T) subject to dpi, > 1 Vi E N (52) pEP(T) A, {0, } Vp P(T) (53) The objective function (51) minimizes the number of selected searchers, and constraints (52) guarantee that each node is covered by at least one searcher within the allowed time frame. We note that a special case of this problem for which a searcher located at node i e N can observe node i and its neighbors, and we need to guarantee immediate detection of the intruder (i.e., when S(i) = A(i), Vi E N and T = 0), is equivalent to the minimum dominating set problem, which is known to be NPhard (Garey and Johnson, 1979). Therefore, the hideandseek problem that we consider is NPhard. 5.2.2 Solution Approach In principle, all walks of length at most T can be enumerated, and HS can be solved directly. This approach might be practical for small values of T, but in general there is an exponential number of such walks, which corresponds to an exponential number of variables. We instead propose a column generation approach to solve HS. Given a subset of walks P'(T) C P(T), we can construct a limited hideandseek (LHS) problem identical to HS, with P(T) replaced by P'(T). The linear programming relaxation of LHS is: LHSLP: minimize A (54) pEP'(T) subject to d > 1 Vi c N (55) pEP'(T) A > 0 Vp P'(T), (56) where upper bounds on the Avariables are not necessary at optimality. Given an optimal dual vector y, the reduced cost of Ap, which we denote by cp, can be calculated as 1  CiN idpi. Since y is an optimal dual vector, c, > 0 for all p P'(T). We can conclude that the current solution of LHSLP is also optimal for the linear programming relaxation of HS if cp > 0 for all p P(T). On the other hand, if cp < 0 for some p E P(T) \ P'(T), then adding p to P'(T) can potentially decrease the value of the objective function (54). We discuss our pricing problem, which seeks such a p, in the next section. 5.2.2.1 Searcher's problem Let <7 be the dual variable associated with the constraint of type (55) corresponding to node i e N. Also, let yi be a decision variable that equals 1 if node i is "seen" by a searcher following a walk that we generate, and 0 otherwise, Vi e N. Given an optimal dual vector y, we solve the following pricing problem to seek a Avariable having a negative reduced cost: max ECEN ^ /, subject to the restriction that (/i,..., yINI) corresponds to a set of nodes observed by a walk of length no more than T. The pricing problem can be formulated as a mixedinteger programming problem on a timeexpanded network consisting of T + 1 stages. In particular, we create a node it for each i E N, t = 0,..., T. We create an arc from node i, Vi E N, t = 0, ., T 1 to nodes Nj(t+l) for all j E A(i). For this problem it is easy to see that an optimal solution exists in which all searchers move at each time period. Therefore, we omit arcs between nodes it and i(t+l) for each i E N. Figure 51 ditpli a simple example graph, and the corresponding timeexpanded network for T = 2. To formulate the pricing problem as a mixedinteger program, we introduce binary variables xt = 1 if the searcher is at node i at time t. Then, an integer programming (a) K > (b) K u Figure 51. (a) An example graph (b) Timeexpanded network for T = 2 formulation of the problem can be given as: maximize ^ I iEN subject to x> t 1 Vt ,...,T iEN (57) (58) (59) (510) x < t1 Vi N,t =,...,T jeA(i) T y, < xt Vi EN t=0 j:ies(j) 0 xJ e {0, 1} Vi e N,t= 0,...,T. Constraints (58) represent the fact that the searcher can visit only one node at a time. Constraints (59) ensure that node i can be visited at time t only if one of its neighbors has been visited at time t 1. Constraints (510) force the value of yi to zero unless node i can be observed by the searcher at some time period. Note that the yvariables will take on binary values in an optimal solution, and therefore we relax them as continuous variables. If the optimal objective function value of (57)(512) is greater than 1, then we have found a variable that has a negative reduced cost, and we therefore add the generated column to LHSLP. 5.2.2.2 Branchandprice algorithm The hideandseek problem can be solved using the following branchandprice algorithm. * Step 0: C'!.... a feasible set of initial search walks, in which every node is seen by at least one searcher. Step 1: Solve LHSLP, and generate columns by solving the searcher's problem until LHSLP has been solved to optimality. If the optimal solution is fractional, then branch, and go back to Step 1 for the subproblems. Else, stop processing the current subproblem with an integral solution. We initialize our algorithm by generating a stationary searcher that stays at node i for T periods, for each i E N. Even though these elementary searchers are not likely to be selected in an optimal solution, they guarantee the feasibility of LHSLP. We discuss several branching strategies that can be used for Step 1 in Section 5.5. 5.3 Pursuit Evasion Problem In this section, we consider a pursuit evasion variant of the search problem. Unlike the hideandseek problem, the intruder is also mobile in this variant, and tries to avoid pursuit by the searchers. We note that this problem also reduces to the minimum dominating set problem for S(i) = A(i), Vi E N and T = 0, and hence it is NPhard. 5.3.1 Mathematical Model Similar to the hideandseek problem, we define P(T) to be the set of all possible walks of length at most T that can be taken by searchers. Similarly, let R(T) denote the set of all possible walks of length T + 1 that can be taken by the intruder, thus potentially evading the searchers for more than T time periods. Let dpr be a parameter whose value is 1 if a searcher following walk p detects an intruder following walk r, and 0 otherwise. This problem can be formulated as a set covering problem, which we denote by PE: PE: minimize A, (513) pEP(T) subject to dpA > 1 Vr e R(T) (514) pEP(T) A, {0,1} Vp P(T), (515) where Ap again equals 1 if and only if a searcher is assigned to follow walk p. The objective function (513) minimizes the number of searchers. Constraints (514) ensure that for each possible intruder walk of length T + 1, at least one searcher is selected to detect it. 5.3.2 Solution Approach Once again, rather than enumerating all possible search patterns of length at most T, and all evasion patterns of length T + 1, we propose a dynamic column and row generation algorithm to solve the problem. We start with a subset of search patterns P'(T) C P(T) and evasion patterns R'(T) C R(T), and solve a limited pursuit evasion (LPE) problem, given P'(T) and R'(T). We next describe how we generate new search and evasion patterns as needed. 5.3.2.1 Searcher's problem Given a subset R'(T) of intruder walks, the searcher's problem is similar to the pricing problem in the hideandseek problem. Let 7, be the dual variable associated with the constraint of type (514) corresponding to intruder walk r E R'(T). Also, let y, be a decision variable that equals 1 if an intruder following walk r is detected by a searcher following a walk that we generate, and 0 otherwise, Vr E R'(T). Given an optimal dual solution y of the linear programming relaxation to LPE, we solve the following pricing problem to seek a Avariable having a negative reduced cost: max reiR'(T) I/ subject to the restriction that (yl,..., YR'(T)r) corresponds to a set of intruder walks detected by a searcher walk of length no more than T. Similar to the previous pricing problem, this pricing problem can also be formulated as a mixedinteger programming problem on a timeexpanded network consisting of T + 1 stages. In particular, we create a node Nit for each i E N, t = 0,..., T. Unlike the previous problem, searchers do not necessarily have to move in each time period. Therefore, we connect each node to its neighbors and the (< li of itself in the next stage. We define a binary variable xt for all i e N, t = 0,..., T, which equals 1 if the searcher is located at node i at time t, and 0 otherwise. We also define a parameter dt = 1 if a searcher located at node i at time t can detect an intruder following walk r. An integer programming formulation of the pricing problem can be given as follows: maximize ^ 1' (516) rER'(T) subject to x 1 Vt 0,...,T (517) iEN x< > t 1 Vi N,t=1,...,T (518) jEA(i) T yr,< Y t Vr R'(T) (519) t=0 iEN 0 < r < 1 Vr e R'(T) (520) x e {0,1} V1 e N,t= 1,..., T. (521) Constraints (517) ensure that the searcher cannot be located at multiple nodes simultaneously. Constraints (518) model the fact that the searcher can either stay at the same node, or can move to an .,.li i:ent node at each period. Constraints (519) represent the condition that the searcher detects intruder r E R'(T) only if it moves to a node where it can detect the intruder during the pursuit. We note that the yvariables can be relaxed as continuous variables in this case, too. This property allows the number of binary variables in the searcher's problem to stay constant as new evasion paths for the intruder are discovered. As before, if the optimal objective function value of (516)(521) is greater than 1, then we have found a variable whose reduced cost is negative. 5.3.2.2 Intruder's problem Given an integer feasible solution A of LPE, in which a subset of the searchers has been selected, we need to solve a subproblem for the intruder to seek a walk that evades the searchers for more than T time units. To solve this problem, we generate a timeexpanded network consisting of T + 1 stages, which contains a node Nit for each i E N, t = 0, ,T. We add a dummy start node s, and connect s to all nodes in the first stage, which corresponds to the initial intrusion at t = 0. Similar to the network generated for the searcher's problem, we connect each node to its neighbors and the copy of itself in the next stage. Finally, we connect all nodes in the last stage to a dummy node q. We then trace each selected searcher's walk, and eliminate the nodes (and the corresponding arcs) from the timeexpanded network that would lead to the detection of the intruder. After constructing the timeexpanded network as described, we seek a feasible sq path on the network by a standard breadthfirstsearch algorithm, which works in O(N2T) time in the worst case if G is dense. If such a path exists, then it corresponds to a walk r that the intruder can take to avoid detection for T + 1 time units. In this case, we add r to R'(T), and generate the associated constraint of type (514). On the other hand, if no such path exists, then A is a feasible solution of PA. 5.3.2.3 Branchcutprice algorithm The pursuit evasion problem can be solved using the following branchcutprice algorithm. * Step 0: C'!..... a set of initial search and evasion walks so that every node is seen by at least one searcher. Step 1: Solve the linear programming relaxation of LPE, and generate columns by solving the searcher's problem until linear programming relaxation of LPE has been solved to optimality. If all columns are integervalued, go to Step 2. Else, branch, and go back to Step 1 for the subproblems. Step 2: Evade by solving the intruder's problem. If the intruder can evade the searchers, then add the evasion walk as a cutting plane of type (514), and go back to Step 1. Else, stop processing the current subproblem with an integral solution. We initialize our algorithm by generating a stationary searcher that stays at node i for T periods, and a stationary evader that stays at node i for T + 1 periods, for each i E N. We discuss several branching strategies that can be used for Step 1 in Section 5.5. 5.4 Patrol Problem 5.4.1 Problem Description The setting for the patrol problem that we consider in this section is as follows. The searchers are assigned to repeated patrol circuits, which they follow indefinitely. We assume that the period of a patrol circuit is bounded from above by a parameter K, where K > 1. Such a restriction might be due to a capacity or range limit of the searchers, or due to desired frequency of visits to individual nodes. Initially there is no intruder in the system. The intruder observes the searchers for a duration of time that is long enough to identify search patterns, and then picks a node and time to enter the system. It then tries to stay in the system as long as possible without being detected. The goal is to find the minimum number of searchers needed, along with the corresponding patrol routes, to ensure that the intruder is detected within T time periods after the intrusion. We note that this problem also reduces to the minimum dominating set problem (for K = 1, S(i) = A(i), Vi E N and T = 0), and hence is also NPhard. 5.4.2 Mathematical Model Let PC(K) denote the set of all possible circuits of period no more than K that can be taken by searchers. Similarly, let R(T) denote the set of all possible walks of length T + 1 that can be taken by the intruder. Let dp, be a parameter whose value is 1 if a searcher following circuit p detects an intruder following walk r, and 0 otherwise. Let us define a binary variable Ap for all p E PC(K), which equals 1 if a searcher is assigned to follow circuit p, and 0 otherwise. The patrol problem can be formulated as a set covering problem as follows. PP: minimize Ap (522) pEPC(K) subject to d,,, > 1 Vr R(T) (523) pEPC(K) Ap {0,1} Vp P(K) (524) We propose a branchcutprice algorithm similar to the pursuit evasion problem for solving this problem. We start with a subset of patrol routes P'`(K) and evasion walks R'(T), and solve the resulting limited patrol problem (LPP). We generate new patrol routes and evasion walks as needed. 5.4.2.1 Searcher's problem The searcher's problem is similar to the pricing problems discussed before. Let 7, be the dual variable associated with the constraint of type (523) corresponding to intruder walk r c R'(T). We define y, to be a decision variable that equals 1 if an intruder following walk r is detected by a searcher following a patrol circuit that we generate, and 0 otherwise, Vr e R'(T). Given an optimal dual solution 7 of the linear programming relaxation to LPP, we solve the following pricing problem to seek a Avariable having a negative reduced cost: max rER'(T) ^ I/ subject to the restriction that (yi,... YR'(T)I) corresponds to a set of intruder walks detected by a searcher following a patrol circuit of period no more than K. We can solve the searcher's problem by solving a series of integer programs as follows. Let T denote the length of the current circuit under consideration. By considering different values of T E {1,..., K} we can find a circuit that optimizes the searcher's problem. Note that some values of T may not correspond to any circuits in G. For each value of , we generate a timeexpanded network containing T + 1 levels, where the first level corresponds to the initial deployment of the searcher, and the last lwr is a dummy li.r that we use to model the recurring patrol patterns. We connect each node to its neighbors in the next stage. As before, we define a binary variable x for all i C N, t = 0,..., T, which equals 1 if the searcher is located at node i at time t. We also define a parameter dr, = 1 if a searcher located at node i at time t can detect an intruder following walk r E R'(T). We solve the following integer program to seek an optimal searcher circuit visiting exactly 'r nodes. maximize ^ n (525) rER'(T) subject to x 1 Vt 0,..., (526) iEN t < x 1 Vi e N, t 1, (527) jEA(i),j i x x Vi EN (528) T1 yr < Vr e R'(T) (529) t=0 iEN 0 < yr < 1 Vr R'(T) (530) x~e {0,1} Vi e N, t 0,...,7. (531) Constraints (526) and (527) ensure that each feasible solution corresponds to a walk. Constraints (528) guarantee that the first and the last nodes visited by the searcher are the same, and hence the searcher's walk forms a circuit. Finally, Constraints (529) relate the x and yvariables, where we can once again relax integrality restrictions on the yvariables. Integer programs corresponding to different values of r can be solved in any sequence. We note that a good solution obtained by solving the searcher problem for a particular value of r can be used to prune problems to be solved later for different values of 7 by bound. Therefore, we can start by solving a searcher problem for the largest value of r, since a searcher following a longer circuit is more likely to detect more intruder walks. Also, we skip any r if we determine that no circuit of length r exists in G. 5.4.2.2 Intruder's problem Given a selected subset of the searcher circuits, the intruder seeks a way of I ii; in the system as long as possible without being detected. Our main observation is that the state of the system with respect to the location of the searchers is cyclic, and its period is equal to the least common multiple of the periods of searchers' circuits, which we denote by L. Therefore, the intruder can stay in the system indefinitely if it can identify a walk that allows it to return to its initial location at the end of a multiple of L steps. To solve the intruder's problem, we can generate a timeexpanded network consisting of L stages similar to the pursuit evasion problem. We connect each node to the copy of itself and its neighbors in the next stage by a directed arc having length 1. We also connect the nodes corresponding to stage L to the nodes to their neighbors in the first stage with directed arc having length 1 (modeling the fact that the overall search pattern repeats after L periods). We add a dummy start node s and a dummy end node q. We connect s to all nodes by a directed arc having length 0 (reflecting our assumption that the intruder can enter the system at any time and location), and connect all nodes to q by a directed arc having length 0. Finally, we trace each selected searcher's circuit, and remove nodes and arcs from the intruder's network that would lead to the detection of the intruder by the searcher. We can solve the intruder's problem on the generated graph by seeking a longest sq path. We first seek a topological ordering of the nodes using a standard depthfirstsearch algorithm, whose complexity is O(N2L) for a dense G. Since a directed graph is .,. iv lic if and only if it is has a topological order, this step identifies whether the graph is cyclic. If there is a cycle in this graph, then the intruder can stay in the system forever without being detected by the searchers. In this case, we generate a cut of type (523), and stop. Otherwise, the graph is .,. iI. ii and given a topological order of the nodes, a longest path can be found in polynomial time by a dynamic programming algorithm whose complexity is O(N2L) (Am!li et al., 1993). If the length of a longest path is greater than T, then the intruder can successfully evade the searchers. We can use such a longest walk to generate a cut of type (523). 5.4.2.3 Branchcutprice algorithm The patrol problem can be solved using a branchcutprice algorithm similar to the algorithm we propose for the pursuit evasion problem. * Step 0: C'! .. .. an initial set of patrol circuits and evasion walks so that so that every node is seen by at least one searcher. Step 1: Solve the linear programming relaxation of LPP, and generate columns by solving the searcher's problem until the linear programming relaxation of LPP has been solved to optimality. If all columns are integervalued, go to Step 2. Else, branch, and go back to Step 1 for the subproblems. Step 2: Evade by solving the intruder's problem. If the intruder can evade the searchers for more than T time periods, then add the evasion walk as a cutting plane of type (523), and go back to Step 1. Else, stop processing the current subproblem with an integer solution. As before, we initialize our algorithm by generating a stationary searcher that r l at node i, and a stationary evader that li , in node i for T + 1 periods, for each i E N. We discuss several branching strategies that can be used in Step 1 in the next section. 5.5 Branching Strategies If an optimal solution of the linear programming relaxation of the master problem is integer feasible after all necessary variables and constraints have been added in the column and rowgeneration phases, then we have found an optimal solution. Else, if some Avariable is fractional, then we need to branch. C'! ....ig a branching rule that does not make the pricing problem too difficult to solve is essential in column generation algorithms. Finding such a branching rule is relatively easy if the master problem is a set packing or set partitioning problem, since in this case it is possible to branch by choosing a subset of variables X and setting x = 0, Vx E X in one branch, and x = 0, Vx X in the other branch. These branching constraints can be added implicitly by removing the corresponding variables from the problem, and adjusting the pricing problem so that those variables are never generated in future iterations. Since no constraints are added explicitly, no new dual variables are created. Therefore, the structure of the pricing problem Il li the same throughout all nodes of the branchandbound tree (Barnhart et al., 1998; Jiinger and Thienel, 2000). However, our master problem is a set cover problem, and the approach described above is not directly applicable. Any constraint on a group of variables for our set cover problem would necessitate adding a branching constraint like :.xe x < A in one branch, and Yxex x > A + 1 (for some suitable A) in the other branch. Since these constraints cannot be added implicitly, we need to handle a new dual variable for each branching constraint in the subproblem. As an alternative, we propose a multitiered branching rule. Given a fractional solution A, we first evaluate the value of each constraint expression vr EpEp(T) dpr~p. If there exists a fractional v, value for some r E R'(T), then we branch on the constraint (523) corresponding to r as follows. In the upbranch, we simply change the righthandside value of the constraint as [vr]. On the downbranch, we set the upper bound of the constraint expression to Lvr] (and convert it to an equality constraint if Lvr] = 1). Note that branching on a constraint in this manner does not introduce any new dual variables or constraints that need to be considered in any of the pricing problems. Another benefit of this branching scheme is the following. If we branch down on a constraint and obtain ,,pc d,,Ap 1, we can then use setpartition type of branching schemes on the corresponding subproblem. This allows us to branch on a group of Avariables in the subsequent branches without d. I i vi.; the pricing problem structure. Note that sign of the dual variable 7, can change after branching down on the constraint (523) corresponding to r, which makes y/ = 0 optimal regardless of the values of xvariables. Therefore, we need to add constraints that force y, to 1 if the searcher's chosen path detects intruder r. Hence, we add constraints y > x Vi e N, j S(i), t 0,...,T (532) yr > dx Vr e R'(T), i N, t 0,...,T (533) yr > d',x4 Vr e R'(T), ie N, t ,...,r 1 to the searcher's pricing problem for the hideandseek, pursuit evasion, and patrol problems, respectively. In general it is possible to have a fractional solution A for which all vvalues are integral, and therefore the branching rule described above cannot be applied. In such cases we can apply a simple variablebased branching rule. If there is a variable Ap whose value is fractional, we can simply create two branches with Ap = 0 and Ap = 1. In the downbranch, we simply eliminate the column corresponding to Ap from the set covering formulation. In the upbranch we need to adjust the righthandside vector of our set covering problem before eliminating Ap. In either case, we need to adjust the pricing problems so that the same variable cannot be regenerated. Recall that the solution methods we propose for all three pricing problems are based on a timeexpanded network formulation. Let us denote by W(p) the set of timeexpanded node indices corresponding to a searcher walk (or circuit) p. We can enforce the condition that a walk p is not generated again by adding the following constraint to the corresponding searcher's pricing problem: x < W(p)l. (535) (i,t)EW(p) Branching on a single variable is likely to be quite weak on the downbranch since only one particular searcher walk (or circuit) is avoided. Hence, we only apply this rule if our constraintbased branching rule fails. Also note that the difficulty of solving the set covering problem does not increase under either branching rule, since no constraints are added explicitly while branching. 5.6 Computational Results We implemented the algorithms discussed in the previous sections in C++ on a Windows XP PC with a 3.4 GHz CPU and 2 GB RAM. We used CPLEX 11.2 to solve the (534) pricing problems and the linear programming relaxations of our set covering formulations. We implemented our algorithms for solving the intruder's problem using Boost Graph Library version 1.36 (Siek et al., 2001). Our base set of test problem instances consists of 150 randomly generated problem instances for which the expected edge density of the graph (measured as iN Vi where we do not consider selfloop edges in calculating edge density) is 10' the number of nodes N ranges from 5 to 25, and the maximum allowed time to detection T ranges from 0 to 5. In generating instances we first picked a random subset of edges so that the edge density is 10' and if necessary added the minimum number of edges needed to make the graph connected (see Siek et al. (2001)). We then added selfloop edges, and we generated five problem instances for each problem size, which is determined by the number of nodes. Finally, we solved each problem instance with different values of T E {0,..., 5} for the hideandseek, pursuit evasion and patrol problems. In each case, we assume that a searcher located at node i E N can observe node i and all nodes .,.1] i:ent to it, and hence we set S(i) = A(i), for all i E N. We imposed a 1200second time limit past which we stopped the execution of an algorithm in all our experiments. Recall that all problems that we consider in this chapter reduce to the minimum dominating set problem for T = 0. We use this property to calculate an initial upper bound by solving our LHS formulation, where we initialize R'(0) by adding a searcher located at each node i E N. Our first experiment focuses on the hideandseek problem. For this experiment, we executed our branchandprice algorithm described in Section 5.2.2.2 on our base data set. All 150 problem instances in our data set were solved to optimality within 12.8 seconds. Table 51 ditpl1i the average number of branchandbound nodes evaluated in our branchandprice algorithm for different values of N and T. Each value represents the average of the results obtained on five randomly generated graphs. We observe that the number of branchandbound nodes that are explored increases as N increases, which Table 51. Average number of branchandbound nodes explored for hideandseek problem N T=0 T=1 T=2 T=3 T=4 T=5 5 1 1 1 1 1 1 10 1 1.4 1.8 1.4 1.4 1 15 1 1 1 1 1 1 20 1 2.6 3.4 1.8 1.8 1 25 1 7.4 2.2 2.6 1.4 1 is expected since the difficulty of the problem increases with increasing N. Table 52 Table 52. Average number of searchers needed for the hideandseek problem N T=0 T=1 T=2 T=3 T 4 T=5 5 2 1.8 1 1 1 1 10 3.4 2.8 2 1.8 1.6 1 15 4.6 3.2 2.2 2 2 1.6 20 4.6 3.6 2.6 2 2 1.8 25 5.8 3.8 2.8 2 2 2 shows the average number of searchers needed for different values of N and T, where once again each value is an average of five problem instances. We note that the T = 0 column corresponds to the cardinality of the minimum dominating set. Table 52 reveals that the number of searchers needed increases as the graph gets larger, and decreases as the maximum allowed time to detect the intruder increases. We analyze the performance of our branchcutprice algorithm described in Section 5.3.2.3 for the pursuit evasion problem. Table 53 shows that our algorithm was able to solve 128 out of the 150 instances within the prescribed time limit. As expected, the difficulty of the problem increases as N and T increase, since this setting allows for more evasion routes for the intruder, and hence requires the searchers to develop more sophisticated routes. Table 53. Number of instances that are solved within time limit for the pursuit evasion problem N T=0 T=1 T=2 T=3 T 4 T=5 5 5 5 5 5 5 5 10 5 5 5 5 5 5 15 5 5 5 5 5 5 20 5 5 5 3 1 2 25 5 5 4 3 0 0 Table 54 dip~'i the average number of branchandbound nodes evaluated in our branchcutprice algorithm for the pursuit evasion problem. We note that for this Table 54. Average number of branchandbound nodes explored for the pursuit evasion problem N T=0 T=1 T=2 T=3 T=4 T=5 5 1 1 1 1 1 1 10 1 1.8 3 1.8 1.8 1 15 1 5 3.4 6.2 4.2 3 20 1 28.2 109 334.6 101 14.2 25 1 23.4 869 324.6 41.4 11.4 problem, processing each node takes longer than the hideandseek problem, and therefore our algorithm can process fewer nodes within the time limit. Finally, Table 55 diipl, the average number of searchers needed for this problem, where we use the best known solutions for instances that were not solved to optimality within the time limit. A Table 55. Average number of searchers needed for the pursuit evasion problem N T=0 T=1 T=2 T=3 T=4 T=5 5 2 1.8 1 1 1 1 10 3.4 3 2.2 2 2 2 15 4.6 3.4 2.6 2 2 2 20 4.6 4 3 2.8 2.8 2.6 25 5.8 4 3.4 3.2 3.2 3 comparison of Tables 52 and 55 reveals that the number of searchers needed for the hideandseek problem is less than that for the pursuit evasion problem. This result is not surprising since the intruder is stationary in the former problem, while it can move to avoid the searchers in the latter problem. Table 56. Number of instances that are solved within time limit for the patrol problem N T=0 T=1 T=2 T=3 T=4 T=5 5 5 5 5 5 5 5 10 5 5 5 5 5 5 15 5 5 5 5 5 5 20 5 1 1 1 1 0 25 5 0 0 0 0 0 Our last experiment evaluates our branchcutprice algorithm described in Section 5.4.2.3 for the patrol problem. Table 56 reveals that our algorithm for the patrol problem can solve fewer instances in our data set within the time limit compared to our algorithms for the other problems. This can be explained by (i) our assumption that the intruder can pick a time to enter the system, and (ii) our solution algorithm for the searcher's problem, which requires solving multiple mixedinteger programs. The first factor makes it easier for the intruder to evade the searchers, while the second factor makes the searcher's problem more difficult to solve. The combined effect of these factors is that processing each node takes longer than the other problems, which can also be seen by considering Table 57. Table 57. Average number of branchandbound nodes explored for the patrol problem N T=0 T=l T=2 T=3 T 4 T=5 5 1 1 1 1 1 1 10 1 1 1.4 1.4 1.4 1.4 15 1 44.6 20.2 6.2 11.4 3 20 1 12.2 5.4 4.2 3.4 1 25 1 5 3.2 2.1 1.2 1 Finally, we report our results on the number of searchers needed for the patrol problem in Table 58. As before, our calculations are based on the best known solutions and do not necessarily correspond to optimal solutions for the instances that were not solved within the time limit. However, we observe that the number of searchers needed for the patrol problem is larger than the other two problems as expected. Table 58. Average number of searchers needed for the patrol problem N T=0 T=1 T=2 T=3 T 4 T=5 5 2 1.8 1.6 1.2 1 1 10 3.4 3 2.8 2.6 2.6 2.6 15 4.6 3.4 3 2.8 2.8 2.6 20 4.6 4.2 3.8 3.1 3 2.8 25 5.8 5.3 4.7 3.8 3.6 3.2 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS In this chapter, we focus on future research areas regarding the problems we have described in the previous chapters. We analyze the techniques we employ. .1 identify associated weaknesses, and ,..i 1 improvements. We also discuss a research topic that is based on a technique for reformulating the master problem for a class of bilevel optimization problems. 6.1 Stochastic EdgePartition Problem For solving the stochastic edgepartition problem (C!i lpter 2, see also Taskm et al. (2009a)) we first developed an integer programming formulation of the problem, and prescribed a bilevel reformulation of the problem that has integer variables in both stages. Our computational tests showed that both the direct solution of the integer programming formulation and our integer programmingbased cutting plane algorithm for the bilevel formulation are capable of solving only small problem instances to optimality. We then designed a hybrid integer programming/constraint programming algorithm to overcome the computational difficulties encountered by the first two approaches. Our hybrid approach first allocates node copies that are to be distributed across configurations using an integer programming formulation, and then assigns nodes to subgraphs using a constraint programming algorithm. After assigning nodes to subgraphs, it partitions edges to subgraphs for each scenario in a third stage, using another integer programming formulation. Our computational experiments show that the hybrid approach significantly outperforms the other approaches. In our study, we have assumed that the number of subgraphs, IKI, is a part of the problem input. In SONET network design application there is no practical limit on the number of subgraphs, but a limit is specified to model the problem (Goldschmidt et al., 2003; Sherali et al., 2000; Smith, 2005). ('!i ..! ig IKI too small may make the problem infeasible or suboptimal, and choosing IKI too large increases the difficulty of the problem as we discussed in Section 2.4. In our experiments, we manually chose IKI sufficiently large to yield a feasible solution in each problem instance. An area for future research is to treat IKI not as a parameter but as a variable, and to find the smallest possible value of IKI that guarantees the existence of an optimal edge partition that minimizes the number of node copies. This problem appears to be very difficult in general, but some upper bounds can be derived for our problem. We first note that choosing IKI = IE is guaranteed to yield a feasible edge partition (and not eliminate any feasible edge partitions from consideration), since each edge can be partitioned into a unique subgraph. Furthermore, if a feasible edge partition having an objective function value z is known, then [L/2J is an upper bound on IKI. This bound follows from the fact that there exists an optimal solution that contains at least two nodes in each nonempty subgraph. Such a z can be calculated by a simple improvement heuristic. We start with IKI = IE, and initially assign each edge to a unique subgraph. We then seek two subgraphs that can be merged without violating any constraints in any scenario, while also improving the objective function value. If we find such subgraphs, we merge them and repeat this process until no more subgraphs can be merged. Since this algorithm starts with a feasible solution, and retains feasibility in each iteration, it yields a feasible solution. However, our preliminary analysis ii. 1 that the bound on IKI that we get using this approach is quite weak. Improving bounds on IKI and finding the smallest possible number of subgraphs that yields an optimal edge partition is a future research area. 6.2 Matrix Decomposition Problem In C'! Ilpter 3 we have described exact decomposition algorithms for solving leaf sequencing problems arising in IMRT treatment planning (also see Taskm et al. (2009b)). Our solution algorithm for the matrix decomposition into apertures satisfying the consecutiveones property is based on an integer programming model for finding a set of intensity values to be assigned to apertures, and a backtracking algorithm that forms apertures by finding compatible leaf positions for each row. Our computational results show that an optimal solution to many clinical problem instances can be found within a few minutes. As such, not only can this algorithm reasonably be used in real clinical settings, but also the bounds obtained from our algorithm can serve as benchmark criteria to compare the performance of several heuristic methods prescribed for this problem. Our solution technique for the consecutiveones matrix decomposition problem and our threestage approach to the stochastic edgepartition problem are based on a similar idea. In both problems, we add ,.'regate vy iil I! to our formulations, which describe important structural properties of solutions, but are not enough by themselves to encode complete solutions. In each case we represent the optimization problem in terms of our . I:. regate variables in a master problem, and provide a subproblem that seeks a complete feasible solution corresponding to the values of the .,. .regate variables chosen by the master problem. In both applications, our master problems are discrete optimization problems, which we solve using integer programming methods, and our subproblems are discrete feasibility problems, which we solve using constraint programming methods. Separating critical optimization decisions from feasibility decisions, and utilizing strengths of integer and constraint programming techniques in a hybrid algorithm has allowed us to obtain significantly better results than other methods. A i, r i theme in our future research is going to be on generalizing this hybrid approach to handle a broader class of problems. In C'!i lter 4 we studied a different variant of the matrix decomposition problem, in which the aperture matrices need to be rectangular in shape (see also Takmn et al. (2008)). Rectangular apertures can be formed by using conventional jaws already integrated into IMRT treatment devices, and do not need an advanced MLC system, which is costly to manufacture and operate. In Chapter 4, we proposed an exact optimization algorithm that can be used to i", i1. whether a jawsonly treatment system can deliver fluence maps efficiently. Our algorithm is based on an integer programming formulation, which we enhance using several valid inequalities and by partitioning the problem into simpler problems. We also derived a bilevel Benders decomposition algorithm for this problem. The master problem of our Benders decomposition approach chooses a subset of the rectangular shapes that can be used in decomposing the input matrix. Later, a subproblem checks whether the selected subset of rectangles can completely decompose the input matrix. Unfortunately, our computational tests showed that our Benders decomposition algorithm is computationally inferior to the integer programming approach. The main reason of slow convergence is the weakness of cuts generated in each iteration. Specifically, given an infeasible subset of rectangles chosen by the master problem, our subproblem (which is a linear programming problem) detects infeasibility, and returns a cut, which is generated based on a dual extreme ray. However, this extreme ray is a mathematical proof of infeasibility, but does not necessarily identify the underlying rea son of infeasibility. In other words, it does not identify which bixels in the input matrix cannot be partitioned with the selected subset of rectangles. Furthermore, there are typically many extreme dual rays for a single infeasible master problem solution, from which several nondominated cuts can be derived. One way of improving the convergence can be applying a twodimensional binary search algorithm on the input matrix to find out which region of the input matrix cannot be partitioned with the selected rectangles. That is, if the entire matrix cannot be partitioned, we try to partition the lefthandside and the righthandside halves of the matrix independently. If one of these submatrices cannot be partitioned, we immediately have a more specific reason for infeasibility (and hence a stronger cut), because this result implies that the infeasibility in a submatrix needs to be fixed using a subset of the rectangles, which cover that part of the matrix. This idea can be applied recursively to find possibly multiple infeasible regions of the matrix. The cuts associated with these infeasible submatrices are much stronger than a single cut derived based on the entire matrix. Furthermore, the information regarding the description of infeasibility generated by analyzing submatrices in a single iteration can only be retrieved after several iterations of the original decomposition algorithm, which is based on solving the entire matrix only. This idea of obtaining a better description of infeasibility by analyzing subsets of variables can be generalized to the general Benders decomposition algorithm, and has the potential of improving the convergence properties in many applications. In our study, we have developed solution techniques for two versions of the matrix decomposition problem, which apply to most available IMRT treatment machinery. However, there are other types of machinery that have different aperture shape restrictions, such as interdigitation or connectedness constraints (see e.g., Lim (2002)). In a related line of research, we are planning to design exact optimization algorithms to solve other variants of the matrix decomposition problem to optimality. Quantifying the effect of several shape constraints enforced by different types of machinery on radiation delivery efficiency would be a valuable contribution to the medical physics field. 6.3 Graph Search Problem In C'!h pter 5 we considered three variants of a graph search problem: (i) a hideandseek problem, where a set of searchers seek a stationary intruder, (ii) a pursuit evasion problem, where the intruder moves to avoid detection, and (iii) a patrol problem, where searchers follow recurring patrol routes. The aim in each problem is to find the minimum number of searchers needed so that the intruder cannot stay in the system without being detected for longer than a prespecified duration of time. We proposed a branchcutprice algorithm, which can be adapted to all three problems with certain modifications. Our main contribution is that we do not make any assumptions on the topology of the input graph, and our algorithms work on general graphs. Even though all three problems that we consider are NPhard, our computational tests show that the hideandseek and pursuit evasion problems can be solved to optimality for modest problems sizes within reasonable computational time. However, our algorithm for the patrol problem can only solve small problem instances to optimality within the limits that we enforced. Our future research will initially focus on improving the performance of our algorithm. In particular, we are planning to (i) add heuristics for the searcher's problem to seek variables having a negative reduced cost before switching to our mixedinteger programming models, (ii) investigate flowbased formulations for the searcher's problem, which might have a tighter linear programming relaxation than the nodebased formulations that we proposed, and (iii) seek valid inequalities to improve dual bounds. We are also planning to extend our models and solution algorithms to handle different types of searchers with different capabilities and costs. For instance, some searchers might be able to detect the intruder from a longer distance, while some others might be stationary but cheaper to operate. A related problem that can be investigated is a setting in which we are given a fixed number of searchers, and which must be coordinated so that the amount of time an intruder can stay in the system is minimized. Our set covering formulation, which is based on the idea that at least one searcher must be chosen for each route the intruder can take, can be generalized to other variants of the graph search problem. In particular, a problem that has been widely studied in the literature assumes that the intruder can reside at the edges of the graph. This problem has been investigated from a theoretical perspective (see, e.g., Dendris et al. (1997); Ellis et al. (1994); Seymour and Thomas (1993)); however, to the best of our knowledge no exact optimization method that works on general graphs has been proposed. We are planning to extend our algorithm so that it can also solve this problem. 6.4 Master Problem Reformulation in BiLevel Cutting Plane Optimization Algorithms In this line of research, we are planning to explore viv of reformulating master problems to improve the coordination between the master and subproblems in bilevel Benders optimization algorithms, resulting in faster convergence to an optimal solution. The key concept behind this line of research is a surprising property of decomposition algorithms, namely the existence of several alternative optimal solutions (or extreme rays) to the dual of the secondstage problems, each resulting in a different Benders inequality. There can be an exponential number of these inequalities, each of which is nondominated. Worse, it is possible that each of these inequalities may need to be generated one at a time after each iteration of the master problem. However, it turns out that it is possible to reformulate the master problem to avoid this ::p. ii!' li I cut" difficulty in some cases. In a stochastic SONET design problem (Smith et al., 2004), and in a product introduction and interdiction game, Smith et al. (2008) consider the addition of a quadraticorder set of variables in the master problem. These new variables are passed to the subproblem, and a Benders cut is generated in terms of the new variables that implies all of the (exponentiallymany) Benders cuts that could have been generated in the original variable space. This master problem reformulation technique has the potential to dramatically reduce the number of iterations required by Benders decomposition to converge to an optimal solution, with only a modest increase in the size of the formulation. In both cases mentioned above, the tradeoff of increasing model size to improve the strength of Benders cuts was computationally beneficial. Let us describe the idea in more detail in the context of a SONET network design problem (Smith et al., 2004), which is similar to the edgepartition problem discussed in C'i Ilpter 2. There exist a set of demand pairs (i,j) E E that may be satisfied on a single communications network (all demand pairs have to be satisfied in our edgepartition problem). The communications network is composed of i i" k = 1,...,K. If both clients i and j have been linked to ring k, then we may choose to satisfy the demand request between i and j on ring k. Define y k to be a continuous variable that represents the fraction of the demand between i and j that is satisfied on ring k in scenario q (these variables are defined to be binary in our edgepartition problem since we assume that each demand pair needs to be satisfied on a single ring). Let Eq be the set of demand pairs (i,j) in scenario q. We also define xik as a binary variable equal to one if and only if client i is assigned to ring k. Setting aside other issues in this problem such as ring capacity, a subset of scenario restrictions is: k < Xik Vq e Q, (i,j) e Et, k e K (61) yk where the existence of scenarios q E Q is due to uncertain demands between clients i and j. Using a straightforward decomposition approach, the problem can be decomposed so that the xvariables are parts of the master problem formulation, while the yqvariables are determined in subproblems corresponding to scenarios q E Q. Constraints (61) and (62) essentially state that in order for the communication demand between customers i and j to be assigned to ring k, both customers i and j have to be assigned to ring k. Unfortunately, cuts enforcing this relationship cannot be represented in the original space of xvariables. Smith et al. (2004) show that there can exist an exponential number of alternative dual solutions associated with a master problem solution represented by k:, each leading to a nondominated Benders cut. Then they reformulate the master problem by adding uijk variables, which represent the minimum of xik and Xjk. In other words, uijk = 1 if both customers i and j are assigned to ring k. Given the uvariables, the constraints (61) and (62) can be replaced by qk < uijk Vq e Q, (i,j) e Eq, k e K. (63) Smith et al. (2004) show that a single Benders cut based on the uvariables dominates an exponential number of cuts based on the original xvariables. In other words, adding a quadratic number of variables to the master problem can save an exponential number of iterations of the Benders decomposition algorithm. In this particular problem, the authors recognized that the yvariables in the subproblem are dependent on min(xik, xjk), and used this nonlinear relationship between the xvariables to reformulate the master problem. This kind of relationship is quite common in bilevel optimization algorithms. Our goal in this line of research will be to generalize this approach to other types relationships between variables so that structures that can be exploited by master problem reformulation are identified automatically. We observe that such nonlinear relationships can be induced using suitable relaxations of the subproblem. For instance, assume that the subproblem contains two constraints like ayI + 0 2 + + al < I (6 4) alyl + ,II_. + + a2yn < X2, (65) where x1 and x2 are variables of the master problem. Since both (64) and (65) are of < type, we can take the componentwise minimum of the two constraints to obtain min(al, a )yi + min(at, a y2 + min(a, a )yn < xi (6 6) min(al, a2)yi + min(a, a),y2 + + min(al, a )yn 2 x. (6 7) Since the lefthandsides of (66) and (67) are the same, we can combine the two constraints into min(a, a l)yi + min(a), ay2 + min(aa )Yn < min(xi,2) (6 8) Constraint (68) describes a nonlinear relationship between the y and xvariables. At this point, the master problem can be reformulated by adding a variable v2 = min(xl, x2), and the subproblem can be reformulated by using this variable as min(a, a + min(a, a minl(a,) v 2 1 (69) a1)yl 1 al)Y2 +''" + n ) n <_ 12" Even though (69) is weaker than (64) and (65), this reformulation might improve the convergence of the algorithm due to the ::II" "i i icut" behavior, especially in the beginning iterations of the Benders process. In our future research we are planning to formalize and generalize this idea, and determine how to apply this reformulation approach to improve computational performance in bilevel cutting plane algorithms. APPENDIX A AN IP MODEL FOR C1MATRIX DECOMPOSITION PROBLEM In this appendix we discuss an integer programming approach to decomposing a fluence map into a number of apertures and corresponding intensities that is based on a model proposed by Langer et al. (2001). Given a maximum number of unitintensity apertures, v T, this formulation determines the positions of the left and right leaves in each row of each of these apertures. We develop the model by separately studying four components: * Fluence map requirements. Define, for each aperture t = 1,..., T and each bixel (i,j) E {1,..., m} x {1,..., n}, a binary variable dt that is equal to one if and only if bixel (i,j) is exposed, i.e., not covered by a left leaf or a right leaf. Since each aperture has unit intensity, the following constraints then ensure that the desired fluence map is delivered: T d bi V Il, ,...,m, j=l,...,n. (Al) t=1 Aperture del.i i.b.iil.hi constraints. Define, for each aperture t = 1,... ,T and each bixel (i,j) E {1,..., m} x {1,..., n}, binary variables p and li that are equal to one if and only if bixel (i,j) is covered by the right leaf or the left leaf in row i of aperture t, respectively. The following set of constraints then ensure that each of the T apertures is deliverable: pt + I+dt 1 Vi 1,...,m, j 1,...,n, t 1,...,T (A 2) Pij < pij+l V i 1,...,m, j,= 1,..., n 1, t= 1,...,T (A3) it < i, V i 1,...,m, j=2,...,n, t= 1,...,T. (A4) In particular, constraints (A2) state that each bixel is either covered by a righthand leaf, covered by a lefthand leaf, or uncovered (where the dvariables are included only for convenience and can be substituted out of the formulation). Constraints (A3) and (A4) state that if any bixel (i,j) is covered by a righthand leaf (resp. lefthand leaf), then bixel (i,j + 1) (resp. (i,j 1)) should be covered by a righthand leaf (resp. lefthand leaf) as well. Beamontime. We associate a binary variable zt with each aperture t = 1,...,T that is equal to one if there are uncovered bixels in aperture t and zero otherwise, so that the beamontime is simply given by T Szt. (A5) t= 1 While Langer et al. (2001) impose the following constraints to ensure that these variables have (at least) their desired value: m n Z di< (mn)zt Vt ,...,IT, (A6) i=1 j 1 we note that the following stronger formulation, which would actually not require enforcing the zvariables to be binary, can be obtained by di.i._r:egating (A6). dt < z V i= 1,..., j 1,...,n, t= 1,...,T. (A6') Note that this model allows zt to be equal to one even if in aperture t no bixels are exposed, so that formally speaking (A5) is an upper bound on the beamontime. The objective function ensures that the zvariables take on their minimum possible value. Number of apertures. We associate a binary variable gt with each aperture t = 1,..., T 1 that is equal to one if aperture t is different from aperture t + 1 and zero otherwise. The number of setups is then given by T Egt. (A7) t=1 (If any aperture is used more than once but separated by another one, we consider the second occurrence of the aperture to be a new setup. However, when minimizing total treatment time there alv,i exists an optimal solution in which identical apertures are delivered sequentially.) Now let c and uj be auxiliary binary variables such that the former is equal to one if bixel (i,j) is exposed in aperture t but not in aperture t + 1 and zero otherwise, and the latter is equal to one if bixel (i,j) is covered in aperture t but not in aperture t + 1. This relationship is stated by cj dij d^ Langer et al. (2001) then use the following constraints to ensure that the variables gt have (at least) their desired value: i (c + U) < (mrn)g Vt ,...,T . (A9) i= 1 j= 1 However, note that again a stronger set of inequalities (that permit g to be equivalently relaxed as continuous variables) is obtained using di,._egation: c + U < g Vi 1,...,m, j 1, ... ,n, t = 1, ... ,T 1. (A9') Similar to the case of the beamontime, this model allows gt to be equal to one even if apertures t and t + 1 are identical, although our objective function ensures that the gvariables are chosen sufficiently small. Langer et al. (2001) then study the problem of minimizing the number of setups (A7) subject to the constraints (A2)(A4), (A6), (A9), the constraint that the beamontime is minimal: T z < (A10) t=1 and binary constraints on the variables, where we recommend determining z via one of the polynomialtime procedures mentioned in Section 3.1. We note that an equivalent model is obtained by simply setting T = 5, which reduces the problem dimension, and hence should be more efficient than adding a beamontime constraint. We wish to minimize the total treatment time as measured by T T Wi gt + W2 Z (A11) t=1 t=1 subject to constraints (A2)(A4), (A6'), (A9'), and binary constraints on the appropriate variables (and hence we do not impose (A10)). APPENDIX B COMPLEXITY OF C1PARTITION Proposition 3. CIPARTITION is strongly NPcomplete. Proof. Let ( be the subset of {1,..., L} such that E c if and only if x > 0. Formally speaking, the problem size is given by log2(L), n, and  (since the zero entries of x need not be encoded). Let KC denote the set of all O(n2) ndimensional binary vectors whose ones appear consecutively, where vk is the binary vector corresponding to k E /C. Consider a guessed solution that consists of Il dimensional nonnegative integer vectors dk, Vk E /C, where dke denotes the number of times leaf position vk, k E /C, is assigned to intensity E c. Since all dkW < L in some feasible solution, we restrict the guessed dvectors as such. The size of the guessed vectors is thus O(r2ll log2(L)). We can verify whether or not CkEkC Z dk3vk = b in 0(n2 ) additions. Therefore, CIPARTITION is in NP. To show that CIPARTITION is NPcomplete, we reduce 3PARTITION to it. 3 PARTITION is a strongly NPcomplete problem and seeks whether a given multiset of integers can be partitioned into triplets having the same sum. Formally, it can be defined as follows (see Garey and Johnson (1979)): 3PARTITION INSTANCE: A multiset A of 3v positive integers al,..., a3, and a positive integer B such that B/4 < ai < B/2 for i = 1,..., 3v and such that E3, a, = vB. QUESTION: Can A be partitioned into v disjoint multisets A1,..., A, such that jA aj = B for i =,..., v? Given an arbitrary instance of 3PARTITION, we construct an instance for CIPARTITION as follows. First, we define x to be an integer vector whose fth component, x^, is equal to the number of indices i for which ai = Furthermore, we let b be a (2v 1)dimensional vector of the form [B 0 B 0 ... 0 B]. We construct a feasible solution to C1 PARTITION that employs only the oddindexed unit vectors of IC. Denote these vectors as el, e3,..., e2~l, and index their associated dvectors as dl, d3,..., d2,l. Assume that the 3PARTITION instance is a yesinstance, and hence there exist multisets A1,..., A, such that EA aj = B. In this case, a feasible solution of the CIPARTITION instance lets d2jl,L be the number of elements of intensity e in Aj, for each j = 1,..., v, and assigns dk = 0 for all other k. Similarly, suppose that the CI PARTITION instance is a yesinstance. 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J. Verhey. 1998. Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments. Medical PhI,'. 25(8) 14241434. BIOGRAPHICAL SKETCH Z. Caner Takmn was born in Bahkesir, Turkey on September 8, 1981. He earned his B.S. and M.S. degrees in Industrial Engineering from Bogazigi University, Istanbul in 2003 and 2005, respectively. Before starting his doctorate study, he worked for ICRON Technologies as a product consultant, where he took role in advanced planning and scheduling projects for customers in several industries including steel, automotive, electronics and glass manufacturing industries. He will finish his Ph.D. degree in Industrial and Systems Engineering at the University of Florida in August 2009. Following graduation, he will join Department of Industrial Engineering at Bogazici University as a faculty member. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 IwouldliketoexpressmydeepestgratitudetoColeSmithforhiswise,enlighteningideas,endlessmotivation,andpatientcounselingduringthewritingofthisdissertation.Hehasbeenagreatteacher,amentorandafriendtomeinthelastfouryears,andworkingwithhimhasbeenaprivilege.IwouldliketothankEdwinRomeijnforintroducingmetotheexcitingeldofoptimizationinhealthcare,andhisinvaluablecontributionstothisstudy.Hisguidanceandsupporthasbeenveryhelpfulthroughoutmygraduateeducation.IwouldalsoliketoacknowledgePanosPardalosandDouglasDankelfortakingpartinmydissertationcommittee,andtheirvaluablesuggestions.Mysincerethanksareduetomyfriendsingraduateschool.Inparticular,Chasehasnotonlybeenagreatcolleague,butalsoaclosefriendforthesefouryears.IcannotbegintocountthethingsthatIlearnedfromhimabouttheculture,thelifestyleandthelanguage.MyexperienceinAmericawouldnothavebeennearlyasenjoyablewithouthimandhiswifeCandace.IamindebtedtomyparentsandmybrotherforguidingandsupportingmeinalllifechoicesIhavemade.Finally,Iamdeeplygratefultomylovelywife,Semra,forherconstantencouragement,support,andunderstanding.Sheisthesourceofmyhappiness,andthesecretofmysuccess.Thisdissertationrepresentstheendofmylifeasagraduatestudent.Italsorepresentsthebeginningofanewstageinmylife,everymomentofwhichIamlookingforwardtosharingwithher. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 13 2STOCHASTICEDGEPARTITIONPROBLEM .................. 19 2.1IntroductionandLiteratureSurvey ...................... 19 2.2FormulationandCuttingPlaneApproach .................. 23 2.3AHybridIP/CPApproach .......................... 33 2.3.1FirstStageProblem ........................... 35 2.3.2SecondStageProblem ......................... 36 2.3.2.1Foundations .......................... 37 2.3.2.2Domainexpansion ...................... 38 2.3.2.3Constraintpropagation .................... 39 2.3.2.4Forwardchecking ....................... 40 2.3.2.5Nodeselectionrule ...................... 41 2.3.2.6Distributionvectororderingrule .............. 41 2.3.3ThirdStageProblem .......................... 42 2.3.4InfeasibilityAnalysis .......................... 42 2.3.5EnhancementsfortheFirstStageProblem .............. 43 2.3.5.1Validinequalities ....................... 43 2.3.5.2Heuristicforobtaininganinitialfeasiblesolution ..... 44 2.3.5.3Processingintegersolutions ................. 45 2.4ComputationalResults ............................. 45 3CONSECUTIVEONESMATRIXDECOMPOSITIONPROBLEM ....... 51 3.1IntroductionandLiteratureSurvey ...................... 51 3.2DecompositionAlgorithm ........................... 55 3.2.1DecompositionFramework ....................... 56 3.2.2MasterProblemFormulationandSolutionApproach ......... 58 3.2.3SubproblemAnalysisandSolutionApproach ............. 61 3.2.4ValidInequalitiesfortheMasterProblem ............... 66 3.2.4.1Beamontimeandnumberofaperturesinequalities .... 66 3.2.4.2Bixelsubsequenceinequalities ................ 68 3.2.5ConstructingaFeasibleMatrixDecomposition ............ 69 3.3ComputationalResultsandComparisons ................... 73 5 PAGE 6 ............................ 73 3.3.2ImplementationIssues ......................... 73 3.3.3ComparisonwithLangeretal.(2001)Model ............. 74 3.3.4RandomProblemInstances ....................... 75 3.3.5ClinicalProblemInstances ....................... 80 4RECTANGULARMATRIXDECOMPOSITIONPROBLEM .......... 84 4.1IntroductionandLiteratureSurvey ...................... 84 4.2AMixedIntegerProgrammingApproach ................... 86 4.2.1ModelDevelopment ........................... 86 4.2.2ValidInequalities ............................ 90 4.2.2.1Adjacentrectangles ...................... 90 4.2.2.2Boundingboxinequalities .................. 90 4.2.2.3Aggregateintensityinequalities ............... 93 4.2.2.4Specialsubmatrices ...................... 94 4.2.2.5Submatrixinequalities .................... 96 4.2.3PartitioningApproach ......................... 97 4.2.3.1Separablecomponents .................... 97 4.2.3.2Independentregions ..................... 98 4.2.3.3Dependentregions ...................... 100 4.2.3.4Upperboundcalculation ................... 102 4.3Extensions .................................... 104 4.3.1MinimizeTotalTreatmentTime .................... 104 4.3.2OptimizationwithBeamonTimeRestrictions ............ 105 4.4ComputationalResults ............................. 106 5GRAPHSEARCHPROBLEM ........................... 113 5.1IntroductionandLiteratureReview ...................... 113 5.2HideandSeekProblem ............................. 115 5.2.1MathematicalModel .......................... 115 5.2.2SolutionApproach ............................ 116 5.2.2.1Searcher'sproblem ...................... 117 5.2.2.2Branchandpricealgorithm ................. 119 5.3PursuitEvasionProblem ............................ 119 5.3.1MathematicalModel .......................... 119 5.3.2SolutionApproach ............................ 120 5.3.2.1Searcher'sproblem ...................... 120 5.3.2.2Intruder'sproblem ...................... 122 5.3.2.3Branchcutpricealgorithm .................. 122 5.4PatrolProblem ................................. 123 5.4.1ProblemDescription .......................... 123 5.4.2MathematicalModel .......................... 123 5.4.2.1Searcher'sproblem ...................... 124 5.4.2.2Intruder'sproblem ...................... 125 6 PAGE 7 .................. 127 5.5BranchingStrategies .............................. 127 5.6ComputationalResults ............................. 129 6CONCLUSIONSANDFUTURERESEARCHDIRECTIONS .......... 134 6.1StochasticEdgePartitionProblem ...................... 134 6.2MatrixDecompositionProblem ........................ 135 6.3GraphSearchProblem ............................. 138 6.4MasterProblemReformulationinBiLevelCuttingPlaneOptimizationAlgorithms ................................... 139 APPENDIX AANIPMODELFORC1MATRIXDECOMPOSITIONPROBLEM ...... 144 BCOMPLEXITYOFC1PARTITION ........................ 147 REFERENCES ....................................... 149 BIOGRAPHICALSKETCH ................................ 158 7 PAGE 8 Table page 21Descriptionsoftheprobleminstancesusedforcomparingalgorithms ...... 46 22Comparisonofthealgorithmsongraphshavingedgedensity=0:2 ....... 47 23Comparisonofthealgorithmsongraphshavingedgedensity=0:3 ....... 48 24Comparisonofthealgorithmsongraphshavingedgedensity=0:4 ....... 49 25Descriptionsoftheprobleminstancesusedforanalyzingthreestagealgorithm 49 26ThreeStagealgorithmongraphshavingedgedensity=0:2 ........... 49 27ThreeStagealgorithmongraphshavingedgedensity=0:3 ........... 50 28ThreeStagealgorithmongraphshavingedgedensity=0:4 ........... 50 31Dimensionsofclinicalprobleminstances ...................... 73 32Comparisonofourbasealgorithmwith Langeretal. ( 2001 )model ........ 75 33EectofrotatingtheMLChead ........................... 81 34Computationalresultsforourbasealgorithm ................... 81 35Comparisonofheuristicalgorithmsonclinicaldata ................ 83 41Eectofvalidinequalitiesandthepartitioningstrategy .............. 108 42Computationalresultsonmodelextensions ..................... 109 43Eectofmaximumintensityvalueonsolvability .................. 111 51Averagenumberofbranchandboundnodesexploredforhideandseekproblem 131 52Averagenumberofsearchersneededforthehideandseekproblem ....... 131 53Numberofinstancesthataresolvedwithintimelimitforthepursuitevasionproblem ........................................ 131 54Averagenumberofbranchandboundnodesexploredforthepursuitevasionproblem ........................................ 132 55Averagenumberofsearchersneededforthepursuitevasionproblem ....... 132 56Numberofinstancesthataresolvedwithintimelimitforthepatrolproblem .. 132 57Averagenumberofbranchandboundnodesexploredforthepatrolproblem .. 133 58Averagenumberofsearchersneededforthepatrolproblem ............ 133 8 PAGE 9 Figure page 21(a)Aninstanceofthedeterministicedgepartitionproblem(b)AsolutionwithjKj=3;r=3;b=20 ................................ 19 31(a)Amultileafcollimatorsystem(b)Theprojectionofanapertureontoapatient 51 32Comparisonoftotaltreatmenttimesonrandomdata ............... 76 33Comparisonofthenumberofaperturesonrandomdata ............. 77 34Comparisonofbeamontimevaluesonrandomdata ............... 78 35ComparisonofTGIvaluesonrandomdata ..................... 79 41Exampleuencemap ................................. 87 42Examplestartindex ................................. 91 43Exampleendindex .................................. 91 44Exampleboundingbox ................................ 92 45Anothernondominatedboundingboxseededat(6,3) ............... 93 46Twocomponentsofauencemap .......................... 98 47Regionsofaconnectedcomponent ......................... 98 48Ecientfrontierfornumberofaperturesandbeamontime ........... 110 51(a)Anexamplegraph(b)TimeexpandednetworkforT=2 ........... 118 9 PAGE 10 Inthisdissertation,weinvestigateaclassofmultileveloptimizationproblems,inwhichdiscretevariablesarepresentatmultiplestages.Suchproblemsariseinmanypracticalsettings,andtheyarenotoriouslydiculttooptimize.Bendersdecomposition,whichisawellknowndecompositionmethodforsolvinglargescalemixedintegerprogrammingproblems,cannotbeutilizedfortheclassofproblemsthatweconsiderduetotheexistenceofdiscretevariablesatlowerlevels.CuttingplanealgorithmssuchasthoseproposedbyLaporteandLouveauxhavebeendesignedforuseinbilevelintegerprogrammingproblemswithbinaryvariablesatbothlevels.However,thesearebasedongenericcuts,whichdonotutilizeanyproblemspecicstructures,andhenceoftenresultinweakconvergence.Ourgoalinthisdissertationistoproposenovelformulationandsolutionstrategiesforseveralmultileveloptimizationproblemstosolvetheseproblemstooptimalitywithinpracticalcomputationallimits. Werstconsideranedgepartitionproblem.ThemotivationforthisstudyisprovidedbyaSynchronousOpticalNetwork(SONET)designapplication.IntheSONETcontext,eachedgerepresentsademandpairbetweentwoclientnodes,andtheweightofeachedgerepresentsthenumberofcommunicationchannelsneededbetweentheclientnodes.Weconsiderastochasticversionoftheproblem,inwhichtheedgeweightsarenotdeterministic,buttheirunderlyingprobabilitydistributionisknown.TheproblemistodesignasetofSONET\rings"atminimumcost,whileensuringthat 10 PAGE 11 Next,wefocusonamatrixdecompositionproblem,whicharisesinIntensityModulatedRadiationTherapy(IMRT)treatmentplanning.Theprobleminputisamatrixofintensityvaluesthatneedstobedeliveredtoapatient,whichmustbedecomposedintoacollectionofaperturesandcorrespondingintensities.Inafeasibledecompositionthesumofbinaryshapematricesmultipliedbycorrespondingintensityvaluesisequaltotheoriginalintensitymatrix.Weconsidertwovariantsoftheproblem:(i)avariantinwhichtheshapematricesusedinthedecompositionhavetosatisfythe\consecutiveones"property,and(ii)avariantinwhichtheshapematriceshavetoberectangular.Fortherstvariant,westartbyinvestigatinganintegerprogrammingmodelproposedintheliterature,andshowhowtheformulationcanbestrengthened.Wethenformulatetheproblemasabileveloptimizationproblemthathasdiscretevariablesatbothstages,andsuggestahybridintegerprogramming/constraintprogrammingdecompositionalgorithmsimilartoouralgorithmfortheedgepartitionproblem.Ourtestsondataobtainedfrompatientsshowthatouralgorithmiscapableofsolvingprobleminstancesofclinicallyrelevantdimensionswithinpracticalcomputationallimits.Wethenturnourattentiontothesecondvariantofthematrixdecompositionproblem.Weformulatetheproblemasamixedintegerprogram,andinvestigateadecompositionmethodforsolvingit.Unliketherstvariant,thesecondlevelproblemturnsouttobealinearprogrammingproblem,andthereforeweareabletoderiveaBendersdecompositionalgorithmforsolvingthisvariant. Finally,weinvestigateaclassofgraphsearchproblems.Inthisclassofproblems,anintruderislocatedatanunknownnodeontheinputgraph,andagroupofsearchersneeds 11 PAGE 12 12 PAGE 13 Inmostcomplexdecisionmakingenvironments,thereexistseveraltypesofinterdependentdecisionsthatneedtobemadetooptimizesomecostorbenetfunction.Asasimpleexample,consideraproductionplanningproblem.Thegoalistodetermine,attheveryleast,thetypesofproductsthataretobeproducedwithinatimeperiod,alongwiththeassociatedproductionquantities.Theremightexistindividualrestrictionsoneachtypeofdecision,suchas\thetotalamountofproductionofproductsAandBcannotexceed,"or\ifproductAisproduced,somustproductB."Theremightalsoexistrestrictionsthatrelatethetwotypesofdecisions,suchas\ifproductAisproduced,thenthebatchsizemustbeatleast."Modelinganoptimizationprobleminvolves(i)deningadecisionvariableforeachindividualdecision,(ii)formulatingtherestrictionsonthedecisionsasconstraints,and(iii)deninganobjectivefunctiontobeoptimized.Theeldofmathematicalprogrammingseekstooptimizesuchmodelsandprovetheoptimalityofthegeneratedsolution,orprovethatnofeasiblesolutionexists. Optimizationproblemsinwhichsomevariablesarerestrictedtotakevaluesfromadiscretesetareclassiedasdiscreteoptimizationproblems.Animportantconcernregardingbuildingandsolvingdiscreteoptimizationproblemsisthattheamountofmemoryandthecomputationaleortneededtosolvesuchproblemsgrowexponentiallywiththenumberofdiscretevariables.Thetraditionalapproach,whichinvolvesmakingalldecisionssimultaneouslybysolvingamonolithicoptimizationproblem,quicklybecomesintractableasthenumberofdiscretevariablesincreases.Multileveloptimizationalgorithms,suchasBendersdecomposition( Benders 1962 ),havebeendevelopedasanalternativesolutionmethodologytoalleviatethisdiculty.Unlikethetraditionalapproach,thesealgorithmsdividethedecisionmakingprocessintoseveralstages.Forinstance,inBendersdecompositionarststagemasterproblemissolvedforasubsetofvariables,andthevaluesoftheremainingvariablesaredeterminedbyasecondstage 13 PAGE 14 Inessence,multileveloptimizationalgorithmssolveaseriesofsmallproblemsinsteadofasinglelargeproblem.Performingmultipleiterationsisusuallyjustiedduetotheexponentiallylargercomputationalresourcerequirementsassociatedwithsolvingalargerproblem.Furthermore,itisoftenthecasethatdecisionsforseveralgroupsofsecondstagevariablescanbemadeindependentlygiventherststagedecisions.Insuchcases,multileveloptimizationalgorithmsareamenabletoparallelimplementations.Theadventofecientparallelcomputinggridshasallowedmodernbileveltechniquestosolveproblemsthatwereregardedasintractablebefore( NtaimoandSen 2005 ).Insomeapplications,solvingproblemsinmultiplestagesallowseorttobeconservedbyavoidingtheexplicitsolutionofproblemsbymathematicalprogramming,suchastheevacuationnetworkdesignalgorithmof AndreasandSmith ( 2009 ).Multileveloptimizationhas 14 PAGE 15 MigdalasandPardalos 1996 ; Migdalasetal. 1997 ),andawideclassofoptimizationproblemscanbereformulatedasmultileveloptimizationproblems( HuangandPardalos 2002 ). Bendersdecompositionhasbeenparticularlysuccessfulinsolvingmixedintegerlinearprogrammingproblemsarisinginawidevarietyofapplications.InBendersdecomposition,discretevariablesoftheproblemarekeptinthemasterproblem,andcontinuousvariablesaremovedtothesubproblem.Ineachiteration,giventhevaluesofthediscretevariablesbythemasterproblem,thesubproblemissolvedasalinearprogram,andacuttingplanetobepassedbacktothemasterproblemisgeneratedusinglinearprogrammingduality.However,thisapproachcannotbeapplieddirectlywhendiscretevariablesalsoappearinthesecondstage.Thereasonisthatnodualinformationcanbeextractedfromthesubproblemifthesecondstageproblemcontainsintegervariables.Inthiscase,onecanemploycuttingplanessuchasthegeneralpurposecutsof LaporteandLouveaux ( 1993 )andcombinatorialBendersinequalities( CodatoandFischetti 2006 ).However,theseinequalitiesareoftenveryweak,andresultinslowalgorithmicconvergence. Ourmainlineofresearchisaboutdesigningecientmultileveloptimizationalgorithmsforproblemsthathavediscretevariablesatmultiplestages.WerstpresentourresultsonanedgepartitionproblemarisinginatelecommunicationsnetworkdesigncontextregardingSynchronousOpticalNetworks(SONET).Theedgepartitionproblemconsidersanundirectedgraphwithweightededges,andsimultaneouslyassignsnodesandedgestosubgraphssuchthateachedgeappearsinexactlyonesubgraph,andsuchthatnoedgeisassignedtoasubgraphunlessbothofitsincidentnodesarealsoassignedtothatsubgraph.Additionally,therearelimitationsonthenumberofnodesandonthesumofedgeweightsthatcanbeassignedtoeachsubgraph( Goldschmidtetal. 2003 ). 15 PAGE 16 2 ,inwhichweassignnodestosubgraphsinarststage,realizeasetofedgeweightsfromamonganitesetofalternatives,andthenassignedgestosubgraphs.Werstformulatetheproblemasamonolithicintegerprogrammingproblem,andshowthatthisapproachisnottractableduetotherapidlyincreasingcomputationalrequirements.Wethenprescribeabilevelcuttingplaneapproachhavingintegervariablesinbothstagesandexaminecomputationaldicultiesassociatedbothwiththegenericinequalitiesby LaporteandLouveaux ( 1993 )andwithourproposedcuttingplanes.Wealsoprescribeathreelevelhybridintegerprogramming/constraintprogrammingalgorithmhavingdiscretevariablesatalllevels,anddiscusshowthishybridapproachresolvessomeofthedicultiesassociatedwiththebilevelcuttingplaneapproach. Chapters 3 and 4 consideraproblemdealingwiththeecientdeliveryofIntensityModulatedRadiationTherapy(IMRT)toindividualpatients.Inparticular,weconsideramatrixdecompositionproblemthatarisesattheleafsequencingstageinIMRTtreatmentplanning.Theprobleminputisanintegermatrixofintensityvaluesthataretobedeliveredtoapatientfromagivenbeamangle.Todeliverthisintensityproletothepatient,wemustdecomposetheinputmatrixintoacollectionofaperturesandcorrespondingintensities.Afeasibledecompositionisoneinwhichtheoriginaldesiredintensityprolematrixisequaltothesumofanumberoffeasiblebinarymatricesmultipliedbycorrespondingintensityvalues.Tomostecientlytreatapatient,wewishtominimizeameasureoftotaltreatmenttime,whichisgivenasaweightedsumofthenumberofaperturesandthesumoftheapertureintensitiesusedinthedecomposition.InChapter 3 ,wedescribeaversionoftheprobleminwhicheachaperturematrixneedstosatisfythe\consecutiveones"property,whichmeansthatallmatrixentrieswithvalue1ineachrowofanaperturematrixmustbeconsecutive.SimilartoChapter 2 ,weprescribeabilevelhybridoptimizationalgorithminwhichthemasterproblemisanintegerprogrammingproblem,andwesolveasubproblemforeachrowofthematrixby 16 PAGE 17 4 dealswithanothervariantofthematrixdecompositionproblem,inwhichonlyrectangularaperturescanbeusedinthedecomposition.WedevelopaBendersdecompositionalgorithmforsolvingthisvariant.Wealsoproposeaschemeforpartitioningtheproblemtoobtainsimultaneousupperandlowerbounds. InChapter 5 ,westudyaclassofgraphsearchproblems,whereagroupofsearchersseekanintruderonagraph.Boththeintruderandthesearchersarelocatedatsomenodesofthegraph,andthesearchercanonly\see"asubsetofthenodesfromeachnode.Ateachtimeperiod,boththeintruderandthesearcherscanmovealonganedgetoanadjacentnode,orstayatthesamenode.Ourgoalistondtheminimumnumberofsearchersneededtolocatetheintruderwithinagiventimelimit.Weinvestigatethreevariantsofthegraphsearchproblem:(i)ahideandseekproblem,inwhichastationaryintruder\hides"atanunknownnode,(ii)apursuitevasionproblem,inwhichtheintrudermovesamongthenodestoavoidbeingdetected,and(iii)apatrolproblem,inwhichnointruderisinitiallyinthegraphandeachsearcherpatrolsthegraphtoprotectitfrompotentialintrusion.Weformulatetheseproblemsasasetcoveringproblemwithanexponentialnumberofvariablesandconstraints,andproposeabranchcutpricealgorithmforsolvingit.Bothourmasterproblemandthesubproblems,whichcorrespondtotheintruderandthesearchers,havediscretevariables.Weformulatetheintruder'ssubproblemasalongestpathproblemonanauxiliarygraph,andthesearcher'ssubproblemsasmixedintegerprogrammingproblems. WeconcludeourdissertationinChapter 6 ,whichexploresfutureresearchdirectionsregardingmultileveloptimizationalgorithms.Werstevaluatetheapproachestakenintheedgepartition,matrixdecomposition,andgraphsearchproblemsdescribedinChapters 2 { 5 ,anddiscussfutureresearchtopicsregardingeachapplication.Wethendescribeourpreliminaryresearchonamasterproblemreformulationtechnique,whichcanbeusedinavarietyofbileveloptimizationalgorithms.Thisreformulationtechnique 17 PAGE 18 18 PAGE 19 Goldschmidtetal. ( 2003 ),whichisdenedonanundirectedgraphG(N;E).Inthedeterministicedgepartitionproblem,wecreateasetKof(possiblyempty)subgraphsofGsuchthateachedgeiscontainedinexactlyonesubgraph,subjecttocertainrestrictionsonthecompositionofthesubgraphs.Theserestrictionsincludethestipulationsthatanedgecannotbeassignedtoasubgraphunlessbothofitsincidentnodesbelongtothesubgraph,andthatnomorethanrnodescanbeassignedtoanysubgraph,forsomer2Z+.Additionally,eachedge(i;j)2Ehasapositiveweightofwij,andthesumofedgeweightsassignedtoeachsubgraphcannotexceedsomegivenpositivenumberb.Theobjectiveoftheproblemistominimizethesumofthenumberofnodesassignedtoeachsubgraph. (a) (b) Figure21. (a)Aninstanceofthedeterministicedgepartitionproblem(b)AsolutionwithjKj=3;r=3;b=20 Figure 21 illustratesthedeterministicedgepartitionproblem.ThegraphGandthecorrespondingedgeweightsareshowninFigure 21 a.Figure 21 bshowsafeasiblesolutionthatpartitionsGintojKj=3subgraphs,wherethenumberofnodesineachsubgraphislimitedbyr=3,andthetotalweightassignedtoeachsubgraphislimitedbyb=20.Notethatthedegreeofnode4isthree,whichimpliesthatitmustbeassignedtoatleasttwosubgraphs,orelsetherewouldbeatleast4>rnodesinasubgraph.Similarly,node5mustbeassignedtoatleasttwosubgraphs.Sincenodes4and5areassignedtotwo 19 PAGE 20 21 bisoptimal. Goldschmidtetal. ( 2003 )discusstheedgepartitionproblem(withdeterministicweights)inthecontextofdesigningSynchronousOpticalNetwork(SONET)rings.IntheSONETcontext,eachedge(i;j)2Erepresentsademandpairbetweentwoclientnodes,andtheweightwijrepresentsthenumberofcommunicationchannelsrequestedbetweennodesiandj.AlltelecommunicationtraciscarriedoverasetofSONETrings,whicharerepresentedbysubgraphsintheedgepartitionproblem.Sinceeachdemandmustbecarriedbyexactlyonering,edgesmustbepartitionedamongtherings.(Notethattheterm\ring"describesonlythephysicalSONETroutingstructure,anddoesnotplaceanyrestrictionsontopologicalpropertiesofdemandedgesincludedonaring.See,e.g., Goldschmidtetal. ( 2003 )formoredetails.)SONETringsarepermittedtocarrycommunicationbetweennodesonlyifthosenodeshavebeenconnectedtotheringbyequipmentcalledAddDropMultiplexers(ADMs).TherearetechnicallimitsonthenumberofADMsthatcanbeassignedtoeachring(e.g.,r),andonthetotalamountofchannels(e.g.,b)thatcanbeassignedtoaring.SinceADMsarequiteexpensive,ringnetworksarepreferredthatemployasfewADMsaspossible,whichechoestheedgepartitionproblem'sobjectiveofminimizingthesumofnodesassignedtoeachsubgraph. Theprimarycontributionby Goldschmidtetal. ( 2003 )isanapproximationalgorithmforaspeciccaseoftheedgepartitionprobleminwhichallwijareequaltoone. Sutteretal. ( 1998 )proposeacolumngenerationalgorithmforthisproblem,and Leeetal. ( 2000b )employabranchandcutalgorithmonaformulationthatweadapt.Forthecaseinwhichtheweightsontheedgescanbesplitamongmultiplerings, Sheralietal. ( 2000 )prescribeamixedintegerprogrammingapproachaugmentedbytheuseofvalidinequalities,antisymmetryconstraints,andvariablebranchingrules. Smith ( 2005 )formulatesthedeterministicversionoftheedgepartitionproblemasaconstraintprogram 20 PAGE 21 2.3 )canimproveperformance. Weconsideraversionoftheedgepartitionproblemwheretheedgeweightsareuncertain,andareonlyrealizedafterthenodetosubgraphassignmentshavebeenmade.AsweshowinSection 2.2 ,thisframeworkallowsustodesignmorerobustsolutionsthanthoseintheliterature,whicharevirtuallyallappliedtodeterministicdata.Weseekaminimumcardinalitysetofnodetosubgraphassignments,suchthatthereexistsanassignmentofedgestosubgraphssatisfyingtheaforementionedsubgraphrestrictionswithaprespeciedhighprobability.Suchaprobabilisticconstraintisextremelyhardtodealwithinanoptimizationframework.Oneapproach,knownasscenarioapproximation(cf. CalaoreandCampi ( 2005 ); LuedtkeandAhmed ( 2008 ); NemirovskiandShapiro ( 2005 ))istodrawindependentidenticallydistributed(i.i.d.)realizationsoftheedgeweights(calledscenarios)andrequirethenodetosubgraphassignmentstoadmitafeasibleedgetosubgraphassignmentineachscenario.Itcanbeshownthat,withasucientlylargescenarioset,asolutiontothisscenarioapproximationsolutionisfeasibletothetrueprobabilisticallyconstrainedproblemwithhighcondence.Inthisstudywedevelopalgorithmicapproachesforsolvingthescenarioapproximationcorrespondingtothediscussedprobabilisticallyconstrainededgepartitionproblem.Werefertothisscenarioapproximationasthestochasticedgepartitionproblem. RelativelylittleworkhasbeendoneinSONETnetworkdesignwhentheedgeweightsareuncertain. Smithetal. ( 2004 )considertheSONETringdesignprobleminwhichedgedemandscanbesplitamongmultipleringsandproposeatwostageintegerprogrammingalgorithm.Thedemandsplittingrelaxationallowsthesecondstageproblemstobesolvedaslinearprograms,andthusstandardBenderscutscaneasilybederivedfromthesecondstagerecourseproblems.However,wehavesecondstageintegerprogramsfromwhichdualinformationcannotbereadilyobtained. 21 PAGE 22 LaporteandLouveaux 1993 ).Thismethodapproximatesthesecondstagevaluefunctionbylinear\cuts"thatareexactatthebinarysolutionwherethecutisgenerated,andareunderestimatesatotherbinarysolutions.Typicalintegerprogrammingalgorithmsprogressbysolvingasequenceofintermediatelinearprogramming(LP)problems.Usingdisjunctiveprogrammingtechniques,itispossibletoderivecutsfromthesolutionstotheseintermediateLPsthatarevalidunderestimatorsofthesecondstagevaluefunctionatallbinaryrststagesolutions( SenandHigle 2005 ; SheraliandFraticelli 2002 ).Thisavoidssolvingdicultintegersecondstageproblemstooptimalityinalliterationsofthealgorithm,providingsignicantcomputationaladvantage.Scenariowisedecompositionmethodshavealsobeenproposed( CareandSchultz 1999 )asanalternativetotheabovestagewisedecompositionapproaches.Herecopiesoftherststagevariablesaremadecorrespondingtoeachscenarioandarelinkedtogethervianonanticipativityconstraints. Ourproposedmethodologydrawsonconstraintprogrammingandstochasticintegerprogrammingtheory.Hybridalgorithmsofthisnaturehaverecentlybeensuccessfullyemployedtosolvenotoriouslydicultproblems. JainandGrossmann ( 2001 )and BockmayrandPisaruk ( 2006 )devisehybridintegerprogramming/constraintprogrammingalgorithmsforsolvingmachineschedulingproblems. Thorsteinsson ( 2001 )proposesaframeworkforintegratingintegerprogrammingandconstraintprogrammingapproaches. HookerandOttosson ( 2003 )extendtheBendersdecompositionframeworksothatconstraintlogicprogramscanbeusedassubproblemstogeneratecutsthatareaddedtoamixedintegerlinearmasterproblem.Arecentworkby Hooker ( 2007 )useslogicbasedBendersdecompositiontosolveseveralplanningandschedulingproblems. 22 PAGE 23 2.2 ,wedevelopamixedintegerprogrammingformulationforthestochasticedgepartitionproblem,andprovidecuttingplanesthatcanbeusedwithinatwostagedecompositionalgorithm.InSection 2.3 ,weprescribeanalternativethreestagealgorithmtoovercomethecomputationaldicultiesassociatedwiththeweaknessoftheproposedcuttingplanes.Finally,wecomparetheecacyofthesealgorithmsinSection 2.4 onasetofrandomlygeneratedtestinstances. 2.4 ).Wedenotethevectorofnodetosubgraphassignmentsbyx.Let~wdenotetherandomvectorofedgeweightswithknowndistribution,andwdenotearealizationwithcomponentswij.Wedenebinarydecisionvariablesyijk=1ifedge(i;j)isassignedtosubgraphk.Givenanallowedviolationprobability2(0;1)theprobabilisticedgepartitionproblemcanbeformulatedasfollows:MinimizeXi2NXk2Kxik whereG(x;w)=Minimizez 23 PAGE 24 Theobjective( 2{1 )minimizesthetotalnumberofnodesassignedtosubgraphs.Constraints( 2{2 )limitthenumberofnodesassignedtoeachsubgraph.Constraints( 2{6 )requirethattheedgesbepartitionedamongthesubgraphs.Constraints( 2{7 )computethemaximumassignedweightoverallsubgraphs.Constraints( 2{8 )requirethatnoedgecanbeassignedtoasubgraphunlessbothofitsincidentnodesareassignedtothatsubgraph,and( 2{3 )and( 2{9 )statelogicalrestrictionsonthevariables.Byconvention,theoptimalvalueG(x;w)oftheintegerprogram( 2{5 ){( 2{9 )is+1iftheproblemisinfeasible.GivenanodetosubgraphassignmentvectorxandedgeweightvectorwthereexistsafeasibleedgetosubgraphassignmentifandonlyifG(x;w)b,i.e.,theweightassignedtoanysubgraphdoesnotexceedb.Thustheprobabilisticedgepartitionproblem( 2{1 ){( 2{4 )seeksaminimumcostnodetosubgraphassignmentsuchthattheprobabilitythattherewillbeafeasibleedgetosubgraphassignmentwhentheedgeweightsarerealizedissucientlyhigh. Tobuildascenarioapproximationoftheprobabilisticedgepartitionproblem( 2{1 ){( 2{4 ),wegenerateani.i.d.sampleof~wdenotedbyfwqgq2Q(wecalleachrealizationascenario).Thescenarioapproximationisthen:MinimizeXi2NXk2Kxik 24 PAGE 25 LuedtkeandAhmed ( 2008 ),providesjusticationforconsideringthescenarioapproximation. 2{10 ){( 2{13 )isfeasibletotheprobabilisticedgepartitionproblem( 2{1 ){( 2{4 )withprobabilityatleast1. Proof. 2{2 )and( 2{3 ),letXdenotethesetoffeasiblesolutionstotheprobabilisticedgepartitionproblem(satisfying( 2{2 ){( 2{4 )),andletXQdenotethesetoffeasiblesolutionstothestochasticedgepartitionproblemcorrespondingtoasampleQ(satisfying( 2{11 ){( 2{13 )).WewanttoboundjQjsuchthatPrfXQXg1. Considerasolutionx2XnX,i.e.,PrfG(x;~w)bg<1.Thenx2XQifandonlyifG(x;wq)bforallq2Q.Sincethewqforq2Qarei.i.d.itfollowsthatPrfx2XQg(1)jQj.NowPrfXQ6Xg=Prf9x2XQs.t.PrfG(x;~w)bg<1gPx2XnXPrfx2XQgjXnXj(1)jQjjXj(1)jQj: 1: 25 PAGE 26 Nextwedescribeanextensiveformmodelofthestochasticedgepartitionproblem.LetEqbethesetofedgeswithnonzeroweightsunderscenarioq.Wedenebinarydecisionvariablesyqijk=1ifedge(i;j)isassignedtosubgraphkinscenarioqand0otherwise,8q2Q;(i;j)2Eq,andk2K.Thestochasticedgepartitionproblemcanthenbeformulatedasfollows:MinimizeXi2NXk2Kxik Observethatifoneweretosolvetheaboveextensiveformproblemgivenby( 2{15 ){( 2{21 ),integralityrestrictionsneedonlybeimposedontheyvariables,whichwouldinturnenforcetheintegralityofthexvariablesatoptimality.Notealsothatgivenaxedsetofxvalues,thisproblemdecomposesintojQjseparableintegerprograms,wherethesubproblemcorrespondingtoscenarioq2Qisgivenby:Sq(x)=Maximize0 (2{22) 26 PAGE 27 2{16 ),( 2{18 ),( 2{19 ),and( 2{21 ): Undertheforegoingmodel,itisusefultodenevijk=minfxik;xjkgasapartoftherststagedecisionvariables,8(i;j)2E;k2K.Thepresenceofthesevariablesallowustoformulatestrongercuttingplanesthanwouldbepossiblewithjustxvariables(seealso Smithetal. ( 2004 )).Assumingthat[q2QEq=E,theextensiveformproblemisnowequivalentto: MinimizeXi2NXk2Kxik subjecttoXi2Nxikr8k2K where subjecttoXk2Kyqijk=18(i;j)2Eq Thevalidinequalities( 2{27 )requirethatforeachedge(i;j)2E,bothiandjmustbeassignedtosomecommonsubgraph,andareusefulinimprovingthecomputationalecacyofthedecompositionalgorithmthatwepropose.Notethatanoptimalsolutionexistsinwhichvijk=minfxik;xjkg8(i;j)2E;k2K,withoutenforcingintegralityrestrictionsorlowerboundsonthevvariables. 27 PAGE 28 SheraliandSmith 2001 ; Sheralietal. 2000 ).Toreducemodelsymmetrywecanrewritethecardinalityconstraints( 2{25 )(or( 2{17 )fortheextensiveformproblem)byusingthefollowinginequalities: Forascenarioqandagivenvector^v,theproblem( 2{30 ){( 2{33 )isessentiallyanidenticalparallelmachineschedulingproblemtominimizemakespan(P==Cmax)(withsomeassignmentrestrictions).Inparticular,therewouldbejKjmachinesandjEqjjobs,whoseprocessingtimesaregivenbywqij;8(i;j)2Eq.Eachjobmustbeassignedtoexactlyonemachine,andthevvariablesimposesomerestrictionsontheassignments.Theintegerprogrammingschemedevelopedin Smith ( 2004 )istailoredforasimilarprobleminwhichthe(weighted)numberofdemandsthatcannotbeplacedononeofthesesubgraphsisminimized(i.e.,minimumweightednumberoftardyjobs).Thisisnotequivalenttosolvingaminimummakespanproblem;however,theoptimalsolutionofFq(^v)isnomorethanbifandonlyiftheminimumnumberoftardyjobsisequalto0.Ifapositivelowerboundtotheproblemofminimizingthenumberoftardyjobsisestablished,onecanterminatethesubproblemalgorithmandconcludeinfeasibility. Wenowpresentacuttingplanealgorithmforsolving( 2{24 ){( 2{29 ).Theschemerelaxesconstraints( 2{29 )andaddscuttingplanesasnecessarytoenforcefeasibilitytothesubproblems.Letuscalltheproblem( 2{24 ){( 2{28 )themasterproblem(MP). 1. SolveMP.IfMPisinfeasiblethenSTOP;theproblemisinfeasible.Otherwiselet^vbeanoptimalsolutionofMP. 2. Forq2Q,computeFq(^v).IfFq(^v)bforallq,thenSTOP;thecurrentsolutionisoptimal.Otherwise,continuetoStep3. 3. UpdateMPbyaddingacuttingplaneoftheform( 2{37 )aspresentedinRemark1,andreturntoStep1. 28 PAGE 29 LaporteandLouveaux ( 1993 )forthisclassofproblemsisgivenby 2{35 )bydividingbothsidesby(F^q(^v)L^q)androundingdowntoobtain 2{36 ): 2{37 )isvalid,considerasolutionv0thatdoesnotsatisfytheaboveinequality,i.e.,v0ijk=0forall(ijk)2O(^v).Therefore,v0ijk^vijkforall(ijk).ThenFq(v0)Fq(^v)>b,andv0isnotfeasible.Inequality( 2{37 )dominates( 2{36 )sincethelefthandsideof( 2{37 )isnotmorethanthatof( 2{36 ),andtherighthandsidesarebothequalto1.Thus,( 2{37 )servesasacuttingplanethatcanbeusedinStep3oftheabovealgorithm. 2{37 ).Intheparlanceofmachinescheduling,insteadoftryingtominimizethemaximummakespan,wemaywishtominimizethetotalsumoftardiness.Letck;8k2Kbeanonnegativevariablethatdenotestheamountofcapacitydecitinsubgraphk.Then,theproblemofminimizingthetotalcapacitydecitcanbeformulated 29 PAGE 30 Clearly,Fq(v)bifandonlyifTq(v)=0,andsowecanreplacemasterproblemconstraints( 2{29 )withtherestrictionsthatTq(v)=0forallscenariosq2Q.IfsubproblemsTq(v)areusedinlieuofFq(v),wewouldobtain( 2{36 )(directly,thistime)fromLaporteandLouveaux'sintegerfeasibilitycut.However,wecanstateastrongercuttingplaneforasolutionvector^vhavingT^q(^v)>0forsomescenario^q,byrequiringthatthetotalamountofadditionalcapacitythatmustbeallocatedtothecollectionofsubgraphsisatleastT^q(^v).Thisinequalityisformallystatedinthefollowingproposition. 2{37 ): 30 PAGE 31 For(i;j)2E^q,ifyijk=1and^vijk=1,thenset^yijk=1aswell. 2. For(i;j)2E^q,ifyijk=1and^vijk=0,thenset^yij^k=1forany^k2Kforwhich(ij^k)2I(^v).(Notethat(ijk)2O(^v)since^vijk=0.) 3. Setallother^yijk=0. Inotherwords,^yisconstructedintwophases.Intherstphase,weensurethatifedge(i;j)wasassignedtosubgraphkinsolutiony,then(i;j)isassignedtokin^yaswell,unless^vijk=0(prohibitingthisassignment).Inthesecondphase,ifyijk=1but^vijk=0,thenweassign(i;j)toany^ksuchthat^vij^k=1.Notethatthisassignmentresultsinasolutionfeasibleto( 2{39 ),( 2{40 ),and( 2{43 ).Next,letusconstruct^c.Observethatintherstphaseofassigningedgestosubgraphsbasedon(ijk)2I(^v)forwhichyijk=1,nosubgraphcapacitiesareviolatedsinceck=0,8k2K,andsoweinitialize^ck=0,8k2K.Inthesecondphase,weguaranteefeasibilityto( 2{41 )(andmaintainfeasibilityto( 2{42 ))byincreasing^c^kbyw^qij.Thus(^y;^c)isafeasiblesolutiontoMTq(^v). Attheendofthesecondphaseofassignments,wehavePk2K^ck=P(ijk)2O(^v)w^qijvijk,sincePk2K^ckisincreasedbyw^qijonlywhenbothvijk=1and(ijk)2O(^v).However,byassumption,wehavethatP(ijk)2O(^v)w^qijvijk PAGE 32 2{44 )mightbenetfromderivingmultiplecutsforeachinfeasiblescenario,sincethesecutscouldbedistinct. ( 2004 )exploretheinclusionof\warmingconstraints"inthemasterproblem,whichenforcesimplenecessaryconditionsforfeasibilitytoSONETproblems.Denotethedegreeofnodei2Nbydeg(i),andthesetofnodesadjacenttoibyA(i). Leeetal. ( 2000b )showthatnodeimustbeassignedtoatleastldeg(i) 2{34 )needtobeadjustedsothattheyareenforcedseparatelyforsubgraphs1;:::;`^{,and`^{+1;:::;jKj. Sheralietal. ( 2000 )showcomputationallythatsuchavariablexingschemeimprovessolvabilityofprobleminstances. Smithetal. ( 2004 )notethatanodeicannotbeassignedtoasubgraphkinanoptimalsolutionunlessanadjacentnodeisalsoassignedtothesamesubgraph.Therefore,wealsoincludethefollowingconstraintsinMP: 32 PAGE 33 ( 2005 )describesvalidinequalitiesthatcanbederivedbyanalyzingthetopologyofthegraph.First,consideranedge(i;j)2Esuchthat`i=`j=1.LetA(i;j)=A(i)[A(j)fi;jgdenotethesetofdistinctnodesthatareadjacenttoiorj.IfjA(i;j)jr1,theniorjmustbeassignedtoatleasttwosubgraphs.Similarly,wedeneWq(i;j)=Pk2A(i;j)(wqik+wqjk)+wqij,andnotethatifWq(i;j)>bforsomeq2Q,thenwecannotfeasiblyassignnodesiandjtoasinglesubgraph.IfA(i;j)r1orWq(i;j)>b,thenwestatethefollowingvalidinequality: Smith ( 2005 )showsthatnodesiandjcollectivelyneedtobeassignedtoatleastfoursubgraphs,whichwestateas: 2.2 arepreferabletosolvingstochasticedgepartitioninstancesbytheextensiveformproblemgivenby( 2{15 ){( 2{21 ),asweshowinSection 2.4 .However,thetwostagecuttingplanealgorithmsstillsuerfromseveralcomputationaldiculties.First,themasterproblem,MP,containsjNjjKjbinaryvariables,jEjjKjcontinuousvariables,andO(jEjjKj)constraints,whichresultsinlargeintegerprograms.Second,thelinearprogrammingrelaxationofMPisquiteweakformanyprobleminstances.Furthermore,thelowerboundimprovesslowlyascutsofthetype( 2{37 )or( 2{44 )areaddedtoMPineachiteration.ThemainreasonforthisslowconvergenceistheexistenceofsymmetryinMP.Inequalities( 2{34 )reduce,butdonotcompletelyeliminate,symmetricsolutionsinMP.Therefore,whenasolutionofMPisfoundtobeinfeasibletoasubproblem,MPoftensimplyswitchestoasymmetricsolution 33 PAGE 34 Inthissectionwedevelopanewdecompositionframeworktoremedythesediculties.Wecombatsymmetryduetoreshuingofsubgraphsbyrepresentingsubgraphsascongurations.AcongurationcisidentiedbyasubgraphnodesetNc(weallowNc=;)andapositiveintegerc,whichgivesthenumberofsubgraphshavingnodesetNc.AsolutionisrepresentedbyacongurationmultisetCwhoseelementsarepairs(Nc;c).Weeliminatesymmetrybyensuringthatnoisomorphiccongurationmultisets(i.e.,thosethatareidenticalafterreindexingcongurationindices)areencounteredinoursearch. AcongurationmultisetCsatisesthefollowingnecessaryfeasibilityconditions. AmultisetCthatsatisesF1,F2,andF3representsafeasiblesolutionifalledgescanbepartitionedonthesetofsubgraphscorrespondingtoCwithoutviolatingtheweightrestrictionsforanyscenario.NotethatthenumberofdistinctcongurationsinC,whichwedenotebyjCj,isdynamicallydeterminedinouralgorithm. Wenowprovideanoverviewofourthreestagehybridalgorithm. 1. Therststageproblemdetermines(viaoptimalsolutionofamixedintegerprogram)thenumberoftimesweassigneachnodetothecongurationsinC.Forinstance,intheexamplegiveninFigure 21 a,wecouldspecifythatwemustusetwocopiesofnodes4and5,andonecopyoftheothernodes. 2. Inthesecondstage,weseekamultisetCthatusesexactlythenumberofnodeassignmentsspeciedintherstphaseandsatisesF1,F2,andF3.Intheexamplementionedabove,amultisetChavingcongurationsf1;2;4g,f3;4;5g,andf5;6g(eachwithmultiplicityone)couldbegeneratedbasedontherststagesolution. 34 PAGE 35 Finally,inthethirdstage,wedeterminewhetherCisfeasible.IfCisfeasiblethenwestopwithanoptimalsolution.Else,wereturntothesecondstage,andgenerateadierentmultisetmeetingthestatedcriteria.Ifnosuchmultisetexists,acutisaddedtotherststageproblem,whichisthenresolved.Fortheexamplegivenabove,themultisetyieldsafeasiblesolution(seeFigure 21 b). (2{51)`izijKj8i2N (2{53) where`iisalowerboundonthenumberofcopiesrequiredfornodei,asgivenin( 2{45 ).Toformulatetherststageproblemasanintegerprogram,werewrite( 2{51 )asanexponentialsetoflinearinequalitiesbyconsideringthezvectorsthatviolateit.Werstneedtointroduceauxiliarybinaryvariablestik;8i2N;k=`i;:::;jKj,sothattik=1ifzi=k.Then,givenavector^zthatdoesnotinduceafeasiblemultiset,wenotethatnozsuchthatzi^zi;8i2N,inducesafeasiblemultiset.Hence,atleastonecomponentof^zmustbeincreased,andso isavalidinequality.Ourrststageproblemcannowbeexpressedasthefollowingintegerprogram:MinimizeXi2Nzi 35 PAGE 36 whereZisthesetofallzvectorsthatdonotinduceafeasiblemultiset.(Thezvariablesareinfactunnecessaryinthisformulation,butwekeepthemforeaseofexposition.)Inouralgorithmwerelaxconstraints( 2{58 )intherststageproblem,andaddtheminacuttingplanefashion.Ineveryiterationwesolvetherststageproblemtond^z,andsolvethesecondandthirdstageproblemstoseekafeasiblemultisetinducedby^z.Ifafeasiblemultisetisfound,then^zinducesanoptimalsolutionandwestop.Otherwise,weaddacutoftype( 2{58 )andresolvetherststageproblem. Smith ( 1995 ), LustigandPuget ( 2001 ),and Rossietal. ( 2006 )forathoroughdiscussionofconstraintprogrammingtechniques. 36 PAGE 37 Forinstance,consideravenodegraph,andletthezvectorobtainedbytherststageproblembe^z=(2;3;1;4;3).Supposethatnodes1,2,and3havebeenprocessed,andthefollowingpartialmultisetwithjCj=3hasbeenobtained: Supposethatweprocessnode4bychoosingitsdistributionvectoras^4=(2;1;1).Addingnode4totwoofthevecopiesofN1createsanewcongurationN01whosenodesetconsistsonlyofnode4(withmultiplicitytwo)andreducesthemultiplicityofN1bytwo.Aftersimilarlyaddingonecopyofnode4toN2andonecopyofnode4toN3,weobtainthefollowingpartialmultisetwithjC0j=5: Ingeneral,whenweprocessnodeibychoosingadistributionvectori,weupdatethepartialmultisetCasfollows.Foreachcongurationc2Cific=0,thennochanges 37 PAGE 38 InthebeginningofthesecondstageweinitializeourmultisetCwithasinglecongurationhavingN1=;and1=jKj.Eachnodecanonlybeaddedtotheloneconguration,andsothedomainfornodeiisinitiallythesingleonedimensionalvectori=(^zi).Ouralgorithmnextprocessessomenodei2Nandupdatestheexistingsetofcongurations:N1=;;1=jKj^ziandN2=fig;2=^zi.Next,thedomainsofallunprocessednodesareupdatedtoreectthechangesinC.Foreachunprocessednodej,weenumerateallpossiblewaysofpartitioning^zjcopiesintonodesetsN1andN2.Thislogicisrepeatedatallfuturestepsaswell.Forinstance,intheexamplegivenabove,supposethat^5=(2;0;1)wastheonlyvectorinthedomainofnode5beforeprocessingnode4.Sinceprocessingnode4modiestherstcongurationbyreducing1andgeneratesanewconguration(N01;01),weexpandthedomainofnode5byenumerating 38 PAGE 39 ToenforceF2,thepropagationalgorithmidentiesallcongurationstowhichrnodeshavebeenassigned.Foreachsuchcongurationc,weremovealldistributionvectorsjhavingjc>0fromthedomainsofallunprocessednodesj2N.ToenforceF3,thepropagationalgorithmiteratesoverthedomainsoftheunprocessednodesjadjacenttoi,andremovesalldistributionvectorsthatdonotaddatleastonecopyofjtoanycongurationinCi.Otherwise,thecongurationscontainingnodeiwouldbedisjointfromthosecontainingnodej,whichviolatesF3. 39 PAGE 40 vanBeek 2006 ). Wecallonesuchtestimpliednodeassignmentanalysis.Supposethatweidentifyaprocessednodeisuchthat^zi=1,andthecongurationctowhichihasbeenassigned.ByconditionF3itfollowsthatallunprocessednodesjadjacenttoimustalsobeassignedtocongurationc.Weusethisanalysistoaugmentpartialcongurationswithimpliednodeassignments,andthencheckwhetheranyaugmentedcongurationcontainsmorethanrnodes,andhenceviolatesF2. Wealsoperformanimpliededgeassignmentanalysisbyndingalledgesthatcanonlybeassignedtoasingleconguration.Foreach(i;j)2E,ifbothnodesiandjhavebeenprocessed,thenwecheckwhetherbothiandjareinasinglecongurationcforwhichc=1.Inthiscaseedge(i;j)canonlybeassignedtocongurationc.Ontheotherhandif(withoutlossofgenerality)nodeihasbeenprocessedbutnodejhasnotyetbeenprocessed,and^zi=1,thenedge(i;j)canonlybeassignedtothecongurationtowhichihasbeenassigned.Afterndingallimpliededgeassignments,wecheckwhetherF3isviolatedforanyscenario. Finally,weconsiderasingletonnodeanalysis,inwhichweensurethateachnodeisadjacenttoatleastoneothernodeineachconguration.Foreachprocessednodei,andforallcongurationsc2Ci,weseekanodejadjacenttoisothateitherj2Nc(ifjalsohasbeenprocessed),orjc>0forsomedistributionvectorinthedomainofj(ifjhasnotbeenprocessed).Ifnosuchjcanbefoundforacongurationc2Ci,thenthecurrentpartialsolutioncannotleadtoanoptimalsolution;nodeicanultimatelyberemoved 40 PAGE 41 LustigandPuget 2001 ; Smith 1995 ).Especiallyforinfeasiblesecondstageprobleminstances,processingthe\problematic"nodesrstcanquicklyleadtothedetectionofinfeasibilityandcanresultinsignicantsavingsincomputationaltime.Weemployadynamicnodeselectionruleinwhichtheorderofnodesconsideredcanvaryindierentsectionsofthesearchtree.Inaccordancewiththe\failrst"principlewidelyusedinconstraintprogrammingalgorithms( HaralickandElliott 1980 ; vanBeek 2006 ),ournodeselectionrulerstpicksanunprocessednodethat 1. hasthefewestnumberofdistributionvectorsinitsdomain, 2. hasthefewestnumberofcopiestobepartitioned,and 3. hasthelargestnumberofunprocessedadjacentnodes, breakingtiesinthegivenorder.Inthismanner,wecanquicklyenumerateallpossibledistributionvectorsofafewkeynodes,allowingconstraintpropagationtoquicklyreducethesizeoftheremainingsearchspace. 41 PAGE 42 2{38 ){( 2{43 )tosolvethethirdstageproblem. Notethatthistransformationreintroducessymmetryintothethirdstageproblem.However,thesolutionofthethirdstageproblemsdoesnotconstituteabottleneckinthealgorithm,andsymmetrybreakingconstraintsappendedtothetransformedsubproblemwillnotimpactthecomputationalecacyoftheoverallalgorithm. 2{58 )statethatthenumberofcopiesofsomenodemustbeincreased,buttheydonotcontainanyinformationaboutwhichnodesneedtobeadded.Weobservethattheprogressofoursecondstagealgorithmcanbeanalyzedtoidentifya\problematic"subsetofnodeswhosecorrespondingzvaluescauseinfeasibilityregardless 42 PAGE 43 Constraints( 2{61 )clearlydominate( 2{58 )foranyPN,andgetstrongerasjPjdecreases.Basedonthisobservation,weupdateournodeselectionrulebygivingpreferencetoselectingnodesthathavealreadybeenaddedtoP.Ourrevisednodeselectionrulerstpicksanodethat 0. hasbeenaddedtoPinapreviousbacktrackingstep, 1. hasthefewestnumberofdistributionvectorsinitsdomain, 2. hasthefewestnumberofcopiestobepartitioned,and 3. hasthelargestnumberofunprocessedadjacentnodes, againbreakingtiesinthestatedorder. 3 canbeadaptedtotherststageproblemtoeliminatethezvectorsthatviolatethecorrespondingnecessaryfeasibilityconditions.Inparticular,constraints( 2{46 )translatetosimplelowerbounds( 2{52 )onthezvariables.Constraints( 2{48 ),whicharewrittenfornodepairsthatsatisfythe 43 PAGE 44 3 ,canbewrittenas: Similarly,eachconstraintoftype( 2{49 )canbeequivalentlyrepresentedasfollowing: Smith ( 2005 )discussesanadditionalvalidinequality,whichcannotberepresentedusingthexvariablesinourtwostagealgorithm,butcanbewrittenintermsofthezandtvariablesintherststageproblemofourhybridalgorithm.Fornodesi2Nandj2N,if(i;j)=2E;deg(i)r1;deg(j)r1;jA(i;j)jr1,andthereexistsacommonneighbork2Nsothatk2A(i);k2A(j);deg(k)r,andifi;j;khavemorethan2r4distinctneighborsintotal,thenzi=1;zj=1implieszk3.Thisconditioncanbewrittenas: whichreducestozk3forzi=zj=1,andisredundantotherwise. 2{62 ){( 2{64 )toobtainaninitialsolution^z,andexecutethesecondandthirdstagealgorithmstoseekafeasiblemultiset.Ifoneisfound,weterminatewithanoptimalsolution.Otherwise,weinvestigatethesetofprocessednodes^PN,andpickanode^{2^Phavingthefewestnumberofcopies(breakingtiesbypickinganodehavingthelargestdegree).Wethenset^z^{=^z^{+1andreinvokethesecondandthirdstagealgorithms.Thisalgorithmeventuallyndsafeasiblemultisetorconcludesthattheentireproblemisinfeasibleaftergeneratingthesolution^zi=jKj;8i2 PAGE 45 2{61 )foreach^zgeneratedbeforeafeasiblemultisetisfound,whichweaddtotherststageproblemtoimprovethelowerbound. 2{61 ),andagainresumethesolutionprocess.Thesameideaisalsoapplicabletothemasterproblem(MP)ofthetwostagealgorithmdiscussedinSection 2.2 Inourtests,thisapproachturnedouttobemoreeectivethansolvingtherststageproblemtooptimalityineachiteration,addingacut,andresolvingit.Thereasonisthattheproblemissolvedusingasinglebranchandboundtree,whichwetightenbyaddingcutsasnecessaryonintegralnodes,insteadofrepeatedlygeneratingabranchandboundtreeineachiteration.Italsoallowsustoobtaingoodfeasiblesolutionsforprobleminstancesthataretoodiculttosolvetooptimality. Wenotethatthisapproachrequiresaminormodicationtothesecondstagealgorithm.Allconstraintpropagation(Section 2.3.2.3 )andforwardcheckingrules(Section 2.3.2.4 )exceptforsingletonnodeanalysisarebasedonnecessaryconditionsforfeasibilityofcongurations,andthereforetheyarevalidforanyintegral^z.However,singletonnodeanalysisisbasedonanoptimalityconditionandhencecanonlybeusedif^zisacandidateoptimalsolutiontotherststageproblem. jNj(jNj1))takesvalues0.2,0.3,and 45 PAGE 46 Goldschmidtetal. ( 2003 ); Sheralietal. ( 2000 ); Smith ( 2005 )).ChoosingjKjtoosmallmaymaketheprobleminfeasible,andlargevaluesofjKjincreasedicultyoftheproblem.Inourtests,wechosejKjsucientlylargetoyieldafeasibleedgepartitionineachprobleminstance.Ingeneratinginstanceswerstpickedarandomsubsetofedgestohaveapositiveweight,andthenweassignedaweightuniformlydistributedbetween1and10toeachedgeineachscenario.Wegeneratedveprobleminstancesforeachproblemsize,whichisdeterminedbytheexpectededgedensity,thenumberofnodes,andthenumberofscenarios.ThedatasetnamesanddetailsusedinourexperimentsaregiveninTable 21 Table21. Descriptionsoftheprobleminstancesusedforcomparingalgorithms 2{27 ),( 2{46 ){( 2{49 ),andthesymmetrybreakingconstraints( 2{34 ),andusesthemodelgivenby( 2{38 ){( 2{43 )forthesubproblem,whichistheformulationthatminimizesthetotaltardiness.Inourbasesettingforthethreestagealgorithm,weusedourheuristictondaninitialfeasiblesolution,generatedvalidinequalities( 2{62 ){( 2{64 ),and(similartothetwostagealgorithm)weusedformulation( 2{38 ){( 2{43 )forthethirdstageproblem.WeusedcallbackfunctionsofCPLEXtogenerateasinglebranchandboundtreeforbothtwostageandthreestagealgorithmsasdiscussedinSection 2.3.5 .Weimposedahalfhour 46 PAGE 47 Ourrstexperimentcomparestheperformanceoftheextensiveform,twostage,andthreestagealgorithms.Table 22 summarizestheresultsofthesethreealgorithmsonlowdensitygraphshavingexpectededgedensity0:2.Foreachproblemsize,wereportthefollowingstatisticscalculatedoververandominstances:(i)thenumberofproblemssolvedtooptimality(\Solved"),(ii)theaverageoptimalitygapobtainedattherootnode(\RootGap"),(iii)theaveragenaloptimalitygapforinstancesthatcouldnotbesolvedwithintheallowedtimelimit(\FinalGap"),(iv)theaverageamountoftimespentbyeachalgorithmontheinstancesthatweresolvedtooptimality(\Time").Outofthe75instancesinthisdataset,CPLEXcouldsolvetheextensiveformtooptimalityfor61instances,whilebothtwostageandthreestagealgorithmssolvedall75instancestooptimalitywithinafewseconds.Theresultsrevealthattheperformanceoftheextensiveformformulationdeterioratesrapidlyasthenumberofscenariosincreases,buttheeectofthenumberofscenariosismitigatedforthetwostageandthreestagealgorithms.Weobservethattheaverageoptimalitygapobtainedbythethreestagealgorithmattherootnodeis1:46%,whichissignicantlylessthantheinitialgapsobtainedusingotherapproaches. Table22. Comparisonofthealgorithmsongraphshavingedgedensity=0:2 TwoStage ThreeStage RootFinal RootFinal RootFinalName SolvedGapGapTime SolvedGapGapTime SolvedGapGapTime 51 50.00%0.1 55.00%0.1 50.00%0.1530 518.33%6.6 54.00%0.2 50.00%0.15100 512.38%5.4 511.00%0.6 50.00%0.381 525.90%0.4 56.67%0.1 50.00%0.1830 512.89%4.0 53.64%0.2 50.00%0.18100 537.61%223.1 514.84%1.2 50.00%0.3101 519.58%0.5 517.80%0.4 50.00%0.11030 557.01%147.3 510.71%0.8 50.00%0.210100 430.35%7.14%684.1 513.94%2.0 50.00%0.4121 547.25%8.1 524.66%2.2 50.00%0.11230 455.09%25.00%507.3 517.99%4.3 53.08%0.312100 262.21%24.88%713.1 536.28%4.7 52.11%0.8151 531.85%33.0 564.38%16.4 54.56%0.21530 165.29%21.65%864.6 539.13%27.1 57.29%0.615100 057.33%28.47%524.49%20.4 54.86%1.2 PAGE 48 23 and 24 comparethethreeapproachesondensergraphshavingedgedensity0:3(mediumdensity)and0:4(highdensity),respectively.Weobservethatperformancesofallthreealgorithmsdeteriorateastheedgedensityincreases,whichisnotsurprisingduetothenatureoftheedgepartitionproblem.Thenumberofinstances Table23. Comparisonofthealgorithmsongraphshavingedgedensity=0:3 TwoStage ThreeStage RootFinal RootFinal RootFinalName SolvedGapGapTime SolvedGapGapTime SolvedGapGapTime 51 50.00%0.1 52.86%0.1 50.00%0.1530 525.76%10.4 56.15%0.5 50.00%0.15100 510.00%3.1 510.77%0.4 50.00%0.281 531.30%0.5 511.20%0.1 50.00%0.1830 542.57%18.0 57.48%0.4 50.00%0.28100 439.37%7.14%110.0 516.19%1.3 51.43%0.3101 532.42%3.7 516.27%0.6 51.18%0.11030 451.33%21.05%953.0 540.82%8.2 55.83%0.310100 261.24%29.05%382.6 535.43%302.7 58.89%0.5121 553.85%312.0 539.49%16.8 54.65%0.21230 063.41%27.06%546.98%120.5 59.31%0.812100 084.24%65.50%442.78%4.35%89.0 511.99%1.4151 446.88%11.54%460.4 572.86%250.5 512.93%0.91530 066.01%42.41%272.20%16.02%30.4 513.51%3.515100 080.76%74.48%053.05%13.21%516.31%4.1 24 clearlyshowsthatthethreestagealgorithmdominatestheotherapproaches,andthetwostagealgorithmprovidesbetterresultsthandirectlysolvingtheextensiveformulation.OuranalysisofoptimalsolutionsobtainedfortheprobleminstancesshowninTables 22 { 24 showedthattheaverageobjectivefunctionvalueforthedeterministic(singlescenario)probleminstancesis14.8.Thisvalueissmallerthantheaverageobjectivefunctionvaluefor30scenarioand100scenarioinstances(15.52and15.6,respectively).Wealsoobservethatseveralsubgraphscanbeemptyinanoptimalsolution. 48 PAGE 49 Comparisonofthealgorithmsongraphshavingedgedensity=0:4 TwoStage ThreeStage RootFinal RootFinal RootFinalName SolvedGapGapTime SolvedGapGapTime SolvedGapGapTime 51 55.00%0.1 50.00%0.1 50.00%0.1530 524.67%2.6 519.79%0.3 50.00%0.15100 512.38%5.6 58.31%0.6 50.00%0.281 541.32%2.0 53.33%0.1 50.00%0.1830 548.89%140.9 517.68%1.1 51.43%0.18100 347.23%22.50%113.0 521.08%8.7 52.50%0.4101 545.08%48.3 532.36%3.5 52.16%0.11030 061.52%20.64%556.55%39.5 58.45%0.410100 064.47%50.91%354.82%7.50%151.7 512.73%1.5121 167.13%14.30%33.2 540.60%327.3 57.86%0.51230 088.61%46.74%542.93%160.9 53.16%0.812100 084.37%68.24%551.54%583.7 513.91%1.7151 260.11%11.21%369.6 353.01%5.56%410.0 511.57%0.91530 085.29%65.29%066.72%22.66%318.03%4.74%120.215100 096.00%86.92%062.62%24.58%319.98%6.45%173.8 Descriptionsoftheprobleminstancesusedforanalyzingthreestagealgorithm 25 .Similartoourpreviousexperiments,we Table26. ThreeStagealgorithmongraphshavingedgedensity=0:2 RootFinalHeuristic RootFinalHeuristicName 5 40750.00%0.82.86% 219450.00%3.50.00%8 83750.00%2.21.54% 434350.00%12.73.33%10 1169510.88%5.65.09% 600651.33%19.61.33%12 172451.11%13.24.19% 877953.00%58.53.33%15 214057.24%22.12.74% 10858512.41%170.94.37%17 2417510.19%41.04.78% 1224559.82%211.18.01%20 2833516.55%79.87.01% 14324512.40%403.74.51%22 3110514.77%128.26.49% 15710515.78%699.96.46% 1 .Hence,inequality( 2{14 )ensuresthatwecanbe95%(99%,respectively)certainthatalldemandscanbesatised95%(99%,respectively)ofthetime.Wegeneratedverandominstancesforeachdataset, 49 PAGE 50 ThreeStagealgorithmongraphshavingedgedensity=0:3 RootFinalHeuristic RootFinalHeuristicName 5 40750.00%0.82.86% 219450.00%3.50.00%8 83750.00%2.62.86% 434353.33%13.02.86%10 116956.58%7.58.99% 6006511.86%27.44.80%12 172458.61%16.14.51% 877958.22%71.74.31%15 2140515.45%45.13.05% 10858418.53%3.45%176.44.25%17 2417513.63%42.43.43% 1224559.93%189.32.68%20 2833517.47%362.53.24% 14324518.13%639.13.32%22 3110416.18%4.76%159.35.82% 15710515.86%738.53.45% 22 ,Tables 26 27 ,and 28 showtherelativegapbetweenthequalityofthesolutionfoundbyourinitialheuristic(Section 2.3.5.2 )andthebestlowerboundobtained(\HeuristicGap").Ouralgorithmcansolve206instancesoutof240tooptimality,andprovidesanaverage Table28. ThreeStagealgorithmongraphshavingedgedensity=0:4 RootFinalHeuristic RootFinalHeuristicName 5 40750.00%0.82.22% 219450.00%3.20.00%8 83757.71%2.72.43% 434357.25%17.35.33%10 1169516.38%9.45.71% 6006515.84%52.19.73%12 1724516.71%63.25.41% 8779514.13%118.45.45%15 2417416.24%2.86%338.84.07% 12245216.55%4.71%549.86.86%17 2140123.46%8.46%993.79.23% 10858124.77%11.44%1515.513.07%20 2833018.83%9.46%9.46% 14324020.01%11.30%11.81%22 3110018.36%11.05%11.47% 15710017.83%9.90%10.29% 50 PAGE 51 31 (a) (b) Figure31. (a)Amultileafcollimatorsystem(b)Theprojectionofanapertureontoapatient 51 PAGE 52 Sincethemid1990's,largescaleoptimizationoftheuenceappliedfromanumberofbeamorientationsaroundapatienthasbeenusedtodesigntreatmentsfromMLCequippedlinearaccelerators.AtypicalapproachtoIMRTtreatmentplanningistorstselectthenumberandorientationsofthebeamstouseaswellasanintensityproleoruencemapforeachofthesebeams,wheretheuencemaptakestheformofamatrixofintensities.Thisproblemhasbeenstudiedextensivelyandcanbesolvedsatisfactorily,inparticularwhen(asiscommoninclinicalpractice)thebeamorientationsareselectedmanuallybythephysicianorclinicianbasedontheirinsightandexpertiseregardingtreatmentplanning.Foroptimizationapproachestotheuencemapoptimizationproblemwithxedbeamorientationswerefertothereviewpaperby Shepardetal. ( 1999 ).Morerecently, Romeijnetal. ( 2006 )proposednewconvexprogrammingmodels,and HamacherandKufer ( 2002 )and Kuferetal. ( 2003 )consideredamulticriteriaapproachtotheproblem. Leeetal. ( 2000a 2003 )studiedmixedintegerprogrammingapproachestotheextensionoftheuencemapoptimizationproblemthatalsooptimizesthenumberandorientationsofthebeamstobeused.However,toenabledeliveryoftheoptimaluencemapsbytheMLCsystem,theyneedtobedecomposedintoacollectionofdeliverableapertures.(Forexamplesofintegratedapproachestouencemapoptimization,alsoreferredtoasaperturemodulation,wereferto Shepardetal. ( 2002 ), PreciadoWaltersetal. ( 2004 ),and Romeijnetal. ( 2005 ).) ThevastmajorityofMLCsystemscontainacollectionofleavesthatcanbemovedinparallel,therebyblockingpartoftheradiationbeam.Thisarchitectureimpliesthatwecanvieweachbeamasamatrixofbeamletsorbixels(thesmallestdeliverablesquarebeamthatcanbecreatedbytheMLC),sothateachaperturecanberepresentedbyacollectionofrows(or,byrotatingtheMLChead,columns)ofbixels,eachofwhich 52 PAGE 53 Baataretal. ( 2005 )):264364215375: 53 PAGE 54 Bortfeldetal. ( 1994 )proposedthesweepmethod,which AhujaandHamacher ( 2005 )(whoderivedanequivalentmethod)showedtoindeedyieldanoptimalsolution;otherexactalgorithmswereproposedby Kamathetal. ( 2003 ),and Siochi ( 1999 ).Inaddition, Baataretal. ( 2005 ), Bolandetal. ( 2004 ), Kalinowski ( 2005a ), Kamathetal. ( 2004a b c d ), Lenzen ( 2000 ),and Siochi ( 1999 )studiedtheproblemofminimizingbeamontimeunderadditionalhardwareconstraints,while Kalinowski ( 2005b )studiedthebenetsofallowingrotationoftheMLChead. AlthoughthetimerequiredbytheMLCsystemtotransitionbetweenaperturesformallydependsontheaperturesthemselves,thefactthatthesetimesaresimilarandthepresenceofsignicant(apertureindependent)vericationandrecordingoverheadtimesjustiestheuseofthetotalnumberofsetups(or,equivalently,thetotalnumberofapertures)tomeasurethetotalsetuptime.Inaddition,deliveringIMRTwithasmallnumberofaperturesprovidestheadditionalbenetsoflesswearandtearonthecollimators(lessstoppingandstarting)andalesserrorpronedeliveryasIMRTdeliveryerrorsareknowntobeproportionaltothenumberofapertures(see Stelletal. ( 2004 )).TheproblemofdecomposingauencemapintotheminimumnumberofrowconvexapertureshasbeenshowntobestronglyNPhard(see Baataretal. ( 2005 )),leadingtothedevelopmentofalargenumberofheuristicsforsolvingthisproblem.Notableexamplesaretheheuristicsproposedby Baataretal. ( 2005 )(whoalsoidentifysomepolynomially 54 PAGE 55 AgazaryanandSolberg ( 2003 ), DaiandZhu ( 2001 ), Que ( 1999 ), Queetal. ( 2004 ), Siochi ( 1999 2004 2007 ), VanSantvoortandHeijmen ( 1996 ), XiaandVerhey ( 1998 ).Inaddition, Engel ( 2005 ), Kalinowski ( 2005a ),and LimandChoi ( 2007 )developedheuristicstominimizethenumberofapertureswhileconstrainingthetotalbeamontimetobeminimal.Finally, Langeretal. ( 2001 )developedamixedintegerprogrammingformulationoftheproblem,while Kalinowski ( 2004 )proposedanexactdynamicprogrammingapproachfortherelatedproblemofminimizingthenumberofaperturesthatyieldstheminimumbeamontime. Baataretal. ( 2007 )describedintegerprogrammingandconstraintprogrammingmodelsforthesameproblem,and Ernstetal. ( 2009 )proposedaconstraintprogrammingapproachforminimizingthenumberofapertures.However,computationalstudiesreportedin Baataretal. ( 2007 ); Ernstetal. ( 2009 ); Kalinowski ( 2004 ); Langeretal. ( 2001 )revealthattheseapproachescanonlybeusedtoecientlysolvesmallprobleminstancestooptimality.Ourprimarycontributionisthatwedeveloptherstalgorithmcapableofsolvingclinicalprobleminstancestooptimality(ortoprovablynearoptimality)withinclinicallyacceptablecomputationaltimelimits. Inthischapter,ourfocusisontheproblemofndingadecompositionofauencemapintorowconvexaperturesthatminimizestotaltreatmenttime,asmeasuredbythesumofthetotalsetuptimeandbeamontime.InSection 3.2 wedevelopourdecompositionbasedsolutionapproach.InSection 3.3 wediscusstheapplicationofouralgorithmonacollectionofclinicalandrandomlygeneratedtestdata,andcompareitsperformancewithalternativeexactandheuristictechniques. 55 PAGE 56 WestartthissectionbydescribingadecompositionframeworkfortheoptimalleafsequencingprobleminSection 3.2.1 andusethistoformulateourmasterprobleminSection 3.2.2 .WeintroduceoursubprobleminSection 3.2.3 ,proveitscomplexity,andprovideacombinatorialsearchalgorithmforitssolution.WethenenhancetheempiricalperformanceofourdecompositionalgorithmbyintroducingclassesofvalidinequalitiestothemasterprobleminSection 3.2.4 ,andnallydescribeanalgorithmforconstructingafeasiblesolutionwithmedicallydesiredpropertiesinSection 3.2.5 Ehrgottetal. ( 2008 )).However,thisapproachisnotdirectlyapplicablewhentheobjectiveistominimizethetotaltreatmenttime. Eventhoughtheoptimalleafsequencingproblemisnotdirectlydecomposablebyrows,thefactthatleavescorrespondingtodierentrowscanbepositionedindependentlycanstillbeexploited.Denoteaparticularpositioningofleftandrightleavesforarowasaleafposition;anapertureiscomposedofaleafpositionforeachrowofB.Ourmainobservationisthatgivenacollectionofintensities,whichcanbeusedinaperturesthatcollectivelycovertheuencemap,therowsareindependentofoneanother.Thatis,wecandeterminetheleafpositionstobeusedforcoveringeachrowindependently,andthenformaperturesforcoveringtheentireuencemapbycombiningindividualleafpositionsforeachrowthatareassignedtothesameintensity. 56 PAGE 57 Considertheallowableintensitymultisetf1,3,5g.Assigningeachofthesevaluestoatmostoneleafposition,therstrowcanbedecomposedas [148]=1[110]+3[011]+5[001]; sothattheallowableintensitymultisetiscompatiblewiththerstrow.Similarly,thesecondrowcanbedecomposedas [385]=3[110]+5[011]: However,therstbixelinthethirdrowmustbecoveredbytwoleafpositionsassignedtointensities1and3,andthesecondbixelmustbecoveredbyasingleleafpositionassignedtointensity5.Therefore,allallowableintensitiesmustbeusedtocoverthersttwobixels,andthethirdbixelwithrequiredintensity3cannotbecovered.Hence,theallowableintensitymultisetisnotcompatiblewiththethirdrow.Alternatively,consideranallowableintensitymultisetthatcontainsthevalues1,3,and4forthesameuencemap.Therowscanbedecomposedas [148]=1[111]+3[011]+4[001]; PAGE 58 (3{4) [453]=1[010]+3[001]+4[110]: Themasterproblemcanthereforesuccinctlybewrittenas minimizew1LX`=1x`+w2LX`=1`x`(3{6)subjectto 58 PAGE 59 whereofcourse~Tcannotbelessthantheminimumachievablebeamontime~z(whichcanbefoundinpolynomialtimeusingthealgorithmsmentionedinSection 3.1 ). Toformulateourmasterproblemasanintegerprogrammingproblem,weintroducebinaryvariablesy`r,8`=1;:::;L,r=1;:::;R`,wherey`r=1ifandonlyifx`=r,andR`isanupperboundonthenumberofapertureshavingintensity`usedinanoptimalsolution.(WecancomputeR`bycomputinganinitialupperboundontheoptimalobjectivefunctionvalueviaanyoftheheuristicsmentionedinSection 3.1 ,andthensettingR`tothelargestvaluesuchthatw1R`+w2`R`isnomorethanthisbound.)Usingthesedecisionvariables,wecanreformulatethemasterproblem(MP)asfollows: minimizew1LX`=1x`+w2LX`=1`x`(3{10)subjectto Wenextformulate( 3{13 )asasetoflinearinequalitiesbyderivingvalidinequalitiesthatcutopreciselythosevectorsxthatviolate( 3{13 ).Tothisend,consideraparticular 59 PAGE 60 Sinceallintegersolutionsexceptfor^xsatisfy( 3{16 ),itisindeedavalidinequality.Constraint( 3{16 )canbetightenedbyobservingthatifthesolution^xisincompatiblewithrowi,thenanysolutionxsuchthatx`^x`,8`=1;:::;L,isalsoincompatiblewithrowi.Therefore,werequirethatxcontainatleastonecomponentthatislargerthanitscorrespondingcomponentin^x,whichyieldsthestrongervalidinequality Constraint( 3{17 )can,inturn,betightenedfurtherbyexplicitlyconsideringtherowsforwhichxisincompatible.LetLi=maxj=1;:::;nbijbethemaximumintensityintheuencemapforrowi.Bythesameargumentasabove,ifthecurrentsolution^xisincompatiblewithrowi,thenanysolutionxsuchthatx`^x`,8`=1;:::;Li,isalsoincompatiblewithrowi,sincenoleafpositionswithintensitygreaterthanLicanbeusedindecomposingrowi.Therefore,werequirethatxislargerthan^xinatleastonecomponent1;:::;Li: Since( 3{18 )isstrongerthan( 3{16 )or( 3{17 ),weusethelatterinequalitiesinourmodel.Notealsothat( 3{18 )statedforrowi1dominatesacutgeneratedforrowi2ifLi1 PAGE 61 3.2.3 Sincethecollection( 3{18 )containsanexponentialnumberofvalidinequalities,weaddthemonlyasneededinacuttingplanefashion.Inparticular,thismeansthatwerelax( 3{18 ),solvetherelaxationof(MP)andgenerateanxsolutionrepresentingacandidateallowableintensitymultiset.Wethensolveasubproblemforeachbixelrowtodetermineiftheallowableintensitymultisetisincompatiblewiththatrow.Ifnot,wehavefoundanoptimalsolutionto(MP).Otherwise,weaddaconstraintoftheform( 3{18 )to(MP)thatcutsothatsolution. Werepresentafeasibledecompositionasacollectionofndimensionalbinaryvectorsv`r(`=1;:::;L;r=1;:::;x`).Thevaluesofv`rthatequal1correspondtothe(consecutive)exposedbixelsintherthaperturehavingintensity`.Forexample,thedecompositioninequation( 3{2 )correspondstov11=(1;1;0),v31=(0;1;1),v51=(0;0;1),andv`r=0forother`;r.(Notethatthisdecompositionwouldbefeasibleaslongasx1;x3;x51.)Thesubproblemcanthenformallybepresentedasfollows: QUESTION:Dothereexistndimensionalbinaryvectorsv`r(`=1;:::;L;r=1;:::;x`)thatsatisfytheconsecutiveonespropertysuchthatPL`=1Px`r=1`v`r=b? PAGE 62 Inprinciple,theC1Partitionproblemcanbeformulatedandsolvedasanintegerprogram.However,wehavedevelopedacomputationallymoreeectivebacktrackingalgorithmthatfocusesonpartitioningintensityrequirementsindividuallyforeachbixel.Anintegervectorpj=(pj1;:::;pjL)providesabixeldecompositionofbixelj2f1;:::;nginrowbifandonlyifbj=PL`=1`pj`.Wethenattempttoformacollectionofleafpositionsthatrealizestheindividualbixelpartitions.Wecallsuchacollectionofleafpositionsaleafdecompositionofb. Tomoreeectivelyconductoursubproblemsearches,wedescribeapropertythatholdsinsomeleafdecomposition(ifoneexists)thatsatisesthegivencollectionofbixeldecompositions. Proof. Wenextderiveanecessaryconditionthatanyfeasiblebixeldecompositionmustsatisfysothatthecorrespondingsetofleafpositionsiscompatiblewithagivenallowableintensitymultisetx.SimilartotheideabehindLemma 1 ,ifpj`>pj+1`,thenpj`pj+1`leafpositionshavingintensity`mustexposebixeljbutnotj+1.Lemma 2 formalizesthisidea. PAGE 63 Wenextdescribeourbacktrackingalgorithm.Inthisalgorithm,werstenumerateallpossiblewaysofdecomposingthebixelintensitiesinbusingasubsetoftheallowableintensitymultisetgivenbyx.WedenotethesetofallcandidatebixeldecompositionsforbixeljbyPj,whereforeachp2[nj=1Pj,wemusthavep`x`;8`=1;:::;L. ThebacktrackingalgorithmforsolvingthesubproblemisstatedformallyinAlgorithm 1 .WebeginbyenumeratingeachpossibleelementofPj,8j=1;:::;n.WedenotethesetofprocessedbixelsbyF(forwhichacandidate\active"bixeldecompositionhasbeenestablished),andthesetofunprocessedbixelsbyR.Ineachiteration,wechecktoseeifthesetofcandidatebixeldecompositionsPjforanyj2Risempty.Ifso,thecurrentactivebixeldecompositionsdonotyieldafeasiblesolution,andthealgorithmbacktracks.Otherwise,weconsideranunprocessedbixel^2R,andchooseanuntriedbixeldecompositionp^2P^tobeactiveforbixel^.Next,wemove^fromRtoF,creatingupdatedsetsR0andF0,andinvokeLemma 2 toupdatethesetofbixeldecompositionsforthebixelsinR0.Specically,foreachj2R0andpj2Pj,wecalculatethenumberofleafpositionsthatwouldberequiredduetoselectingpjastheactivebixeldecompositionforbixelj,inadditiontothosealreadyselectedforbixelsinF0.Weeliminatepjifaconditionoftype( 3{19 )isviolated.Wethenrecursivelycalltheproceduretocontinuewithanewbixelj02R0. Westopeitherwhenwendafeasiblebixeldecompositionforallbixels,orwhenweexhaustallbixeldecompositionswithoutndingafeasiblesolution.Intheformercase,aleafdecompositionthatrealizesthebixeldecompositionsforbixelsj2f1;:::;ngcanbe 63 PAGE 64 3 ,whichisbasedontherepeatedapplicationofLemma 1 .ToseethatAlgorithm 3 recoversafeasibleleafdecomposition,notethatAlgorithms 1 and 2 providebixeldecompositionsthatsatisfyLemma 2 ,andinparticular,thecondition Algorithm 3 recoversafeasibleleafdecompositionif,intheouterwhileloopcorrespondingtoeach`=1;:::;L,thecounterrisneverincrementedmorethanx`times.Notethatrisincrementedeachtimetheinnerwhileloopterminates,whichoccurseitherwhen~>n(atotalofpn`times),orwhenp~`=0(p~1`p~`times)for~=2;:::;n.Thetotalnumberoftimesthatrisincrementedintheouterwhileloopfor`=1;:::;Listhusthelefthandsideof( 3{20 ),whichisnomorethanx`,asrequired. Ifweexhaustallbixeldecompositionswithoutndingafeasiblesolution,weconcludethatthecurrentallowableintensitymultisetisincompatiblewiththecurrentrow. fThisalgorithmndswhetherthereexistsaC1Partitionofbcompatiblewithxg F;fFisthesetofprocessedbixelsg Rf1;:::;ngfRisthesetofunprocessedbixelsg SinceAlgorithm 1 isabacktrackingalgorithm,andthereforeintheworstcaseinvestigatesallpossiblebixeldecompositions,itisofexponentialtimecomplexity(asexpected,duetoProposition 3 ).However,theempiricalrunningtimeofthealgorithmcanbereducedusingthefollowingobservations: (i) Iftwoadjacentbixelsinarowhavethesamerequiredintensityvalue,theremustexistanoptimalsolutioninwhichtheyareexposedbythesameleafpositions.ThisresultcanbeproveninasimilarwayasLemma 1 ,andisthereforeomittedforbrevity.Thisobservationimpliesthatwecanpreprocessthedatabymerging 64 PAGE 65 returntruefallbixelshavebeenprocessed,Prepresentsafeasiblesolutiong if9j2R:Pj=;then returnfalsefthereisnoremainingwayofdecomposingbixeljg F0F[f^g,R0Rnf^g 2 ,giventheactivedecompositionsp~for~2F0g returntruefafeasiblesolutionthatusesp^todecomposebixel^isfoundg v`r0 forall`2f1;:::;Lgdo ifpj`>0then PAGE 66 (ii) Inchoosingthenextbixeltobeprocessed,wepickabixelj2Rhavingthesmallestnumberofremainingcandidatebixeldecompositions.Inthismanner,wecanquicklyenumerateallpossiblebixeldecompositionsforafewkeybixelsandeliminateasignicantportionofbixeldecompositionsfortheremainingbixelswithoutwastingeortbyunnecessarybacktrackingsteps. (iii) Inchoosingthenextcandidatebixeldecompositionpj2Pjforachosenbixelj2R,weselectanuntriedbixeldecompositionhavingthefewestnumberofintensityvalues.Sinceeachintensityvalueusedindecomposingabixelneedstobeassignedtoadierentaperture,thisrulefavorsabixeldecompositionusingthefewestnumberofaperturestodecomposethechosenbixel.Therefore,ittendstoretaintheavailabilityofmoreelementsoftheallowableintensitymultiset(andhenceapertures)fortheremainingbixels,makingiteasiertondafeasiblesolution(ifoneexists). 3{18 )haveyetbeenaddedtothemodelwillsetallvariablesequaltozero,whichisclearlyincompatiblewithallrows.Inthissection,wederivesomecharacteristicsofallfeasiblesolutionsandusethesetodenevalidinequalitiesfor(MP).Inthisway,weattempttoimprovetheconvergencerateofthedecompositionalgorithmbyeliminatingsomeclearlyinfeasiblesolutionsbeforetheinitialexecutionofthemasterproblem. 66 PAGE 67 WeformulateanintegerprogrammingmodeltodetermineTi(w01;w02)foragivenrowi.First,denotethesetofpossibleleafpositionsforthatrowbyK,anddenendimensionalbinaryvectorsvkfork2K(wherejKj=O(n2)),suchthatvkj=1ifandonlyifbixeljisexposedbyleafpositionk.Inadditiontodecisionvariablesx`asin(MP),denebinarydecisionvariableszk`,8k2K,`=1;:::;Lksuchthatzk`=1ifandonlyifleafpositionkisusedwithintensity`(whereLk=minj:vkj=1bjisanupperboundontheintensityofleafpositionk.)ThenTi(w01;w02)istheoptimalobjectivefunctionvalueofthefollowingoptimizationproblem,(SR): minimizew01LX`=1x`+w02LX`=1`x`(3{22)subjectto Constraints( 3{23 )ensurethateachbixelreceivesexactlyitsrequiredamountofdosewhileconstraints( 3{24 )guaranteethateachleafpositioniseithernotusedorisassignedtoasingleintensityvalue.Finally,constraints( 3{25 )relatethexandzvariables. Apracticaldicultyinimplementingthevalidinequalitiesoftheform( 3{21 )isthatwemustdetermineappropriatevaluesfortheweightsw01andw02.However, Baatar ( 2005 )showsthat,whendecomposingasinglebixelrow,thereexistsasetofleafpositions 67 PAGE 68 3{21 )by Itiseasytoseethatwecancaptureallofthesevalidinequalitiesbyrestrictingourselvestothecoecientpairs(w01;w02)=(1;0)and(0;1)only: Wecangeneralizethisideaasfollows.LetR(L)denotethesetofrowsforwhichthemaximumintensityrequirementisboundedbyLforsomeL2f1;:::;Lg,i.e.,R(L)=fi2f1;:::;mg:LiLg.SinceintensityvaluesgreaterthanLcannotbeusedindecomposingtherowsinR(L),asimilarapproachtotheoneabovecanbeusedtoderivethefollowingfamilyofvalidinequalities Finally,notethatthevaluesofNiand~zicanbefoundbysolving(SR)withw01=1;w02=1orbyusingthemethodof Kalinowski ( 2004 ),sincethereexistsasolutionthatminimizesbothbeamontimeandthenumberofapertures( Baatar 2005 ). 3{16 ){( 3{18 )representnecessaryconditionsforfeasibilityofanallowableintensitymultisetwithrespecttoaparticularrow.Itispossibletodevelopstronger 68 PAGE 69 3 showsthat,ifagivenallowableintensitymultisetisincompatiblewithasubsequencesofrowi,thenitalsomustbeincompatiblewithrowi. Proof. NotethatwecaninvokeLemma 3 toassociateasubproblemwitheachoftheO(2n)subsequencesofabixelrowb.Eachofthesesubproblemscanthenbeusedtogeneratecuttingplanesoftheform( 3{18 ),aswellasvalidinequalitiesoftheform( 3{31 )and( 3{32 ).However,sincethestrengthof( 3{18 ),( 3{31 )and( 3{32 )dependonthelargestintensityvalueinabixelrow,weformsubsequencesofeachbixelrowby,forL=1;:::;L,consideringonlythosebixelshavingrequiredintensitylessthanorequaltoL.ThevalidinequalitiesgeneratedbytheO(min(n;L))subsequencesgeneratedinthisfashionimplyallO(2n)validinequalitiesassociatedwithallpossiblesubsequences. 3 tondaleafdecompositionforeachrow.Wecanthengenerateaperturematricesbyarbitrarilycombiningleafpositionsusingthesameintensityvaluesindierent 69 PAGE 70 Sinceanypairofleafpositionsassignedtothesameintensityvalueindierentrowscanbecombined,thereareuptoQL`=1(x`!)maperturematricesthatcanbeconstructedfromagivenfeasibleleafdecompositionforeachrow.Eventhougheachsuchchoicerepresentsanalternativeoptimalsolutiontotheoptimalleafsequencingproblem,somematrixdecompositionsmayclinicallybepreferabletoothersbasedontheirstructuralproperties.Perhapsthemostchallengingstructuralconsiderationpertainstothesocalled\tongueandgroove"eectobservedinMLCs.Wereferthereadertotheworksof Dengetal. ( 2001 )and Queetal. ( 2004 )fortechnicaldetailsofthetongueandgrooveeectindynamicMLCdosedelivery.Forthepurposesofthisstudy,itissucienttounderstandthatleavesinadjacentrowsofteninterlockwithatongueonthebottomofonerowslidingalongagrooveinthetopofanotherrow.Tongueandgrooveunderdosageoccurssincealeaf'stongueblocksdosageintendedforcellsbeneathit.Therefore,itisdesirabletolimitsuchunderdosages. Tomeasuretheamountoftongueandgrooveeectinatreatmentplan, Queetal. ( 2004 )notethatitisgenerallynotdesirabletodeliveroneapertureinwhichsomebixel(i;j)isblockedbyaleafwhilebixel(i+1;j)isnotblocked,ifanotherapertureisbeingdeliveredwhere(i;j)isnotblockedbyaleafwhile(i+1;j)isblocked.Basedonthisobservation, Queetal. ( 2004 )derivethefollowingtongueandgrooveindex(TGI).SupposeatreatmentplanconsistsofKaperturesdescribedbybinaryvaluesvikj,wherevikj=0ifcell(i;j)isblockedbyaleafinaperturekandvikj=1otherwise,foreachi=1;:::;m,j=1;:::;n,k=1;:::;K.LetIkbetheintensitydeliveredinaperturek=1;:::;K.ThentheTGIofamatrixdecompositionisdenedas:m1Xi=1nXj=1K1Xk=1KX`=k+1minfIk;I`ghvikj(1vi+1;kj)(1vi`j)vi+1;`j+(1vikj)vi+1;kjvi`j(1vi+1;`j)i: 70 PAGE 71 Givenapairofadjacentrows,weattempttomatchindividualleafpositionsinthetworowstominimizetheTGIinducedbytheadjacentrowpair.Tolimitcomputationaloverheadinthisphaseofouralgorithm,wereduceTGIindirectlybythefollowingscheme.Letusdenotealeafpositionforrowibyabinarynvectorvi,wherevij=1iftheleafpositionexposesbixeljinrowi.Wemeasuretheoverlapbetweentwoleafpositionshavingthesameintensityvalueinconsecutiverowsbycountingthenumberofcolumnsthatbothleafpositionsexposesimultaneously.Formally,wedenetheoverlapbetweenleafpositionsviandvi+1as(vi;vi+1)=Pnj=1vijvi+1j.OurapproachistoheuristicallyminimizeTGIbymaximizingthetotaloverlapbetweenallleafpositionpairs,whichcanecientlybesolvedasanassignmentproblem.Theeciencyoftheassignmentproblemscanbefurtherimprovedbynotingthattheproblemdecomposesovertheintensityvalues`2f1;:::;Lg,sinceonlyleafpositionshavingthesameintensityvaluecanbecombined.Therefore,wecangenerateamatrixdecompositionbyndingaleafdecompositionforeachrow,andthenmatchingleafpositionsinadjacentrowshavingthesameintensityvaluebysolvinganassignmentproblemsothatthetotaloverlapismaximized. TheTGIminimizationstepdescribedinthepreviousparagraphcanbeimprovedasfollows.Typically,multiplebixeldecompositionsexistforeachrowthatarecompatiblewithagivenfeasibleintensitymultiset.Algorithm 2 canbemodiedinastraightforwardmannersothatitndsallleafdecompositionsofarow,insteadofstoppingoncetherstfeasiblebixeldecompositionforallbixelsisfound.Sincedierentbixeldecompositionsforabixelrowcorrespondtodierentleafdecompositions,consideringalternativebixeldecompositionscanleadtoamatrixdecompositionhavingasmallerTGI. 71 PAGE 72 2 byterminatingonce250feasiblebixeldecompositionshavebeenidentied.Next,notethatastraightforwardacyclicshortestpathimplementationprocesseslayersoneatatime,anddoesnotgenerateafeasibleS{Fpathbeforeprocessingthelastlayer.Sincebeingabletospecifyatimelimitisadesiredfeatureinapracticalsetting,weuseahybridalgorithmforsolvingtheshortestpathproblem.Ouralgorithmstartsbyprocessinglayersonebyone,updatingnodelabelsasusual.Ifashortestpathisnotfoundwhenagiveninitialtimelimitexpires,ouralgorithmswitchestoadepthrstsearch(DFS)procedure,whichweterminateafteragivennaltimelimit.WestartDFSfromanunprocessednodeNidhavingasmallestlabel,selectaminimumcostarc(Nid;N(i+1)d0)exitingthatnode,andupdatethelabelofN(i+1)d0ifwehavefoundanewshortestS{N(i+1)d0path.Else,the PAGE 73 3.3.1ProblemInstances 31 reportstheproblemcharacteristicsfortheseprobleminstancesintermsofthematrixdimensionsmandn.ThemaximumintensityvalueisL=20foralltheseinstances.Inaddition,toallowcomparisonofourresultswithpublishedresultsonotherapproachestotheproblem,wegenerated100randomprobleminstancesofdimensions2020havingmaximumintensityvalueL=10. However,sincetheseprobleminstancesaregenerallytoolargetobesolvablebytheintegerprogrammingmodelfrom Langeretal. ( 2001 )anditsmodicationdescribedinAppendixA,wealsorandomlygeneratedeightinstances(\test5x5a",:::,\test6x7b")todemonstratethecomputationallimitationsofthelatterapproaches.Unlessotherwisespecied,weusedw1=7andw2=1astheobjectiveweightsforthenumberofaperturesandbeamontime,respectively. Table31. Dimensionsofclinicalprobleminstances c2b11820c3b12217c4b11922c5b11516c1b21115 c2b21719c3b21519c4b21324c5b21317c1b31515 c2b31818c3b32017c4b31823c5b31416c1b41515 c2b41818c3b41917c4b41723c5b41416c1b51115 c2b51718c3b51519c4b51224c5b51217 73 PAGE 74 3{31 )and( 3{32 )describedinSection 3.2.4.2 .WealsouseEngel'sheuristic( Engel 2005 ),whichexecutesinwellunderoneCPUsecondforeachinstanceandgeneratesasolutionhavingminimumbeamontime,to(i)obtainaninitialupperboundand(ii)computetheupperboundsR`(`=1;:::;L). Langeretal. ( 2001 )andtothemodicationoftheirmodelasdescribedinAppendixA.Wechooserandomlygeneratedtestinstancesofvariousdimensionstoidentifytheproblemsizesthatcanbesolvedbyeachalgorithm,aswellasfourofthesmallestclinicalinstancestocomparetheeectivenessofthealgorithmsonclinicalinstances.Weimposedaonehourtimelimitpastwhichwehaltedtheexecutionofanalgorithm.FortheseexperimentswedisabledtheuseofEngel'sheuristicasaninitialheuristictotesttheabilityofthesemodelstoecientlyndgoodqualityupperbounds. Table 32 summarizestheresultsofthesethreealgorithmsintermsoftheexecutiontime,thebestupperandlowerboundsfoundwithinthetimelimit,andtheoptimalitygap(calculatedasthedierencebetweentheupperandlowerboundasapercentageoftheupperbound).Ourdecompositionalgorithmcansolveall15instancesinthisdatasetwithinafewseconds,whereasonlysixinstancescanbesolvedtooptimalitywithinanhourbyeitherintegerprogrammingformulation.Weconcludethat,eventhoughtheintegerprogrammingformulationgivenin( Langeretal. 2001 )cansolvesmallinstancestooptimality,itcannotbeusedtosolveclinicalprobleminstancestooptimalitywithinpracticalcomputationtimelimits. 74 PAGE 75 Comparisonofourbasealgorithmwith Langeretal. ( 2001 )model Langer ModiedLangerNamemnL CPUUBLBGap CPUUBLBGap test3x3338 0.129 1.62929.000.0% 0.92929.000.0%test3x4348 0.137 5.23737.000.0% 1.63737.000.0%test4x4448 0.136 30.43636.000.0% 10.73636.000.0%test5x5a5510 0.245 2069.64545.000.0% 86.44545.000.0%test5x5b5515 0.250 198.25050.000.0% 92.65050.000.0%test5x6a5610 0.255 36006133.5345.0% 36005540.9525.5%test5x6b5618 0.471 36008451.5838.6% 36007758.6723.8%test6x6a6613 0.355 36005545.6317.0% 36005548.0012.7%test6x6b6613 0.352 36005743.8223.1% 36005750.0012.3%test6x7a7610 0.245 690.04545.000.0% 435.14545.000.0%test6x7b6715 0.474 36009435.6962.0% 36008047.8840.1%c1b1151420 1.3111 360033648.5885.5% 360027342.0084.6%c1b2111520 0.8104 360028038.2686.3% 360013239.5570.0%c1b5111520 3.1104 360028046.2083.5% 360014049.2964.8%c5b4141620 2.5124 360036034.0090.6% 360036039.1189.1% Siochi ( 2007 ), Engel ( 2005 ),and XiaandVerhey ( 1998 ),whichweexecutedonthesamedataset.(Theresultswepresentfrom Siochi ( 2007 )refertotheVariableDepthRecursion(VDR)algorithmwithouttongueandgrooveconstraints,usingtheparametersrecommendedinthepaper.Wediscusstheeectofincludingtongueandgrooveconsiderationsinthealgorithmbelow.) Figure 32 summarizesthetotaltreatmenttimesassociatedwiththesolutionsgeneratedbythesixalgorithmswetested.Eachalgorithmisrepresentedbyacurvethatdepictsqualityofthesolutionsobtainedbythecorrespondingalgorithm.ForeachvalueToftotaltreatmenttimeonthehorizontalaxis,eachcurveplotsthenumberofprobleminstancesforwhichthecorrespondingalgorithmwasabletondasolutionhavingtotaltreatmenttimenomorethanT.Forinstance,Figure 32 showsthatSiochi'sheuristicfoundasolutionwithatotaltreatmenttimeofatmost175timeunitsin5%oftheprobleminstances,whileanoptimalsolution(representedby\TotalTime")has 75 PAGE 76 Comparisonoftotaltreatmenttimesonrandomdata thesamequalitylevelin97%oftheprobleminstances.Weobservethatallthreeexactalgorithmsndsolutionshavingsimilartreatmenttimes.SolutionqualitiesgeneratedbytheEngelandSiochiheuristicsaresimilar,withtheSiochiheuristicbeingslightlybetter.AcomparisonoftheheuristicsolutionswithoptimalsolutionsrevealsthataverageoptimalitygapsforSiochi,EngelandXiaVerheyheuristicsare10.1%,12.0%,and51.5%,respectively. Figure 33 comparesthealgorithmswithrespecttothenumberofaperturesusedintheirrespectivesolutions.Wenotethatouralgorithmthatminimizestotaltreatmenttime(\TotalTime")ndsasolutionthatalsominimizesthenumberofaperturesformostprobleminstances.Asexpected,lexicographicminimizationofthetwoobjectivefunctions 76 PAGE 77 Comparisonofthenumberofaperturesonrandomdata resultsinanincreasednumberofapertures.Forthisobjectivethe\#Apertures"algorithmndsoptimalsolutions.AverageoptimalitygapsfortheheuristicsofSiochi,Engel,andXiaVerheyare15.6%,18.9%,and62.3%,respectively. WeanalyzethebeamontimevaluesofthesolutionsgeneratedbyeachalgorithminFigure 34 .SincebothEngel'sheuristicandour\Lexicographic"algorithmndoptimalsolutionshavingminimumbeamontime,theircurvesoverlap.WeobservethattheSiochiheuristicandour\TotalTime"algorithmtendtogeneratesolutionshavingsmallbeamontimevalues,butthesolutionsgeneratedbyour\#Apertures"algorithm,andbytheXiaVerheyheuristichavehigherbeamontimevalues.Wecalculatedtheaverageoptimalitygapsforthelattertwoalgorithmsas12.6%and32.1%,respectively. 77 PAGE 78 Comparisonofbeamontimevaluesonrandomdata ThenalmeasureofsolutionqualitythatweconsiderisTGI,whichisameasureofthetongueandgrooveeectgivenby( 3{33 ).Figure 35 revealsthatthesolutionsobtainedbyallthreevariantsofourdecompositionalgorithmhavesignicantlylowerTGIvaluesthantheheuristicprocedures.Thisresultimpliesthat,eventhoughourTGIreductionalgorithmdescribedinSection 3.2.5 doesnotguaranteeaminimumTGI,itishighlyeectiveinndingsolutionswithTGIvaluessuperiortotheotherheuristicapproaches.Toestimateoptimalitygapsfortheheuristicswecompareheuristicsolutionswiththesolutionsgeneratedbyour\Lexicographic"algorithm,whichprovidesthebestTGIamongallmethodsmentionedabove.WenotethataveragegapsforSiochi,EngelandXiaVerheyheuristicsare162.1%,164.4%,and205.4%,respectively.Wealsonote 78 PAGE 79 ComparisonofTGIvaluesonrandomdata thattheseheuristicsdonotattempttominimizeTGI,anditmightbepossibletomodifythemtoobtainsolutionswithlowerTGIvalues.ItisinterestingtonotethatavariantofSiochi'salgorithm( Siochi 2007 )iscapableofcompletelyeliminatingTGIattheexpenseofcreatingadditionalapertures.Thisvariantisreportedtoincreasethenumberofaperturesby10%to30%relativetothevariantthatdoesnotremoveTGI( Siochi 2007 ). Finally,theEngelandXiaVerheyheuristicstooklessthanonesecondofCPUtimeinallinstanceswetested.TheaverageCPUtimeforSiochi'sheuristic,\TotalTime"algorithm,\#Apertures"algorithm,and\Lexicographic"algorithmwere31.5,963.1,414.8,and421.4seconds,respectively.Wenotethatallvariantsofourtwostagealgorithmshoweda\heavytail"behavior,whereabout80%oftheprobleminstancesweresolvedto 79 PAGE 80 3.1 thatinclinicalpractice,wecandelivereachuencemapusingadecompositionintoeitherrowconvexorcolumnconvexapertures,wherethelatterrequiresrotationoftheMLChead.Ournalsetofexperimentscomparesthealgorithmsonclinicalprobleminstancesinourdataset,allowingforMLCheadrotation. Werstshowtheresultsofapplyingourdecompositionalgorithmtodecomposeeachofthe25clinicaluencemapsintorowconvexapertures,andcolumnconvexapertures,wherethelatterisachievedbyapplyingouralgorithmtothetransposeofeachuencemap.Table 33 reportstheperformanceofouralgorithmwhentheobjectivefunctionissettominimizetotaltreatmenttime,anddisplaysthenumberofapertures(\nAper"),beamontime(\BOT"),totaltreatmenttime(\Time"),tongueandgrooveindex(\TGI"),andCPUtimeused(\CPU")forthealgorithm. Ouralgorithmndsanoptimalsolutiontoseveralinstanceswithinafewsecondswhilefourinstancestakemorethan10minutesofCPUtimetobesolvedtooptimality.Comparingthesolutionsobtainedforrowconvexandcolumnconvexdecompositions,weobservethatrotatingtheMLCheadismostbenecial(intermsoftreatmenttime)forinstancesinwhichthenumberofrowsismuchsmallerthanthenumberofcolumns.Thesebenetsaremostapparentoninstancesc4b2andc4b5,whererotatingtheMLCheadcanresultinmorethan50%reductionintotaltreatmenttime.Wealsonotethatseveralprobleminstancesrequiremuchlesscomputationaltimetosolveforacolumnconvexdecompositioncomparedtoarowconvexdecomposition. 80 PAGE 81 EectofrotatingtheMLChead ColumnConvexName nAperBOTTimeTGICPU nAperBOTTimeTGICPU c1b1 10411111021.1 1138115505.5c1b2 1034104800.8 82379140.7c1b3 11311089711.4 92891591.0c1b4 11331107437.0 11371141467.0c1b5 10341041334.3 83288491.2c2b1 143413213426.5 123011418711.5c2b2 134113215920.1 11331101928.0c2b3 134914024514.7 11281051513.1c2b4 145114931687.3 12341181488.3c2b5 1341132217395.6 1027971202.0c3b1 1341132323310.0 144013825423.0c3b2 14461443204759.8 82379861.1c3b3 134914053310373.9 124012436018.6c3b4 1244128481524.9 1240124327428.2c3b5 13341251333.3 92790752.6c4b1 164015221634.9 124613024410.6c4b2 166918145020901.0 9279014915.8c4b3 144113913044.7 10321021293.3c4b4 1444142246164.3 1027971638.0c4b5 177619547014511.4 92487484.0c5b1 102696680.5 1035105410.5c5b2 12411255914.3 82581270.6c5b3 10341041553.1 92386421.0c5b4 12401241052.2 1032102874.3c5b5 124613015151.9 83187170.8 Computationalresultsforourbasealgorithm #Apertures LexicographicName nAperBOTTGICPU nAperBOTTGICPU nAperBOTTGICPU c1b1 10411024.7 10411022.3 1138504.8c1b2 823141.1 823141.1 823141.1c1b3 928593.0 928594.5 928594.5c1b4 11337441.2 113712827.1 11337412.2c1b5 832492.1 834561.3 92691.9c2b1 123018715.6 123018714.9 123018714.4c2b2 113319210.8 11381616.9 11331467.8c2b3 11281498.9 11281139.9 112819710.8c2b4 123414816.8 123414816.8 123414817.1c2b5 10271206.1 10311556.2 10271206.2c3b1 1341323315.0 125152162.1 134132531.4c3b2 823864.4 826874.5 823625.6c3b3 124036027.4 1240360894.7 124036520.1c3b4 1240327442.2 1246284548.8 133892855.1c3b5 927755.6 927755.4 927755.7c4b1 124624416.8 124622710.6 124622711.3c4b2 92714945.5 93215056.2 92713535.0c4b3 103212915.7 103410814.9 103212915.6c4b4 102716332.0 102811232.6 11267229.9c4b5 9244827.8 9244827.7 9244827.0c5b1 1026681.2 1026681.2 1026681.2c5b2 825271.1 825271.0 92381.1c5b3 923423.6 924453.2 923833.1c5b4 1032875.8 10411012.7 1032872.8c5b5 831171.4 833161.2 831711.1 PAGE 82 34 showsthenAper,BOT,TGI,andCPUmetricsobtainedfromouralgorithmenhancedwiththeaboveboundingscheme,correspondingtothe\TotalTime,"\#Apertures,"and\Lexicographic"objectives.Observethatall25instances,underanymetric,terminateinunder15minutesofCPUtimewithasolutionthatisoptimalwithrespecttothecorrespondingobjective,andallinstancesaresolvedtooptimalitywithinaminuteusingthe\Lexicographic"algorithm. Recallthatthe\BOT"columnin\Lexicographic"reportstheminimumachievablebeamontime,andthe\nAper"columnundertheobjective\#Apertures"reportstheminimumnumberofaperturesneededtodecomposeeachinstance.Perhapssurprisingly,incomparingthesevalueswiththeresultsof\TotalTime,"weobservethatthereexistsasolutionthatminimizesboththenumberofshapesandthebeamontimesimultaneouslyin19ofthe25instances. Finally,weanalyzeperformanceofthethreeheuristicsonclinicaldata,whereweexecuteeachheuristiconeachprobleminstanceanditstranspose(correspondingtorowconvexandcolumnconvexdecompositions),andpickthesolutionyieldingthesmallesttreatmenttime.Table 35 showsthenumberofapertures,beamontime,TGI 82 PAGE 83 Comparisonofheuristicalgorithmsonclinicaldata Engel XiaVerheyName nAperBOTTGICPU nAperBOTTGICPU nAperBOTTGICPU c1b1 113824514.0 1238261<1 1340219<1c1b2 8231093.0 823127<1 1032133<1c1b3 9282134.0 1028192<1 1234198<1c1b4 12343069.5 1137398<1 1442355<1c1b5 9261033.6 926175<1 1235124<1c2b1 123065211.2 1230738<1 1545635<1c2b2 123339517.8 1233464<1 1545460<1c2b3 122862534.8 1228429<1 1543459<1c2b4 123462843.2 1234723<1 1856417<1c2b5 112746315.3 1127465<1 1441375<1c3b1 144382836.3 15401054<1 1755765<1c3b2 92314311.1 923127<1 1236289<1c3b3 1440131640.8 1440869<1 19601038<1c3b4 134867833.3 1438765<1 1755553<1c3b5 9282637.0 927325<1 1345261<1c4b1 134661729.4 1446625<1 1862531<1c4b2 102929573.8 1027466<1 1444350<1c4b3 113233919.4 1132365<1 1448428<1c4b4 112648913.5 1126540<1 1546424<1c4b5 92423689.8 924328<1 1544328<1c5b1 11261884.6 1226176<1 1238185<1c5b2 9231296.9 923100<1 1033145<1c5b3 9262015.1 1023293<1 1232189<1c5b4 113221811.2 1132322<1 1346243<1c5b5 8322177.2 931211<1 1135138<1 34 revealsthateventhoughtheheuristicsconsistentlygeneratedhighqualitysolutions,theSiochiandEngelheuristicswereabletondanoptimalsolutioninonlyveprobleminstances,andXiaVerheyheuristiccouldnotndanoptimalsolutiontoanyinstance. 83 PAGE 84 Externalbeamradiationtherapyisdeliveredfrommultipleanglesbyadevicethatcanrotatearoundapatient.Theuseofmultiple(typically3{9)anglesisoneofthetoolsthatallowforthetreatmentofdeepseatedtumorswhilelimitingtheradiationdosetosurroundingfunctioningorgans.Conventionalconformalradiationtherapythenfurtherusesblocksandwedgestoshapethebeams(see,e.g., Lim ( 2002 )and Limetal. ( 2004 2007 )).IMRTisamorepowerfultherapythatinsteadmodulatesbeamintensity.Themostcommontechniqueforachievingthismodulationistodynamicallyshapebeamswiththehelpofamultileafcollimator(MLC)system.Suchsystemscandynamicallyformmanycomplexaperturesbyindependentlymovingleafpairsthatblockpartoftheradiationbeam.Unfortunately,MLCsystemsareverycostlyandtechnologicallyadvanced,andarethereforedicultandexpensivetooperateandmaintain.Moreover,MLCsystemsarecurrentlyonlyavailableforusewithasocalledlinearacceleratorthatgenerateshighenergyphotonbeamsfortreatment.However,theuseofradioactive60Co(Cobalt)sourcesforradiationtherapyisstillubiquitousinmanypartsoftheworldandis 84 PAGE 85 Earletal. ( 2007 )).Recently,researchershavebeguntoexploretheclinicalfeasibilityofdeliveringIMRTusingconventionaljawsthatarealreadyintegratedintoradiationdeliverydevicesandcancreateaperturesthatarerectangularinshape(see,e.g., Earletal. ( 2007 ), Kimetal. ( 2007 ),and Menetal. ( 2007 )).Successfulapplicationofthismuchsimplerdeliverytechniquedependscriticallyontheabilitytoecientlydeliverhighqualitytreatmentplans.Wethereforedevelopandtestnewoptimizationapproachestominimizethetreatmenttimerequiredforaparticulartreatmentplanusingrectangularaperturesonly. SolvingasocalleduencemapoptimizationproblemyieldsanoptimalIMRTtreatmentplanthatresolvesdierent,andconicting,clinicalmeasuresoftreatmentplanqualityrelatedtotumorcontrolandsideeects(see,e.g., Shepardetal. ( 1999 )forareview; Leeetal. ( 2000a 2003 )formixedintegerprogrammingapproaches; Romeijnetal. ( 2006 )forconvexprogrammingmodels;and HamacherandKufer ( 2002 )and Kuferetal. ( 2003 )foramulticriteriaapproach).Atreatmentplanthenconsistsofacollectionofnonnegativeintensitymatrices,oftenreferredtoasuencemaps,onecorrespondingtoeachbeamangle.Tolimittreatmenttime,eachofthesematricesisthenexpressedasamultipleofanintegraluencemapinwhichthemaximumelementisontheorderof10{20.Toallowdeliveryofthetreatmentplan,eachoftheseuencemapsshouldbe 85 PAGE 86 AhujaandHamacher ( 2005 ), Bolandetal. ( 2004 ), Kamathetal. ( 2003 ), Engel ( 2005 ), Kalinowski ( 2005a ),and Tasknetal. ( 2009b ).(Notethatintegratedapproachestouencemapoptimization,alsoreferredtoasaperturemodulation,havebeenproposedaswell;wereferto,e.g., PreciadoWaltersetal. ( 2004 ), Romeijnetal. ( 2005 ),and Menetal. ( 2007 ).) Theproblemthatwestudyisthedecompositionofanintegraluencemapintorectangularaperturesandcorrespondingintensities.While DaiandHu ( 1999 )proposedastraightforwardheuristicforavariantofthisdecompositionproblem,wedeveloptherstcomputationallyviableoptimizationapproachtothisproblem.InSection 4.2 weconsiderthecoreproblemofdecomposingan(integral)uencemapwhileminimizingthenumberofrectangularapertures.InSection 4.3 wethenextendourmodelstotheproblemsof(i)minimizingtotaltreatmenttime(asmeasuredbythesumoftherequiredaperturesetuptimesandthebeamontime,i.e.,theactualtimethatradiationisbeingdelivered);and(ii)minimizingthenumberofaperturessubjecttobeamontimebeingminimal.Finally,Section 4.4 discussesourcomputationalresultsonacollectionofclinicaluencemaps. 4.2.1 byformallydescribingtheoptimizationmodelunderinvestigationandmodelingitwithamixedintegerprogrammingformulation.WenextdescribeseveralclassesofvalidinequalitiesinSection 4.2.2 .Finally,wediscussmethodsforpartitioningtheinputmatrixinSection 4.2.3 ,whichleadstoeectivelowerandupperboundingtechniques. 86 PAGE 87 41 showsanexampleuencemap,whichweusethroughoutthischapter. Figure41. Exampleuencemap LetRbethesetofallO(n2m2)possiblerectangularapertures(i.e.,submatricesofBhavingcontiguousrowsandcolumns)thatcanbeusedtodecomposeB,excludingthosethatcontainazerobixel.Foreachrectangler2Rwedeneacontinuousvariablexrthatrepresentstheintensityassignedtorectangler,andabinaryvariableyrthatequals1ifrectanglerisusedindecomposingB(i.e.,ifxr>0),andequals0otherwise.LetCrbethesetofbixelsthatisexposedbyrectangler.WedeneMr=min(i;j)2Crfbijgtobetheminimumintensityrequirementamongthebixelscoveredbyrectangler.Furthermore,wedenotethesetofrectanglesthatcoverbixel(i;j)byR(i;j).Giventhesedenitions,wecanformulatetheproblemasfollows:IPR:MinimizeXr2Ryr 87 PAGE 88 4{1 )minimizesthenumberofrectanglesusedinthedecomposition.Constraints( 4{2 )guaranteethateachbixelreceivesexactlytherequireddose.Constraints( 4{3 )enforcetheconditionthatxrcannotbepositiveunlessyr=1.Finally,( 4{4 )statesboundsandlogicalrestrictionsonthevariables.Notethattheobjective( 4{1 )guaranteesthatyr=0whenxr=0inanyoptimalsolutionofIPR. FormulationIPRcontainstwovariablesandaconstraintforeachrectangle,resultinginalargescalemixedintegerprogramforprobleminstancesofclinicallyrelevantsizes.Furthermore,theMrtermsinconstraints( 4{3 )leadtoaweaklinearprogrammingrelaxation;withnovalidinequalitiesorbranchingyetperformedontheproblem,wehavethatyr=xr=MratoptimalitytothelinearprogrammingrelaxationofIPR.AnalternativeformulationthatdoesnotrequireMrtermsemploysadecompositionmethod.Recallthatweinvestigatedtheproblemofdecomposinganintegermatrixinto\consecutiveones"matricesinChapter 3 ,whereineachdecomposedmatrixallnonzerovaluestakethesamevalueandappearconsecutivelyoneachrow.Ourcomputationalresultsshowedthatsolvabilityoftheproblemissignicantlyimprovedbyapplyingabileveloptimizationalgorithm.Asimilarapproachfortheproblemweconsiderinthischapterwouldformulateamasterproblemas:MP:MinimizeXr2Ryr (4{6)yrbinary8r2R; whereweaddresstheformof( 4{6 )inthesequel.Givenavector^y,wecancheckwhetherconstraint( 4{6 )issatisedbysolvingthefollowinglinearprogram:SP(^y):Minimize0 (4{8)subjectto:Xr2R(i;j)xr=bij8i=1;:::;m;j=1;:::;n 88 PAGE 89 Associatingvariablesijwith( 4{9 ),andrwith( 4{10 ),weobtainthedualformulation:DSP(^y):MaximizemXi=1nXj=1bijij+Xr2RMr^yrr OurBendersdecompositionstrategyrstsolvesMP,whichyields^y.IfSP(^y)isfeasible,then^ycorrespondstoafeasibledecompositionandisoptimal.Else,DSP(^y)isunbounded(sincethetrivialallzerosolutionguaranteesitsfeasibility).Let(^,^)beanextremedualrayofDSP(^y)suchthatPmi=1Pnj=1bij^ij+Pr2RMr^yr^r>0.Then,allyvectorsthatarefeasiblewithrespectto( 4{6 )mustsatisfymXi=1nXj=1bij^ij+Xr2R(Mr^r)yr0: Weadd( 4{16 )inacuttingplanefashionasnecessary. PAGE 90 4.2.1 (a) (b) Weobservethatthereexistsanoptimalsolutioninwhichnotwoadjacentrectanglesareusedinthedecomposition.Toseethis,assumethatadjacentrectanglesr1andr2haveintensitiesxr1andxr2,respectively,wherexr1xr2withoutlossofgenerality.Inthiscase,analternativeoptimalsolutioncanbeconstructedbyextendingr1intor2.Specically,letr0betherectangleforwhichCr0=Cr1[Cr2.Analternativeoptimalsolutionthatdoesnotcontainanyadjacentrectanglesusesr2havingintensityxr2xr1,andr0havingintensityxr1.Thisdominancecriterioncanbewrittenas:yr1+yr218adjacentrectanglesr1;r2; whichstatesthatnopairofadjacentrectanglescanbeselectedinanoptimalsolution. 90 PAGE 91 42 .Sinceb43=2,thetotalintensitydeliveredto(5;3)byallrectanglesthatstartinrowsi=1;:::;4cannotexceed2.However,b53=14>2,andhenceatleastonerectanglethatstartsinrow5isrequiredtocoverbixel(5;3).Similarly,b53>b52impliesthatatleastonerectanglethatstartsincolumn3isrequiredtocoverthesamebixel.Theseresultscanbestrengthenedbyconsideringboth(4;3)and(5;2)simultaneously.Sinceb53>b43+b52,weconcludethatatleastonerectanglethatstartsatbixel(5;3)isrequiredinanyfeasibledecompositionoftheuencemap.Ingeneral,arectanglemuststartat(i;j)ifbij>b(i1)j+bi(j1)issatised.Figure 43 illustratesasimilaridea,wherewe Figure42. Examplestartindex comparetheintensityrequirementofbixel(6;4)withthebixelbelowit,andtheoneonitsright.Usingargumentssimilartotheonesregardingstartingindices,weconcludethatarectanglemustend(i.e.,havealowerrightcorner)at(6;4)sinceb64>b74+b65. Figure43. Exampleendindex Startingandendingindexconditionscanbegeneralizedfurtherasfollows.Assumethatthereexistintegersu2[0;i1],d2[i+1;m+1],l2[0;j1],andr2[j+1;n+1] 91 PAGE 92 44 illustratesaboundingboxforbixel(6;3)(markedindarkgray),whichcorrespondsto(l;u;r;d)=(2;4;5;7).Thefourbixelsthatrepresentthebordersofaboundingboxaremarkedinlightgray.Wenotethatanyrectanglethat Figure44. Exampleboundingbox containsbixel(6;3),anddoesnotstartinsidetheboundingbox(at(5,3)or(6,3))orendinsidetheboundingbox(at(6,3)or(6,4)),hastocontainatleastoneofthefourbixelsontheborder.Therefore,thesumofintensitiesofthoserectanglesisboundedbythetotalrequiredintensityofthebixelsinlightgray.Sincetheintensityofthedarkgraybixelcannotbesatisedbythoserectanglesalone,itfollowsthatatleastonerectanglecontainedwithintheboundingboxmustbeusedtocoverbixel(6;3).LetBBijrepresenttheinteriorofaboundingboxforbixel(i;j),i.e.,given(l;u;r;d)allbixelsattheintersectionofrowsu+1;:::;d1andcolumnsl+1;:::;r1.WedenotethesetofrectanglesinR(i;j)thatarecontainedwithinBBijbyR(BBij).Inthiscase,thefollowinginequalityisvalid:Xr2R(BBij)yr1: Notethat(0;0;n+1;m+1),whichcorrespondstotheinputmatrix,isaboundingboxforanybixel.Thereforetherecanbemultipleboundingboxesassociatedwitheachbixel.LetBBijandBB0ijbetwoboundingboxesforbixel(i;j).WesaythatBBijdominatesBB0ijifR(BBij)R(BB0ij).Sincetheinequality( 4{18 )thatcorrespondstoadominated 92 PAGE 93 45 displaysanothernondominatedboundingboxforthebixelconsideredinFigure 44 Figure45. Anothernondominatedboundingboxseededat(6,3) Togeneratenondominatedboundingboxes,werstmakethefollowingobservation.Anondominatedboundingboxforbixel(i;j)isminimalinthesensethatnoneofitsedgescanbeshiftedcloserto(i;j)withoutviolatingtheboundingboxintensityproperty.Weusethisobservationtodesignanalgorithmthatndsseveralnondominatedboundingboxesassociatedwithagivenbixel.Inouralgorithm,westartatabixel(i;j),andrstmoveinaverticalorhorizontaldirectionuntilweencounterabixel(i0;j0)havingbi0j0 PAGE 94 Wenotethatinequalities( 4{19 )areimpliedby( 4{2 )and( 4{3 )inIPR.However,( 4{19 )canbeusedtotightenthemasterproblemoftheBendersdecompositionapproachdiscussedinSection 4.2.1 .Furthermore,varioustighteningprocedurescanbeappliedto( 4{19 )foruseineitherthedirectsolutionofIPRorintheBendersmasterproblem.Inourimplementation,weapplyaChvatalGomoryroundingprocedure(see,e.g., NemhauserandWolsey ( 1988 ))inwhichwedividebothsidesoftheinequalitybythesmallestMrcoecientonthelefthandside(unlessbijisdivisiblebythatnumber),androundupcoecientsonbothsidesoftheinequality.IfbijisdivisiblebythesmallestMrcoecientonthelefthandsideof( 4{19 ),thentheroundingprocedureyieldsaninequalityimpliedby( 4{19 ),andhencewedonotgenerateit. 4.2.2.3 dividesbothsidesof( 4{19 )bybij1,providedthatbij2,andthenroundsupallcoecientsandtherighthandside.Notingthatallcoecientsonthelefthandsideareboundedfromabovebybij,thisprocessyields:Xr2R(i;j):Mr PAGE 95 4{21 ),andthusallvcorrespondingtoafeasiblesolutionmustalsosatisfy( 4{21 ). Similarly,considersubmatricesoftheformqLqR; 4 revealsthatthefollowinginequalityisvalid:2Xr2A=Lyr+2Xr2A=Ryr+2Xr2A=2yr+Xr2A PAGE 96 4.2.2.4 forothersubmatricesaswell.However,thisprocessisverytedious,andthereisalargenumberofpossiblesubmatrixcombinations.Inthissectionwedescribeasimilarsetofinequalities,whichareweakerthanthosedescribedintheprevioussection,butareeasiertogenerate.WerstobservethattheformulationIPRcanbesolvedquicklyforsmallinputmatrices.LetSdenoteasubmatrixoftheinputmatrix,andR(S)representthesetofrectanglesthatcoveratleastonebixelinS.LetLB(S)bealowerboundonthenumberofrectanglesrequiredtodecomposeS.SinceLB(S)constitutesalowerboundonthetotalnumberofrectanglesrequired,thefollowinginequalityisvalidforanysubmatrixS:Xr2R(S)yrdLB(S)e: WecanobtainLB(S)byformulatinganauxiliaryintegerprogrammingproblemoftypeIPRforS,andsettingalimitonthemaximumsolutiontime. 96 PAGE 97 4.2.2 Weobserveonclinicaldatathatseveralregionsoftheinputmatrixarecompletelysurroundedbyzerobixels.Sincenorectanglecancoverazerobixel,eachoftheseregionscanbesolvedindependently.Aconnectedsubsetoftheinputmatrixobeysthepropertythatarectilinearpathexistsbetweenanytwononzerobixelsofthesubset,suchthateachbixelinthepathisalsoanonzerobixelthatbelongstothesubset.Wecallaconnectedsetofnonzerobixelsacomponentoftheinputmatrixifitisadjacenttozerobixelsacrossallofitsboundaries(i.e.,ifthesubsetisnotcontainedwithinalargerconnectedsubset). Toidentifythecomponentsoftheinputmatrix,wegenerateagraphGinwhicheachnonzerobixelhasacorrespondingnode.Weaddanarcbetweenapairofnodesifandonlyifthecorrespondingbixelsareadjacentintheinputmatrix.WethenidentifyconnectedcomponentsonGbyrunningastandarddepthrstsearchalgorithm.EachconnectedcomponentonGcorrespondstoacomponentoftheinputmatrix,whichcanbesolvedindependentlyofothercomponents.Figure 46 depictsthecomponentsoftheuencemapgiveninFigure 41 97 PAGE 98 Twocomponentsofauencemap 47 ,theregionswithlightanddarkgraybackgroundareindependent.IfwesolveIPRseparatelyoverallindependentregions,thesumofrectanglesrequiredtodecomposeeachindependentregionyieldsalowerboundontheobjectivefunctionforthecorrespondingcomponent. Figure47. Regionsofaconnectedcomponent Ingeneral,therearemultiplewaysofpartitioningacomponentintoindependentregions,witheachyieldingpossiblydierentlowerbounds.Theproblemofndingapartitionthatyieldsthebestlowerboundcanbethoughtofasa\dual"ofndingtheminimumnumberofrectanglestodecomposeacomponent.Tosolvethisdualproblem,weneedtobalancetwoconictingcriteria: 98 PAGE 99 Weuseaheuristicproceduretopartitionacomponentintoindependentregions,whichemploysanauxiliaryobjectiveofmaximizingthenumberofcomponentbixelscoveredbyanindependentregion.Eachbixel(i;j)iscalled\committed"ifiteitherbelongstoanindependentregion,orif(i;j)iscontainedwithinsomerectangleinRthatalsocoversbixelsinanindependentregion(andhence,(i;j)cannotbelongtoanotherindependentregion).Allotherbixelsarecalled\uncommitted."Weselectourindependentregionsoneatatime,untilnomoreuncommittedbixelsremain.Theprocedure'sdetailsaredescribedasfollows. 99 PAGE 100 Inouralgorithmforsolvingacomponent,weexecutetheforegoingheuristictondasetofindependentregions.WeformulateIPRforeachregion,withalimitonthemaximumsolutiontime.Wethenusethelowerboundobtainedforeachregiontogenerateaninequalityoftype( 4{24 ).(Itisoftenprudenttoskipthisstepifonlyoneregioniscomputedforacomponent.) 47 isadependentregion.LetDrepresentthesetofbixelsinadependentregion,andletR(D)representthesetofrectanglesthatcoveronlyasubsetofthebixelsinD. Toimproveourlowerbound,wewishtocomputetheminimumnumberofrectanglesrequiredtocoverD;however,wewishtoavoiddoublecountingthoserectanglesusedtocoverbixelsinindependentregions.Accordingly,weseektheminimumnumberofrectanglesinR(D),perhapsinconcertwithrectanglesoutsideR(D),requiredtocoverthebixelsinD.Usingthexandyvariablesasbefore,weformulatethefollowingvariationofIPRtondtheminimumnumberofrectanglesinR(D)requiredtopartition 100 PAGE 101 (4{28)xr08r2R;yrbinary8r2R(D) (4{29) Objective( 4{25 )minimizesthenumberofrectanglesinR(D)usedinthesolution.Constraints( 4{26 )ensurethatthebixelsinDgetpartitionedexactly,where( 4{27 )limittheintensitydeliveredtotheremainingbixels.Constraints( 4{28 )relatethexandyvariablesasdoneinIPR,andnally( 4{29 )denevariabletypes.Asbefore,wesetatimelimitforthesolutionofDPR,andobtainalowerboundontheobjectivefunctionvalue,whichwedenotebyLB(D).Giventhisvalue,thefollowinginequalityisvalid:Xr2R(D)yrdLB(D)e: Inourexample,theoptimalvalueofDPRfortheblack(dependent)regionis1sincetheintensityrequirementofbixel(1;4)cannotbesatisedcompletelybyrectanglesthatcoverbixelsinthegray(independent)regions(infact,thisresultcanalsobeseenduetotheboundingboxconstraintimplyingthatonerectanglerepresentingthesingletonbixel(1,4)mustappearinanyfeasiblesolution).WenotethattherectanglesinR(D),bydenition,donotintersectanyother(dependentorindependent)regions.Therefore,thelowerboundsobtainedforallregionscanbesummedtoobtainalowerboundontheminimumnumberofrectanglesrequiredtodecomposeacomponent. 101 PAGE 102 4.2.3.2 .FeasiblesolutionsfordependentregionscanbeextractedfromsolutionsoftheformulationgivenbyDPR.However,sinceDPRminimizesthenumberofrectanglesthatarecontainedwithinadependentregion,andnotnecessarilythetotalnumberofrectanglesrequiredtodecomposeadependentregion,thesolutionsobtainedfromDPRpotentiallyuseanunnecessarilylargenumberofrectanglesnotcontainedinR(D). AbetterwayofobtainingfeasiblesolutionsfordependentregionsistoformulatetheproblemIPRforeachdependentregion.SinceIPRexplicitlyminimizesthetotalnumberofrectanglesrequired,weexpectthisapproachtoresultinfeasiblesolutionsofhigherquality.However,thisapproachdoesnotconsiderthefactthatsomeoftherectanglesthatarealreadyusedfordecomposingindependentregionscanbeextendedintodependentregionswithoutincreasingthetotalnumberofrectangles.Topermittheuseofrectanglesthatintersectindependentanddependentregions,werequirearevisedintegerprogrammingformulation. Inourapproach,wesolvetheintegerprogrammingformulationsfordecomposingtheindependentregionsrst,andstorethebestfeasiblesolutionsfoundwithintheallowedtimelimit.Letxrrepresenttheintensityassignedtorectanglerfordecomposingindependentregions.Next,wegenerateafeasiblesolutionforeachdependentregion,oneatatime,asfollows.Werstndthesetofrectanglesthatcanbeextendedintothecurrentdependentregion,anddeterminehowthoserectanglescanbeextended.LetE(D;r)representthesetofrectanglesinRthatextendrectanglerintodependentregion 102 PAGE 103 (4{34)xr0;yrbinary8r2R(D) (4{35)zrebinary8r2R;e2E(D;r): Wegenerateafeasiblesolutionbycombiningthreetypesofrectangles:(i)rectanglesusedtodecomposeindependentregionsthatarenotextendedbyEPR;(ii)rectanglesobtainedbyextendingrectanglesfromindependentregionsintodependentregionsbyEPR;and(iii)rectanglesinR(D)usedbyEPR. NotethattheoptimalvalueofEPRforthedependentregiongiveninFigure 47 is1.Thiscanbeseenbyobservingthattherectangle(s)thatcoverbixel(3;4)canbeextendeduptofullysatisfytheintensityrequirementofbixel(2;4)withoutanypenaltyontheobjectivefunctionofEPRformulatedforthedependentregion.Therefore,asinglerectanglecontainedinthedependentregionsolvesEPRoptimally.SincetheoptimalvalueofDPRforthedependentregionisalso1,ourpartitionsolvestheproblemofndingtheminimumnumberofrectanglestooptimality. 103 PAGE 104 wherewisaparameterthatrepresentstheaveragesetuptimeperaperturerelativetothetimerequiredtodeliveraunitofintensity. TheBendersdecompositionprocedurediscussedinSection 4.2.1 alsoneedstobeadjustedaccordingly.WerstaddacontinuousvariablettoMP,which\predicts"theminimumbeamontimethatcanbeobtainedbythesetofrectangleschosenbyMP.Theupdatedmasterproblemcanbewrittenasfollows.MPTT:MinimizewXr2Ryr+t (4{39)tminimumbeamontimecorrespondingtoy 104 PAGE 105 NotethatSPTTisobtainedbysimplychangingtheobjectivefunctionofSP.IfSPTT(^y)isinfeasible,thenweaddaBendersfeasibilitycutoftype( 4{16 )asbefore,andresolveMPTT.Otherwise,letthevalueoftinMPTTbe^t,andtheoptimalobjectivefunctionvalueofSPTTbet?.If^t=t?,then(^y;^t)isanoptimalsolutionofMPTTthatminimizesthetotaltreatmenttime.However,if^t>t?,thenweaddthefollowingBendersoptimalitycuttmXi=1nXj=1bij^ij+Xr2R(Mr^r)yr; where^ijand^rareoptimaldualmultipliersassociatedwithconstraints( 4{43 )and( 4{44 ),respectively. 105 PAGE 106 ThemodicationsrequiredfortheBendersdecompositionalgorithmarealsostraightforward.Toenforcetheminimumbeamontimerestriction,weadd( 4{47 )toSP,whichcheckswhetheragivensetofrectanglescandecomposetheuencemap.TheupdatedfeasibilitycutisgivenbymXi=1nXj=1bij^ij+Xr2R(Mr^r)yr+T?^0; whereisthedualvariableassociatedwith( 4{47 )inSP.Finally,weneedtocheckwhetherthesolutiongeneratedbyourheuristicdiscussedinSection 4.2.3.4 satisesconstraint( 4{47 );ifso,thenitcanbeusedasaninitialupperbound. OurpreliminarycomputationaltestsshowedthatthenaiveimplementationofourBendersdecompositionapproach,inwhichweaddacutandresolvethemasterproblemineachiteration,wasnotcomputationallycompetitivewithsolvingtheexplicitintegerprogrammingformulation.Thisisduetothefactthatrepetitivelysolvingthemasterproblem,whichisanintegerprogrammingproblem,iscomputationallyveryexpensive.WeinsteadusedcallbackfunctionsofCPLEXtogenerateasinglebranchandbound 106 PAGE 107 OurrstexperimentquantiestheeectsofthevalidinequalitiesdiscussedinSection 4.2.2 ,andthepartitioningapproachdiscussedinSection 4.2.3 onsolutionqualityandexecutiontime.InTable 41 ,thesetofcolumnslabeled\DefaultCPLEX"showstheresultsweobtainedbysolvingtheformulationIPRoneachprobleminstanceusingdefaultCPLEXoptions.The\+Validinequalities"columnsrepresenttheIPRformulationenhancedwiththeadjacentrectangleinequalities( 4{17 ),boundingboxinequalities( 4{18 ),strengthenedaggregateintensityinequalities( 4{19 )and( 4{20 ),and12submatrixinequalities( 4{21 )and( 4{22 ).(Additionalcomputationalresultsshowedthatthe22submatrixinequalities( 4{23 )andthearbitrarysubmatrixinequalities( 4{24 )didnotimprovethesolvabilityofthemodel.)Thesetofcolumnslabeled\+Partitions"showstheresultsweobtainedbypartitioningtheproblemintoseparablecomponents(Section 4.2.3.1 ),furtherpartitioningeachcomponentintoindependentanddependentregions(Sections 4.2.3.2 and 4.2.3.3 ),andusingourupperboundingheuristic(Section 4.2.3.4 )inadditiontothevalidinequalitiesusedforthetestsintheprevioussetofcolumns.Werefertothelattersettingsasourbasealgorithmintheremainingcomputationaltests. EachsetofcolumnsinTable 41 displaysthetimespentforeachprobleminstance(\CPU"),andupperbound(\UB"),lowerbound(\LB"),andoptimalitygap(\GAP")obtained.Wealsoreporttheaverageandmaximumgapsoverallprobleminstances.WeobservethatnoneoftheprobleminstancesweresolvedtooptimalityusingthedefaultCPLEXoptions,whereasc1b2andc5b2weresolvedtooptimalityafteraddingthevalidinequalitiesofSection 4.2.2 .Anadditionalinstance(c5b5)wassolvedusingthepartitioningstrategydescribedinSection 4.2.3 .Wenotethateventhoughourapproach 107 PAGE 108 Eectofvalidinequalitiesandthepartitioningstrategy +Validinequalities +PartitionsNamemnL CPUUBLBGap CPUUBLBGap c1b1151420 180066600.09 180063620.02 180048470.02 138.248480 180057540.05 180057540.05 180061520.15 180061530.13 180059550.07c1b5111520 180047450.04 180046450.02 1800114790.31 1800119850.29 1800103870.16c2b2171920 180095690.27 180096810.16 180094820.13c2b3181820 180098730.26 1800103770.25 180094770.18c2b4181820 1800114800.3 1800115840.27 1800105880.16c2b5171820 180094640.32 180098720.27 180091720.21c3b1221720 1800121690.43 1800134790.41 1800119790.34c3b2151920 180073460.37 180071520.27 180070520.26c3b3201720 1800119690.42 1800119750.37 1800107770.28c3b4191720 1800103690.33 1800106730.31 180099780.21c3b5151920 180073550.25 180071580.18 1800106790.25 1800107890.17 180088540.39 180099580.41 180091580.36c4b3182320 180095710.25 180099750.24 180093770.17c4b4172320 1800103780.24 1800102810.21 180098830.15c4b5182420 180093620.33 180093660.29 180087670.23c5b1151620 180066640.03 180066650.02 180058570.02 102.158580 180063540.14 180068560.18 180065570.12c5b4141620 180063570.1 180064590.08 180062590.05c5b5121720 180053470.11 180051480.06 36.249490 OurnextexperimenttestsourbasealgorithmundertheextensionsdiscussedinSection 4.3 .Thesetofcolumnslabeledas\TotalTime"inTable 42 presentstheextensioninwhichtheobjectivefunctionisdenedasalinearcombinationofthebeamontimeandthenumberofrectangles.Theactualvalueofwdependsontheparticulartreatmentdeliveryequipmentusedintheclinic,wherevaluesofwintherange1{10aretypical(see,e.g., DaiandHu ( 1999 ),and Tasknetal. ( 2009b )).Inourexperiments,wethereforeusedw=7asarepresentativevalue.Thenextsetofcolumns(\Lexicographic")isdedicatedtotheextensioninwhichwerstminimizebeamontime,T?,andthenndtheminimumnumberofrectanglesthatyieldstheminimumbeamontime.Thecolumn\BOT"representsthevalueofT?,and\TotalTime"representsthetotaltreatmenttimeassociatedwiththesolutionfound,whereweagainusew=7astheaveragesetuptimeperrectangle.Weobservethatouralgorithmcouldsolvemoreprobleminstancestooptimalityforbothextensionscomparedtothe 108 PAGE 109 Computationalresultsonmodelextensions LexicographicNamemnL CPUUBLBGapBOTTotalTime c1b1151420 255.96216210 36.566660176638c1b2111520 330.34594590 132.650500121471c1b3151520 1800548542.720.01 130.462620147581c1b4151520 1800557542.490.03 186.962620136570c1b5111520 1800451443.630.02 30.953530115486c2b1182020 1800962814.240.15 18001071040.03194943c2b2171920 1800883797.740.1 180096920.04207879c2b3181820 1800918797.60.13 180096880.08237909c2b4181820 18001028889.360.13 18001111060.052581035c2b5171820 1800890721.130.19 180092830.1207851c3b1221720 18001161858.90.26 18001161030.112661078c3b2151920 1800668533.240.2 180070640.09151641c3b3201720 18001066847.090.21 1800111950.142781055c3b4191720 18001023857.910.16 1800103950.082871008c3b5151920 1800722610.010.16 204.476760182714c4b1192220 18001044918.570.12 18001081050.032751031c4b2132420 1800895656.150.27 180095760.2232897c4b3182320 1800858743.620.13 180092890.03189833c4b4172320 1800943834.320.12 1800101960.05235942c4b5182420 1800913740.190.19 180086770.1260862c5b1151620 271.46266260 5.571710158655c5b2131720 33.45975970 19.963630156597c5b3141620 1800623597.960.04 869.268680180656c5b4141620 1800584571.150.02 192.466660145607c5b5121720 90.45035030 37.257570147546 Baataretal. ( 2005 )).Ontheotherhand,minimizingthebeamontimeisapolynomiallysolvableproblem(seeSection 4.3 ).Therefore,weexpectthattheproblemshouldbecomeeasierastheweightofthebeamontimetermintheobjectivefunctionincreases.Thereasonthelexicographicminimizationproblemiseasiertosolvethantheothertwovariationsisbecausetheadditionalbeamontimeconstraintconsiderablyshrinksthefeasiblesolutionspace. Anotherwayoflookingattheproblemofbalancingthenumberofaperturesandthebeamontimeistoviewtheproblemasamulticriteriaoptimizationproblem.Inthissetting,weareinterestedinconstructingtheParetoecientfrontierofsolutionswiththepropertythatneitherofthetwocriteriacanbeimprovedwithoutdeterioratingtheother.NotethatthelexicographicapproachthatweconsideredabovedeterminesaparticularParetooptimalsolutiontothemulticriteriaproblem.Togenerateothernondominated 109 PAGE 110 41 )withacorrespondingbeamontimeof160,whiletheminimumbeamontimeforthisprobleminstanceis147(seeTable 42 )whichrequires57apertures.Figure 48 thendepicts(i)thenondominatedsolutions;(ii)theParetoecientfrontierforvaluesof2[49;57],and(iii)the(boundaryofthe)convexhulloftheParetoset.Thesolutionsonthelatteraretheoptimalsolutionstotheproblemofminimizingtotaltreatmenttimethatcanbeobtainedwithdierentvaluesofw. Figure48. Ecientfrontierfornumberofaperturesandbeamontime OurnalexperimentanalyzestheeectofthemaximumintensityvalueL.Usuallyuencemapsareobtainedbysolvinganonlinearoptimizationproblemforeachbeamangletodetermineanintensityproleforeachbeamangle,whichisrepresentedbya 110 PAGE 111 Eectofmaximumintensityvalueonsolvability CPUUBLBGap CPUUBLBGap c1b11514 4.43053050 22.94414410 1800539536.990c1b21115 1.42382380 4.43203200 498.53943940c1b31515 9.62872870 228.63773770 1800495487.670.01c1b41515 5.82692690 1800393377.420.04 1800513493.130.04c1b51115 2.42162160 28.93263260 316.74114110c2b11820 31.34404400 1800648635.10.02 1800826732.680.11c2b21719 51.54484480 1800625599.840.04 1800759690.670.09c2b31818 144.44284280 1800645615.640.05 1800800704.610.12c2b41818 1593.14874870 1800755678.490.1 1800919796.130.13c2b51718 197.94294290 1800606538.620.11 1800728621.90.15c3b12217 1359.44804800 1800747662.470.11 18001080779.160.28c3b21519 1800280274.970.02 1800414376.010.09 1800532461.60.13c3b32017 1800461446.770.03 1800731641.20.12 1800908755.250.17c3b41917 1800463456.420.01 1800713634.260.11 1800900762.020.15c3b51519 1800332325.870.02 1800481466.60.03 1800582532.20.09c4b11922 39.95295290 1800758719.530.05 1800899827.730.08c4b21324 1800422408.140.03 1800595503.920.15 1800764582.690.24c4b31823 126.34094090 1800579564.50.03 1800695666.230.04c4b41723 321.44444440 1800662649.10.02 1800815742.810.09c4b51824 1194.74144140 1800636573.020.1 1800794662.740.17c5b11516 5.93423420 5.64424420 28.95845840c5b21317 4.32892890 5.54394390 49.35165160c5b31416 1574.42942940 1800453441.440.03 1800566538.220.05c5b41416 3.62392390 1800473468.490.01 1800534524.310.02c5b51217 2.12522520 9.33543540 63.94414410 43 showstheresultsofourexperiments.WeobservethatouralgorithmproducessmalleroptimalitygapsasLdecreases,whichisnotsurprisingsinceIPRbecomestighterastheMrcoecients(whichareboundedbyL)decrease.Furthermore,deliveryeciencyisalsohigherforsmallvaluesofL.Theaveragetreatmenttime(calculatedoverthelowerbounds)forallprobleminstancesincreasesfrom366:05forL=5to513:79forL=10,609:79forL=15,and684:96forL=20,whichiscalculatedusingthesetofcolumnslabeled\TotalTime"inTable 42 .Ourresultsshowthatthechoiceofgranularitychosenforroundinghasasignicanteectonthetreatmenttime.Foreachindividualpatient,therisksassociatedwiththedeteriorationintreatmentplanqualityduetotheroundingofintensitiesneeds 111 PAGE 112 112 PAGE 113 Thegraphsearchproblemwasinitiallydenedby Parsons ( 1978 )inthecontextofseekingapersonlostinacave.Thecaveisrepresentedasagraph,wheretunnelsofthecavecorrespondtoedgesofthegraph.Searchershavetosweepedgesofthegraphtolocatethemissingperson,whoisassumedtobewanderingunpredictablyorispurposefullytryingtoevadesearchers.Thesearchnumbers(G)ofagraphGisdenedtobetheminimumnumberofsearchersneededsothatthemissingpersoncanbefoundevenifhecouldmoveinnitelyfastalonganypathnotoccupiedbysearchers( Parsons 1978 ).Computings(G)isNPhardforgeneralgraphs( BienstockandSeymour 1991 ; LaPaugh 1993 ; Megiddoetal. 1988 ),butitcanbecomputedinlineartimefortrees( Alspach 2004 ; Megiddoetal. 1988 ; Pengetal. 2000 ).Thesearchnumberofagraphhasbeenshowntoberelatedtootherimportantparameterssuchastreewidth,pathwidth,andvertexseparation( Dendrisetal. 1997 ; Ellisetal. 1994 ; SeymourandThomas 1993 ). 113 PAGE 114 Flocchinietal. 2008 ; LaPaugh 1993 ; PenuelandSmith 2009 ).Inrendezvousproblemsdierentplayers,whoarenotawareofthelocationofothers,trytomeetatacommonnodeasquicklyaspossible( Alpern 1995 ; AlpernandGal 2003 ; KikutaandRuckle 2007 ).Hideandseekproblemsconsideranintruderthat\hides"inastationarylocation,whilethesearcherstrytolocatetheintruderinminimumtime( Alpern 2008 ; JotshiandBatta 2008 ).Suchproblemsalsoariseinsearchandrescuesettings( Benkoskietal. 1991 ).Pursuitevasion(or\copsandrobber")gamesmodelanintruderthattriestoavoidbeingcapturedbysearchers( AignerandFromme 1984 ; Alspachetal. 2008 ; Hahn 2007 ; IslerandKarnad 2008 ).Insomeapplicationsnodesofagraphneedtobepatrolledforprotectionorsupervision( Chevaleyreetal. 2004 ; Saketal. 2008 ).Inparticular,oneinterestingapplicationcoordinatesautomatedsoftwaresearcherssothattheypatroltheInternettondwebsitesthatexploitbrowservulnerabilities( Wangetal. 2005 ).Wereferthereaderto AlpernandGal ( 2003 ); Alspach ( 2004 ); FominandThilikos ( 2008 )fordetailedsurveysoftheliteratureonsearchproblemsandapplicationsinvariouspracticalsettings. Mostofthepreviousresearchongraphsearchproblemshasfocusedontheoreticalaspectsoftheproblems(e.g. Chevaleyreetal. ( 2004 ); Dendrisetal. ( 1997 ); Ellisetal. ( 1994 ); GoldsteinandReingold ( 1995 ); SeymourandThomas ( 1993 ))ordesigningalgorithmsforsolvingtheproblemsonspecialgraphstructures(e.g. Alpern ( 2008 ); Flocchinietal. ( 2008 ); KikutaandRuckle ( 2007 ); Pengetal. ( 2000 )).Ourcontributionisanexactoptimizationalgorithmforsolvingseveralvariantsofthesearchproblemongeneralgraphs(seealso PenuelandSmith ( 2009 )foradecontaminationprobleminwhichtheintruderlocationhasbeendetermined).Inparticular,weconsiderthreespecicgraphsearchproblems:(i)ahideandseekproblem,(ii)apursuitevasionproblem,and(iii)a 114 PAGE 115 Avariantofthebranchandboundalgorithm,whichaddscuttingplanestolinearprogrammingrelaxationstotightendualboundsiscalledbranchandcut,andisemployedinmostcommercialsolversforsolvingintegerprograms( Marchandetal. 2002 ; NemhauserandWolsey 1988 ; Wolsey 1998 ).Aneectivemethodforsolvingintegerprogramshavingalargenumberofvariablesisbranchandprice,whichisbasedondynamiccolumngeneration( Barnhartetal. 1998 ).Branchcutpriceisessentiallyanalgorithmthatcombinesdynamiccolumngenerationwithdynamicrowgeneration( JungerandThienel 2000 ). Theremainderofthischapterisorganizedasfollows.InSection 5.2 wedescribeahideandseekproblemandproposeacolumngenerationalgorithmforsolvingitslinearprogrammingrelaxation.Similarly,Sections 5.3 and 5.4 analyzethepursuitevasionandpatrolproblems,respectively.WedescribesomebranchingrulesthatcanbeusedinallthreealgorithmstoobtainanoptimalsolutiontotheseproblemsinSection 5.5 .Finally,wegivecomputationalresultsinSection 5.6 Ahujaetal. 1993 ).Wedenethelengthofawalkasthenumberoftraversededgesonthewalk.LetP(T)denotethesetofallpossiblewalksof 115 PAGE 116 (5{3) Theobjectivefunction( 5{1 )minimizesthenumberofselectedsearchers,andconstraints( 5{2 )guaranteethateachnodeiscoveredbyatleastonesearcherwithintheallowedtimeframe.Wenotethataspecialcaseofthisproblemforwhichasearcherlocatedatnodei2Ncanobservenodeianditsneighbors,andweneedtoguaranteeimmediatedetectionoftheintruder(i.e.,whenS(i)=A(i);8i2NandT=0),isequivalenttotheminimumdominatingsetproblem,whichisknowntobeNPhard( GareyandJohnson 1979 ).Therefore,thehideandseekproblemthatweconsiderisNPhard. 116 PAGE 117 whereupperboundsonthevariablesarenotnecessaryatoptimality.Givenanoptimaldualvector^,thereducedcostofp,whichwedenotebycp,canbecalculatedas1Pi2N^idpi.Since^isanoptimaldualvector,cp0forallp2P0(T).WecanconcludethatthecurrentsolutionofLHSLPisalsooptimalforthelinearprogrammingrelaxationofHSifcp0forallp2P(T).Ontheotherhand,ifc^p<0forsome^p2P(T)nP0(T),thenadding^ptoP0(T)canpotentiallydecreasethevalueoftheobjectivefunction( 5{4 ).Wediscussourpricingproblem,whichseekssucha^p,inthenextsection. 5{5 )correspondingtonodei2N.Also,letyibeadecisionvariablethatequals1ifnodeiis\seen"byasearcherfollowingawalkthatwegenerate,and0otherwise,8i2N.Givenanoptimaldualvector^,wesolvethefollowingpricingproblemtoseekavariablehavinganegativereducedcost:maxPi2N^iyi,subjecttotherestrictionthat(y1;:::;yjNj)correspondstoasetofnodesobservedbyawalkoflengthnomorethanT. ThepricingproblemcanbeformulatedasamixedintegerprogrammingproblemonatimeexpandednetworkconsistingofT+1stages.Inparticular,wecreateanodeNitforeachi2N;t=0;:::;T.WecreateanarcfromnodeNit,8i2N;t=0;:::;T1tonodesNj(t+1)forallj2A(i).Forthisproblemitiseasytoseethatanoptimalsolutionexistsinwhichallsearchersmoveateachtimeperiod.Therefore,weomitarcsbetweennodesNitandNi(t+1)foreachi2N.Figure 51 displaysasimpleexamplegraph,andthecorrespondingtimeexpandednetworkforT=2. Toformulatethepricingproblemasamixedintegerprogram,weintroducebinaryvariablesxti=1ifthesearcherisatnodeiattimet.Then,anintegerprogramming 117 PAGE 118 (b) Figure51. (a)Anexamplegraph(b)TimeexpandednetworkforT=2 formulationoftheproblemcanbegivenas:maximizeXi2N^iyi Constraints( 5{8 )representthefactthatthesearchercanvisitonlyonenodeatatime.Constraints( 5{9 )ensurethatnodeicanbevisitedattimetonlyifoneofitsneighborshasbeenvisitedattimet1.Constraints( 5{10 )forcethevalueofyitozerounlessnodeicanbeobservedbythesearcheratsometimeperiod.Notethattheyvariableswilltakeonbinaryvaluesinanoptimalsolution,andthereforewerelaxthemascontinuousvariables.Iftheoptimalobjectivefunctionvalueof( 5{7 ){( 5{12 )isgreaterthan1,then 118 PAGE 119 WeinitializeouralgorithmbygeneratingastationarysearcherthatstaysatnodeiforTperiods,foreachi2N.Eventhoughtheseelementarysearchersarenotlikelytobeselectedinanoptimalsolution,theyguaranteethefeasibilityofLHSLP.WediscussseveralbranchingstrategiesthatcanbeusedforStep1inSection 5.5 119 PAGE 120 (5{14)p2f0;1g8p2P(T); wherepagainequals1ifandonlyifasearcherisassignedtofollowwalkp.Theobjectivefunction( 5{13 )minimizesthenumberofsearchers.Constraints( 5{14 )ensurethatforeachpossibleintruderwalkoflengthT+1,atleastonesearcherisselectedtodetectit. 5{14 )correspondingtointruderwalkr2R0(T).Also,letyrbeadecisionvariablethatequals1ifanintruderfollowingwalkrisdetectedbyasearcherfollowingawalkthatwegenerate,and0otherwise,8r2R0(T).Givenanoptimaldualsolution^ofthelinearprogrammingrelaxationtoLPE,wesolvethefollowingpricingproblemtoseekavariablehavinganegativereducedcost:maxPr2R0(T)^ryr,subjecttotherestrictionthat(y1;:::;yjR0(T)j)correspondstoasetofintruderwalksdetectedbyasearcherwalkoflengthnomorethanT. 120 PAGE 121 (5{19)0yr18r2R0(T) (5{20)xti2f0;1g8i2N;t=1;:::;T: Constraints( 5{17 )ensurethatthesearchercannotbelocatedatmultiplenodessimultaneously.Constraints( 5{18 )modelthefactthatthesearchercaneitherstayatthesamenode,orcanmovetoanadjacentnodeateachperiod.Constraints( 5{19 )representtheconditionthatthesearcherdetectsintruderr2R0(T)onlyifitmovestoanodewhereitcandetecttheintruderduringthepursuit.Wenotethattheyvariablescanberelaxedascontinuousvariablesinthiscase,too.Thispropertyallowsthenumberofbinaryvariablesinthesearcher'sproblemtostayconstantasnewevasionpathsfortheintruderarediscovered.Asbefore,iftheoptimalobjectivefunctionvalueof( 5{16 ){( 5{21 )isgreaterthan1,thenwehavefoundavariablewhosereducedcostisnegative. 121 PAGE 122 Afterconstructingthetimeexpandednetworkasdescribed,weseekafeasibles{qpathonthenetworkbyastandardbreadthrstsearchalgorithm,whichworksinO(N2T)timeintheworstcaseifGisdense.Ifsuchapathexists,thenitcorrespondstoawalkrthattheintrudercantaketoavoiddetectionforT+1timeunits.Inthiscase,weaddrtoR0(T),andgeneratetheassociatedconstraintoftype( 5{14 ).Ontheotherhand,ifnosuchpathexists,then^isafeasiblesolutionofPA. 5{14 ),andgobacktoStep1.Else,stopprocessingthecurrentsubproblemwithanintegralsolution. 122 PAGE 123 5.5 5.4.1ProblemDescription 123 PAGE 124 (5{23)p2f0;1g8p2Pc(K) (5{24) Weproposeabranchcutpricealgorithmsimilartothepursuitevasionproblemforsolvingthisproblem.WestartwithasubsetofpatrolroutesP0c(K)andevasionwalksR0(T),andsolvetheresultinglimitedpatrolproblem(LPP).Wegeneratenewpatrolroutesandevasionwalksasneeded. 5{23 )correspondingtointruderwalkr2R0(T).Wedeneyrtobeadecisionvariablethatequals1ifanintruderfollowingwalkrisdetectedbyasearcherfollowingapatrolcircuitthatwegenerate,and0otherwise,8r2R0(T).Givenanoptimaldualsolution^ofthelinearprogrammingrelaxationtoLPP,wesolvethefollowingpricingproblemtoseekavariablehavinganegativereducedcost:maxPr2R0(T)^ryr,subjecttotherestrictionthat(y1;:::;yjR0(T)j)correspondstoasetofintruderwalksdetectedbyasearcherfollowingapatrolcircuitofperiodnomorethanK. Wecansolvethesearcher'sproblembysolvingaseriesofintegerprogramsasfollows.Letdenotethelengthofthecurrentcircuitunderconsideration.Byconsideringdierentvaluesof2f1;:::;Kgwecanndacircuitthatoptimizesthesearcher'sproblem.NotethatsomevaluesofmaynotcorrespondtoanycircuitsinG.Foreachvalueof,wegenerateatimeexpandednetworkcontaining+1levels,wheretherstlevelcorrespondstotheinitialdeploymentofthesearcher,andthelastlayerisadummylayerthatweusetomodeltherecurringpatrolpatterns.Weconnecteachnodetoitsneighborsinthenextstage.Asbefore,wedeneabinaryvariablextiforalli2N;t=0;:::;,whichequals1ifthesearcherislocatedatnodeiattimet.Wealsodeneaparameterdtir=1ifasearcherlocatedatnodeiattimetcandetectanintruderfollowingwalkr2R0(T).We 124 PAGE 125 (5{29)0yr18r2R0(T) (5{30)xti2f0;1g8i2N;t=0;:::;: Constraints( 5{26 )and( 5{27 )ensurethateachfeasiblesolutioncorrespondstoawalk.Constraints( 5{28 )guaranteethattherstandthelastnodesvisitedbythesearcherarethesame,andhencethesearcher'swalkformsacircuit.Finally,Constraints( 5{29 )relatethexandyvariables,wherewecanonceagainrelaxintegralityrestrictionsontheyvariables. Integerprogramscorrespondingtodierentvaluesofcanbesolvedinanysequence.Wenotethatagoodsolutionobtainedbysolvingthesearcherproblemforaparticularvalueofcanbeusedtopruneproblemstobesolvedlaterfordierentvaluesofbybound.Therefore,wecanstartbysolvingasearcherproblemforthelargestvalueof,sinceasearcherfollowingalongercircuitismorelikelytodetectmoreintruderwalks.Also,weskipany~ifwedeterminethatnocircuitoflength~existsinG. 125 PAGE 126 Tosolvetheintruder'sproblem,wecangenerateatimeexpandednetworkconsistingofLstagessimilartothepursuitevasionproblem.Weconnecteachnodetothecopyofitselfanditsneighborsinthenextstagebyadirectedarchavinglength1.WealsoconnectthenodescorrespondingtostageLtothenodestotheirneighborsintherststagewithdirectedarchavinglength1(modelingthefactthattheoverallsearchpatternrepeatsafterLperiods).Weaddadummystartnodesandadummyendnodeq.Weconnectstoallnodesbyadirectedarchavinglength0(reectingourassumptionthattheintrudercanenterthesystematanytimeandlocation),andconnectallnodestoqbyadirectedarchavinglength0.Finally,wetraceeachselectedsearcher'scircuit,andremovenodesandarcsfromtheintruder'snetworkthatwouldleadtothedetectionoftheintruderbythesearcher. Wecansolvetheintruder'sproblemonthegeneratedgraphbyseekingalongests{qpath.Werstseekatopologicalorderingofthenodesusingastandarddepthrstsearchalgorithm,whosecomplexityisO(N2L)foradenseG.Sinceadirectedgraphisacyclicifandonlyifitishasatopologicalorder,thisstepidentieswhetherthegraphiscyclic.Ifthereisacycleinthisgraph,thentheintrudercanstayinthesystemforeverwithoutbeingdetectedbythesearchers.Inthiscase,wegenerateacutoftype( 5{23 ),andstop.Otherwise,thegraphisacyclic,andgivenatopologicalorderofthenodes,alongestpathcanbefoundinpolynomialtimebyadynamicprogrammingalgorithmwhosecomplexityisO(N2L)( Ahujaetal. 1993 ).IfthelengthofalongestpathisgreaterthanT,thentheintrudercansuccessfullyevadethesearchers.Wecanusesuchalongestwalktogenerateacutoftype( 5{23 ). 126 PAGE 127 5{23 ),andgobacktoStep1.Else,stopprocessingthecurrentsubproblemwithanintegersolution. Asbefore,weinitializeouralgorithmbygeneratingastationarysearcherthatstaysatnodei,andastationaryevaderthatstaysinnodeiforT+1periods,foreachi2N.WediscussseveralbranchingstrategiesthatcanbeusedinStep1inthenextsection. 127 PAGE 128 Barnhartetal. 1998 ; JungerandThienel 2000 ). However,ourmasterproblemisasetcoverproblem,andtheapproachdescribedaboveisnotdirectlyapplicable.AnyconstraintonagroupofvariablesforoursetcoverproblemwouldnecessitateaddingabranchingconstraintlikePx2Xx~inonebranch,andPx2Xx~+1(forsomesuitable~)intheotherbranch.Sincetheseconstraintscannotbeaddedimplicitly,weneedtohandleanewdualvariableforeachbranchingconstraintinthesubproblem. Asanalternative,weproposeamultitieredbranchingrule.Givenafractionalsolution^,werstevaluatethevalueofeachconstraintexpressionvr=Pp2P0(T)dpr^p.Ifthereexistsafractionalvrvalueforsomer2R0(T),thenwebranchontheconstraint( 5{23 )correspondingtorasfollows.Intheupbranch,wesimplychangetherighthandsidevalueoftheconstraintasdvre.Onthedownbranch,wesettheupperboundoftheconstraintexpressiontobvrc(andconvertittoanequalityconstraintifbvrc=1).Notethatbranchingonaconstraintinthismannerdoesnotintroduceanynewdualvariablesorconstraintsthatneedtobeconsideredinanyofthepricingproblems.Anotherbenetofthisbranchingschemeisthefollowing.IfwebranchdownonaconstraintandobtainPp2Pdprp=1,wecanthenusesetpartitiontypeofbranchingschemesonthecorrespondingsubproblem.Thisallowsustobranchonagroupofvariablesinthesubsequentbrancheswithoutdestroyingthepricingproblemstructure. Notethatsignofthedualvariablercanchangeafterbranchingdownontheconstraint( 5{23 )correspondingtor,whichmakesyr=0optimalregardlessofthevaluesofxvariables.Therefore,weneedtoaddconstraintsthatforceyrto1ifthesearcher'schosenpathdetectsintruderr.Hence,weaddconstraintsyixtj8i2N;j2S(i);t=0;:::;T 128 PAGE 129 (5{34) tothesearcher'spricingproblemforthehideandseek,pursuitevasion,andpatrolproblems,respectively. Ingeneralitispossibletohaveafractionalsolution^forwhichallvvaluesareintegral,andthereforethebranchingruledescribedabovecannotbeapplied.Insuchcaseswecanapplyasimplevariablebasedbranchingrule.Ifthereisavariable^pwhosevalueisfractional,wecansimplycreatetwobrancheswith^p=0and^p=1.Inthedownbranch,wesimplyeliminatethecolumncorrespondingto^pfromthesetcoveringformulation.Intheupbranchweneedtoadjusttherighthandsidevectorofoursetcoveringproblembeforeeliminating^p.Ineithercase,weneedtoadjustthepricingproblemssothatthesamevariablecannotberegenerated.Recallthatthesolutionmethodsweproposeforallthreepricingproblemsarebasedonatimeexpandednetworkformulation.LetusdenotebyW(p)thesetoftimeexpandednodeindicescorrespondingtoasearcherwalk(orcircuit)p.Wecanenforcetheconditionthatawalkpisnotgeneratedagainbyaddingthefollowingconstrainttothecorrespondingsearcher'spricingproblem:X(i;t)2W(p)xtijW(p)j1: Branchingonasinglevariableislikelytobequiteweakonthedownbranchsinceonlyoneparticularsearcherwalk(orcircuit)isavoided.Hence,weonlyapplythisruleifourconstraintbasedbranchingrulefails.Alsonotethatthedicultyofsolvingthesetcoveringproblemdoesnotincreaseundereitherbranchingrule,sincenoconstraintsareaddedexplicitlywhilebranching. 129 PAGE 130 Sieketal. 2001 ).Ourbasesetoftestprobleminstancesconsistsof150randomlygeneratedprobleminstancesforwhichtheexpectededgedensityofthegraph(measuredasjEjjNj jNj(jNj1),wherewedonotconsiderselfloopedgesincalculatingedgedensity)is10%,thenumberofnodesNrangesfrom5to25,andthemaximumallowedtimetodetectionTrangesfrom0to5.Ingeneratinginstanceswerstpickedarandomsubsetofedgessothattheedgedensityis10%,andifnecessaryaddedtheminimumnumberofedgesneededtomakethegraphconnected(see Sieketal. ( 2001 )).Wethenaddedselfloopedges,andwegeneratedveprobleminstancesforeachproblemsize,whichisdeterminedbythenumberofnodes.Finally,wesolvedeachprobleminstancewithdierentvaluesofT2f0;:::;5gforthehideandseek,pursuitevasionandpatrolproblems.Ineachcase,weassumethatasearcherlocatedatnodei2Ncanobservenodeiandallnodesadjacenttoit,andhencewesetS(i)=A(i),foralli2N.Weimposeda1200secondtimelimitpastwhichwestoppedtheexecutionofanalgorithminallourexperiments. RecallthatallproblemsthatweconsiderinthischapterreducetotheminimumdominatingsetproblemforT=0.WeusethispropertytocalculateaninitialupperboundbysolvingourLHSformulation,whereweinitializeR0(0)byaddingasearcherlocatedateachnodei2N. Ourrstexperimentfocusesonthehideandseekproblem.Forthisexperiment,weexecutedourbranchandpricealgorithmdescribedinSection 5.2.2.2 onourbasedataset.All150probleminstancesinourdatasetweresolvedtooptimalitywithin12.8seconds.Table 51 displaystheaveragenumberofbranchandboundnodesevaluatedinourbranchandpricealgorithmfordierentvaluesofNandT.Eachvaluerepresentstheaverageoftheresultsobtainedonverandomlygeneratedgraphs.WeobservethatthenumberofbranchandboundnodesthatareexploredincreasesasNincreases,which 130 PAGE 131 Averagenumberofbranchandboundnodesexploredforhideandseekproblem 51111111011.41.81.41.41151111112012.63.41.81.812517.42.22.61.41 52 Table52. Averagenumberofsearchersneededforthehideandseekproblem 521.81111103.42.821.81.61154.63.22.2221.6204.63.62.6221.8255.83.82.8222 52 revealsthatthenumberofsearchersneededincreasesasthegraphgetslarger,anddecreasesasthemaximumallowedtimetodetecttheintruderincreases. WeanalyzetheperformanceofourbranchcutpricealgorithmdescribedinSection 5.3.2.3 forthepursuitevasionproblem.Table 53 showsthatouralgorithmwasabletosolve128outofthe150instanceswithintheprescribedtimelimit.Asexpected,thedicultyoftheproblemincreasesasNandTincrease,sincethissettingallowsformoreevasionroutesfortheintruder,andhencerequiresthesearcherstodevelopmoresophisticatedroutes. Table53. Numberofinstancesthataresolvedwithintimelimitforthepursuitevasionproblem 555555510555555155555552055531225554300 54 displaystheaveragenumberofbranchandboundnodesevaluatedinourbranchcutpricealgorithmforthepursuitevasionproblem.Wenotethatforthis 131 PAGE 132 Averagenumberofbranchandboundnodesexploredforthepursuitevasionproblem 51111111011.831.81.8115153.46.24.2320128.2109334.610114.225123.4869324.641.411.4 55 displaystheaveragenumberofsearchersneededforthisproblem,whereweusethebestknownsolutionsforinstancesthatwerenotsolvedtooptimalitywithinthetimelimit.A Table55. Averagenumberofsearchersneededforthepursuitevasionproblem 521.81111103.432.2222154.63.42.6222204.6432.82.82.6255.843.43.23.23 52 and 55 revealsthatthenumberofsearchersneededforthehideandseekproblemislessthanthatforthepursuitevasionproblem.Thisresultisnotsurprisingsincetheintruderisstationaryintheformerproblem,whileitcanmovetoavoidthesearchersinthelatterproblem. Table56. Numberofinstancesthataresolvedwithintimelimitforthepatrolproblem 555555510555555155555552051111025500000 5.4.2.3 forthepatrolproblem.Table 56 revealsthatouralgorithmforthepatrolproblemcansolvefewerinstancesinourdatasetwithinthetimelimitcomparedtoouralgorithmsfortheotherproblems.Thiscanbeexplainedby(i)ourassumptionthattheintrudercanpickatimetoenterthesystem,and(ii)oursolutionalgorithmforthesearcher'sproblem,whichrequiressolvingmultiplemixedintegerprograms.Therstfactormakesiteasierfor 132 PAGE 133 57 Table57. Averagenumberofbranchandboundnodesexploredforthepatrolproblem 511111110111.41.41.41.415144.620.26.211.4320112.25.44.23.4125153.22.11.21 58 .Asbefore,ourcalculationsarebasedonthebestknownsolutionsanddonotnecessarilycorrespondtooptimalsolutionsfortheinstancesthatwerenotsolvedwithinthetimelimit.However,weobservethatthenumberofsearchersneededforthepatrolproblemislargerthantheothertwoproblemsasexpected. Table58. Averagenumberofsearchersneededforthepatrolproblem 521.81.61.211103.432.82.62.62.6154.63.432.82.82.6204.64.23.83.132.8255.85.34.73.83.63.2 PAGE 134 Inthischapter,wefocusonfutureresearchareasregardingtheproblemswehavedescribedinthepreviouschapters.Weanalyzethetechniquesweemployed,identifyassociatedweaknesses,andsuggestimprovements.Wealsodiscussaresearchtopicthatisbasedonatechniqueforreformulatingthemasterproblemforaclassofbileveloptimizationproblems. 2 ,seealso Tasknetal. ( 2009a ))werstdevelopedanintegerprogrammingformulationoftheproblem,andprescribedabilevelreformulationoftheproblemthathasintegervariablesinbothstages.Ourcomputationaltestsshowedthatboththedirectsolutionoftheintegerprogrammingformulationandourintegerprogrammingbasedcuttingplanealgorithmforthebilevelformulationarecapableofsolvingonlysmallprobleminstancestooptimality.Wethendesignedahybridintegerprogramming/constraintprogrammingalgorithmtoovercomethecomputationaldicultiesencounteredbythersttwoapproaches.Ourhybridapproachrstallocatesnodecopiesthataretobedistributedacrosscongurationsusinganintegerprogrammingformulation,andthenassignsnodestosubgraphsusingaconstraintprogrammingalgorithm.Afterassigningnodestosubgraphs,itpartitionsedgestosubgraphsforeachscenarioinathirdstage,usinganotherintegerprogrammingformulation.Ourcomputationalexperimentsshowthatthehybridapproachsignicantlyoutperformstheotherapproaches. Inourstudy,wehaveassumedthatthenumberofsubgraphs,jKj,isapartoftheprobleminput.InSONETnetworkdesignapplicationthereisnopracticallimitonthenumberofsubgraphs,butalimitisspeciedtomodeltheproblem( Goldschmidtetal. 2003 ; Sheralietal. 2000 ; Smith 2005 ).ChoosingjKjtoosmallmaymaketheprobleminfeasibleorsuboptimal,andchoosingjKjtoolargeincreasesthedicultyoftheproblem 134 PAGE 135 2.4 .Inourexperiments,wemanuallychosejKjsucientlylargetoyieldafeasiblesolutionineachprobleminstance.AnareaforfutureresearchistotreatjKjnotasaparameterbutasavariable,andtondthesmallestpossiblevalueofjKjthatguaranteestheexistenceofanoptimaledgepartitionthatminimizesthenumberofnodecopies.Thisproblemappearstobeverydicultingeneral,butsomeupperboundscanbederivedforourproblem.WerstnotethatchoosingjKj=jEjisguaranteedtoyieldafeasibleedgepartition(andnoteliminateanyfeasibleedgepartitionsfromconsideration),sinceeachedgecanbepartitionedintoauniquesubgraph.Furthermore,ifafeasibleedgepartitionhavinganobjectivefunctionvalue^zisknown,thenb^z=2cisanupperboundonjKj.Thisboundfollowsfromthefactthatthereexistsanoptimalsolutionthatcontainsatleasttwonodesineachnonemptysubgraph.Sucha^zcanbecalculatedbyasimpleimprovementheuristic.WestartwithjKj=jEj,andinitiallyassigneachedgetoauniquesubgraph.Wethenseektwosubgraphsthatcanbemergedwithoutviolatinganyconstraintsinanyscenario,whilealsoimprovingtheobjectivefunctionvalue.Ifwendsuchsubgraphs,wemergethemandrepeatthisprocessuntilnomoresubgraphscanbemerged.Sincethisalgorithmstartswithafeasiblesolution,andretainsfeasibilityineachiteration,ityieldsafeasiblesolution.However,ourpreliminaryanalysissuggeststhattheboundonjKjthatwegetusingthisapproachisquiteweak.ImprovingboundsonjKjandndingthesmallestpossiblenumberofsubgraphsthatyieldsanoptimaledgepartitionisafutureresearcharea. 3 wehavedescribedexactdecompositionalgorithmsforsolvingleafsequencingproblemsarisinginIMRTtreatmentplanning(alsosee Tasknetal. ( 2009b )).Oursolutionalgorithmforthematrixdecompositionintoaperturessatisfyingtheconsecutiveonespropertyisbasedonanintegerprogrammingmodelforndingasetofintensityvaluestobeassignedtoapertures,andabacktrackingalgorithmthatformsaperturesbyndingcompatibleleafpositionsforeachrow.Ourcomputationalresults 135 PAGE 136 Oursolutiontechniquefortheconsecutiveonesmatrixdecompositionproblemandourthreestageapproachtothestochasticedgepartitionproblemarebasedonasimilaridea.Inbothproblems,weadd\aggregatevariables"toourformulations,whichdescribeimportantstructuralpropertiesofsolutions,butarenotenoughbythemselvestoencodecompletesolutions.Ineachcasewerepresenttheoptimizationproblemintermsofouraggregatevariablesinamasterproblem,andprovideasubproblemthatseeksacompletefeasiblesolutioncorrespondingtothevaluesoftheaggregatevariableschosenbythemasterproblem.Inbothapplications,ourmasterproblemsarediscreteoptimizationproblems,whichwesolveusingintegerprogrammingmethods,andoursubproblemsarediscretefeasibilityproblems,whichwesolveusingconstraintprogrammingmethods.Separatingcriticaloptimizationdecisionsfromfeasibilitydecisions,andutilizingstrengthsofintegerandconstraintprogrammingtechniquesinahybridalgorithmhasallowedustoobtainsignicantlybetterresultsthanothermethods.Amajorthemeinourfutureresearchisgoingtobeongeneralizingthishybridapproachtohandleabroaderclassofproblems. InChapter 4 westudiedadierentvariantofthematrixdecompositionproblem,inwhichtheaperturematricesneedtoberectangularinshape(seealso Tasknetal. ( 2008 )).RectangularaperturescanbeformedbyusingconventionaljawsalreadyintegratedintoIMRTtreatmentdevices,anddonotneedanadvancedMLCsystem,whichiscostlytomanufactureandoperate.InChapter 4 ,weproposedanexactoptimizationalgorithmthatcanbeusedtoanalyzewhetherajawsonlytreatmentsystemcandeliveruencemapseciently.Ouralgorithmisbasedonanintegerprogramming 136 PAGE 137 WealsoderivedabilevelBendersdecompositionalgorithmforthisproblem.ThemasterproblemofourBendersdecompositionapproachchoosesasubsetoftherectangularshapesthatcanbeusedindecomposingtheinputmatrix.Later,asubproblemcheckswhethertheselectedsubsetofrectanglescancompletelydecomposetheinputmatrix.Unfortunately,ourcomputationaltestsshowedthatourBendersdecompositionalgorithmiscomputationallyinferiortotheintegerprogrammingapproach.Themainreasonofslowconvergenceistheweaknessofcutsgeneratedineachiteration.Specically,givenaninfeasiblesubsetofrectangleschosenbythemasterproblem,oursubproblem(whichisalinearprogrammingproblem)detectsinfeasibility,andreturnsacut,whichisgeneratedbasedonadualextremeray.However,thisextremerayisamathematicalproofofinfeasibility,butdoesnotnecessarilyidentifytheunderlyingreasonofinfeasibility.Inotherwords,itdoesnotidentifywhichbixelsintheinputmatrixcannotbepartitionedwiththeselectedsubsetofrectangles.Furthermore,therearetypicallymanyextremedualraysforasingleinfeasiblemasterproblemsolution,fromwhichseveralnondominatedcutscanbederived.Onewayofimprovingtheconvergencecanbeapplyingatwodimensionalbinarysearchalgorithmontheinputmatrixtondoutwhichregionoftheinputmatrixcannotbepartitionedwiththeselectedrectangles.Thatis,iftheentirematrixcannotbepartitioned,wetrytopartitionthelefthandsideandtherighthandsidehalvesofthematrixindependently.Ifoneofthesesubmatricescannotbepartitioned,weimmediatelyhaveamorespecicreasonforinfeasibility(andhenceastrongercut),becausethisresultimpliesthattheinfeasibilityinasubmatrixneedstobexedusingasubsetoftherectangles,whichcoverthatpartofthematrix.Thisideacanbeappliedrecursivelytondpossiblymultipleinfeasibleregionsofthematrix.Thecutsassociatedwiththeseinfeasiblesubmatricesaremuchstrongerthanasinglecutderivedbasedontheentirematrix.Furthermore,theinformationregardingthe 137 PAGE 138 Inourstudy,wehavedevelopedsolutiontechniquesfortwoversionsofthematrixdecompositionproblem,whichapplytomostavailableIMRTtreatmentmachinery.However,thereareothertypesofmachinerythathavedierentapertureshaperestrictions,suchasinterdigitationorconnectednessconstraints(seee.g., Lim ( 2002 )).Inarelatedlineofresearch,weareplanningtodesignexactoptimizationalgorithmstosolveothervariantsofthematrixdecompositionproblemtooptimality.Quantifyingtheeectofseveralshapeconstraintsenforcedbydierenttypesofmachineryonradiationdeliveryeciencywouldbeavaluablecontributiontothemedicalphysicseld. 5 weconsideredthreevariantsofagraphsearchproblem:(i)ahideandseekproblem,whereasetofsearchersseekastationaryintruder,(ii)apursuitevasionproblem,wheretheintrudermovestoavoiddetection,and(iii)apatrolproblem,wheresearchersfollowrecurringpatrolroutes.Theaimineachproblemistondtheminimumnumberofsearchersneededsothattheintrudercannotstayinthesystemwithoutbeingdetectedforlongerthanaprespecieddurationoftime.Weproposedabranchcutpricealgorithm,whichcanbeadaptedtoallthreeproblemswithcertainmodications.Ourmaincontributionisthatwedonotmakeanyassumptionsonthetopologyoftheinputgraph,andouralgorithmsworkongeneralgraphs. EventhoughallthreeproblemsthatweconsiderareNPhard,ourcomputationaltestsshowthatthehideandseekandpursuitevasionproblemscanbesolvedtooptimalityformodestproblemssizeswithinreasonablecomputationaltime.However, 138 PAGE 139 Oursetcoveringformulation,whichisbasedontheideathatatleastonesearchermustbechosenforeachroutetheintrudercantake,canbegeneralizedtoothervariantsofthegraphsearchproblem.Inparticular,aproblemthathasbeenwidelystudiedintheliteratureassumesthattheintrudercanresideattheedgesofthegraph.Thisproblemhasbeeninvestigatedfromatheoreticalperspective(see,e.g., Dendrisetal. ( 1997 ); Ellisetal. ( 1994 ); SeymourandThomas ( 1993 ));however,tothebestofourknowledgenoexactoptimizationmethodthatworksongeneralgraphshasbeenproposed.Weareplanningtoextendouralgorithmsothatitcanalsosolvethisproblem. 139 PAGE 140 Smithetal. 2004 ),andinaproductintroductionandinterdictiongame, Smithetal. ( 2008 )considertheadditionofaquadraticordersetofvariablesinthemasterproblem.Thesenewvariablesarepassedtothesubproblem,andaBenderscutisgeneratedintermsofthenewvariablesthatimpliesallofthe(exponentiallymany)Benderscutsthatcouldhavebeengeneratedintheoriginalvariablespace.ThismasterproblemreformulationtechniquehasthepotentialtodramaticallyreducethenumberofiterationsrequiredbyBendersdecompositiontoconvergetoanoptimalsolution,withonlyamodestincreaseinthesizeoftheformulation.Inbothcasesmentionedabove,thetradeoofincreasingmodelsizetoimprovethestrengthofBenderscutswascomputationallybenecial. LetusdescribetheideainmoredetailinthecontextofaSONETnetworkdesignproblem( Smithetal. 2004 ),whichissimilartotheedgepartitionproblemdiscussedinChapter 2 .Thereexistasetofdemandpairs(i;j)2Ethatmaybesatisedonasinglecommunicationsnetwork(alldemandpairshavetobesatisedinouredgepartitionproblem).Thecommunicationsnetworkiscomposedof\rings"k=1;:::;K.Ifbothclientsiandjhavebeenlinkedtoringk,thenwemaychoosetosatisfythedemandrequestbetweeniandjonringk.Deneyqijktobeacontinuousvariablethatrepresentsthefractionofthedemandbetweeniandjthatissatisedonringkinscenarioq(thesevariablesaredenedtobebinaryinouredgepartitionproblemsinceweassumethateachdemandpairneedstobesatisedonasinglering).LetEqbethesetofdemandpairs 140 PAGE 141 wheretheexistenceofscenariosq2Qisduetouncertaindemandsbetweenclientsiandj.Usingastraightforwarddecompositionapproach,theproblemcanbedecomposedsothatthexvariablesarepartsofthemasterproblemformulation,whiletheyqvariablesaredeterminedinsubproblemscorrespondingtoscenariosq2Q.Constraints( 6{1 )and( 6{2 )essentiallystatethatinorderforthecommunicationdemandbetweencustomersiandjtobeassignedtoringk,bothcustomersiandjhavetobeassignedtoringk.Unfortunately,cutsenforcingthisrelationshipcannotberepresentedintheoriginalspaceofxvariables. Smithetal. ( 2004 )showthattherecanexistanexponentialnumberofalternativedualsolutionsassociatedwithamasterproblemsolutionrepresentedby^x,eachleadingtoanondominatedBenderscut.Thentheyreformulatethemasterproblembyaddinguijkvariables,whichrepresenttheminimumofxikandxjk.Inotherwords,uijk=1ifbothcustomersiandjareassignedtoringk.Giventheuvariables,theconstraints( 6{1 )and( 6{2 )canbereplacedbyyqijkuijk8q2Q;(i;j)2Eq;k2K: Smithetal. ( 2004 )showthatasingleBenderscutbasedontheuvariablesdominatesanexponentialnumberofcutsbasedontheoriginalxvariables.Inotherwords,addingaquadraticnumberofvariablestothemasterproblemcansaveanexponentialnumberofiterationsoftheBendersdecompositionalgorithm.Inthisparticularproblem,theauthorsrecognizedthattheyvariablesinthesubproblemaredependentonmin(xik;xjk),andusedthisnonlinearrelationshipbetweenthexvariablestoreformulatethemaster 141 PAGE 142 wherex1andx2arevariablesofthemasterproblem.Sinceboth( 6{4 )and( 6{5 )areoftype,wecantakethecomponentwiseminimumofthetwoconstraintstoobtainmin(a11;a21)y1+min(a11;a21)y2++min(a1n;a2n)ynx1 Sincethelefthandsidesof( 6{6 )and( 6{7 )arethesame,wecancombinethetwoconstraintsintomin(a11;a21)y1+min(a11;a21)y2++min(a1n;a2n)ynmin(x1;x2): Constraint( 6{8 )describesanonlinearrelationshipbetweentheyandxvariables.Atthispoint,themasterproblemcanbereformulatedbyaddingavariablev12=min(x1;x2),andthesubproblemcanbereformulatedbyusingthisvariableasmin(a11;a21)y1+min(a11;a21)y2++min(a1n;a2n)ynv12: Eventhough( 6{9 )isweakerthan( 6{4 )and( 6{5 ),thisreformulationmightimprovetheconvergenceofthealgorithmduetothe\exponentialcut"behavior,especiallyinthebeginningiterationsoftheBendersprocess.Inourfutureresearchweareplanning 142 PAGE 143 143 PAGE 144 Inthisappendixwediscussanintegerprogrammingapproachtodecomposingauencemapintoanumberofaperturesandcorrespondingintensitiesthatisbasedonamodelproposedby Langeretal. ( 2001 ).Givenamaximumnumberofunitintensityapertures,sayT,thisformulationdeterminesthepositionsoftheleftandrightleavesineachrowofeachoftheseapertures.Wedevelopthemodelbyseparatelystudyingfourcomponents: Inparticular,constraints( A{2 )statethateachbixeliseithercoveredbyarighthandleaf,coveredbyalefthandleaf,oruncovered(wherethedvariablesareincludedonlyforconvenienceandcanbesubstitutedoutoftheformulation).Constraints( A{3 )and( A{4 )statethatifanybixel(i;j)iscoveredbyarighthandleaf(resp.lefthandleaf),thenbixel(i;j+1)(resp.(i;j1))shouldbecoveredbyarighthandleaf(resp.lefthandleaf)aswell. 144 PAGE 145 While Langeretal. ( 2001 )imposethefollowingconstraintstoensurethatthesevariableshave(atleast)theirdesiredvalue: wenotethatthefollowingstrongerformulation,whichwouldactuallynotrequireenforcingthezvariablestobebinary,canbeobtainedbydisaggregating( A{6 ).dtijzt8i=1;:::;m;j=1;:::;n;t=1;:::;T:( A{6 Notethatthismodelallowszttobeequaltooneevenifinaperturetnobixelsareexposed,sothatformallyspeaking( A{5 )isanupperboundonthebeamontime.Theobjectivefunctionensuresthatthezvariablestakeontheirminimumpossiblevalue. (Ifanyapertureisusedmorethanoncebutseparatedbyanotherone,weconsiderthesecondoccurrenceoftheaperturetobeanewsetup.However,whenminimizingtotaltreatmenttimetherealwaysexistsanoptimalsolutioninwhichidenticalaperturesaredeliveredsequentially.)Nowletctijandutijbeauxiliarybinaryvariablessuchthattheformerisequaltooneifbixel(i;j)isexposedinaperturetbutnotinaperturet+1andzerootherwise,andthelatterisequaltooneifbixel(i;j)iscoveredinaperturetbutnotinaperturet+1.Thisrelationshipisstatedby Langeretal. ( 2001 )thenusethefollowingconstraintstoensurethatthevariablesgthave(atleast)theirdesiredvalue: 145 PAGE 146 A{9 Similartothecaseofthebeamontime,thismodelallowsgttobeequaltooneevenifaperturestandt+1areidentical,althoughourobjectivefunctionensuresthatthegvariablesarechosensucientlysmall. Langeretal. ( 2001 )thenstudytheproblemofminimizingthenumberofsetups( A{7 )subjecttotheconstraints( A{2 ){( A{4 ),( A{6 ),( A{9 ),theconstraintthatthebeamontimeisminimal: andbinaryconstraintsonthevariables,wherewerecommenddetermining~zviaoneofthepolynomialtimeproceduresmentionedinSection 3.1 .WenotethatanequivalentmodelisobtainedbysimplysettingT=~z,whichreducestheproblemdimension,andhenceshouldbemoreecientthanaddingabeamontimeconstraint. Wewishtominimizethetotaltreatmenttimeasmeasuredby subjecttoconstraints( A{2 ){( A{4 ),( A{6 A{9 A{10 )). 146 PAGE 147 3 .C1PartitionisstronglyNPcomplete. Proof. LetKdenotethesetofallO(n2)ndimensionalbinaryvectorswhoseonesappearconsecutively,wherevkisthebinaryvectorcorrespondingtok2K.Consideraguessedsolutionthatconsistsofjjdimensionalnonnegativeintegervectorsdk,8k2K,wheredk`denotesthenumberoftimesleafpositionvk,k2K,isassignedtointensity`2.Sincealldk`Linsomefeasiblesolution,werestricttheguesseddvectorsassuch.ThesizeoftheguessedvectorsisthusO(n2jjlog2(L)).WecanverifywhetherornotPk2KP`2dk`vk=binO(n2jj)additions.Therefore,C1PartitionisinNP. ToshowthatC1PartitionisNPcomplete,wereduce3Partitiontoit.3PartitionisastronglyNPcompleteproblemandseekswhetheragivenmultisetofintegerscanbepartitionedintotripletshavingthesamesum.Formally,itcanbedenedasfollows(see GareyandJohnson ( 1979 )): QUESTION:CanAbepartitionedintodisjointmultisetsA1;:::;AsuchthatPj2Aiaj=Bfori=1;:::;? Givenanarbitraryinstanceof3Partition,weconstructaninstanceforC1Partitionasfollows.First,wedene^xtobeanintegervectorwhose`thcomponent,^x`,isequaltothenumberofindicesiforwhichai=`.Furthermore,weletbbea(21)dimensional 147 PAGE 148 Assumethatthe3Partitioninstanceisayesinstance,andhencethereexistmultisetsA1;:::;AsuchthatPj2Aiaj=B.Inthiscase,afeasiblesolutionoftheC1Partitioninstanceletsd2j1;`bethenumberofelementsofintensity`inAj,foreachj=1;:::;,andassignsdk`=0forallotherk.Similarly,supposethattheC1Partitioninstanceisayesinstance.Sinceallpositivevaluesinbareadjacentto0,inanyfeasiblesolutiontotheinstanceofC1Partition,wemayonlyuseleafpositionsthatexposeasingleoddindexbixel.Also,sinceB=4 PAGE 149 Agazaryan,N.,T.D.Solberg.2003.Segmentalanddynamicintensitymodulatedradiotherapydeliverytechniquesformicromultileafcollimator.MedicalPhysics30(7)1758{1765. 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