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# Graph and combinatorial algorithms for geometric constraint solving

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GRAPH AND COMBINA TORIAL ALGORITHMS F OR GEOMETRIC CONSTRAINT SOL VING By ANDREW LOMONOSO V A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2004

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Cop yrigh t 2004 b y Andrew Lomonoso v

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Dedicated to H.

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A CKNO WLEDGMENTS I am mostly indebted to m y advisor, Dr. Meera Sitharam, who has b een sup ervising me with patience and ev erlasting in terest. In the b eginning she oered me a v ery extensiv e c hoice of researc h problems and w as willing to sp end considerable time explaining the nature and bac kground of these problems, so I w as able to c ho ose the area that p ersonally app ealed to me. While all these problems w ere in teresting and c hallenging, the problem statemen ts w ere easy to understand, so one could start thinking ab out them righ t a w a y learning necessary tec hnical skills along the w a y My advisor has giv en me ev ery opp ortunit y to acquire these tec hnical skills b y sp ending a lot of her time meeting me, running relev an t sp ecial topics seminars, in tro ducing me to exp erts in the eld and sending me to plen t y of conferences and w orkshops. While allo wing for m y indep enden t gro wth, m y advisor w as trying to mak e the en tire researc h pro cess as ecien t as p ossible. When w orking on one problem she alw a ys k ept another one on \bac kburner", so if the progress on one problem slo w ed do wn somewhat then I could alw a ys turn to the second problem. This arrangemen t w as made p ossible b y her con tagious en th usiasm ab out the researc h topics, en th usiasm that w as highly motiv ating whenev er I w ould feel temp orarily b ogged do wn. Throughout the researc h pro cess m y advisor w as mindful of m y future career, and w as steering me to w ard learning prop er mix of theoretical and practical skills. Also in all the classes where I w as her T eac hing Assistan t, she alw a ys to ok trouble to explain the in tricacies and p oten tial pitfalls of the teac hing pro cess. iv

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T o some advisors the dissertation is not the only nal pro duct, the individual gro wth of a studen t is at least as imp ortan t. I am v ery happ y that Dr. Meera Sitharam is one suc h advisor. I w ould also lik e to thank m y committee mem b ers Drs. Sarta j Sahni, Gerhardt Ritter, Tim Da vis, Sanja y Rank a and Ra vi Ah uja for discussions w e had and their helpful suggestions regarding m y researc h, esp ecially concerning net w ork ro w related issues and relationships b et w een v arious problems considered. I w ould lik e to thank Dr. Christopher Homan for v arious assistance he has pro vided during our join t w ork. I w ould lik e to thank Dr. P ardalos as w ell as Burak and Sandra Eksioglu for helping me with step-cost function approac hes to obtaining appro ximate solution of minim um dense problem. Finally I w ould lik e to thank p eople with whom I ha v e b een w orking on Geometric Constrain t Solving T eam: Jianjun Oung, Naganandhini Koharesw aran, Y ong Zhou, Aditee Kum thek ar, Ramesh Balasubramaniam and Heping Gao for helping me with v arious asp ects of this large researc h area. v

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T ABLE OF CONTENTS page A CKNO WLEDGMENTS . . . . . . . . . . . . . . iv LIST OF FIGURES . . . . . . . . . . . . . . . . viii ABSTRA CT . . . . . . . . . . . . . . . . . . xi CHAPTER1 INTR ODUCTION . . . . . . . . . . . . . . . 1 1.1 Problem Denitions . . . . . . . . . . . . . 1 1.1.1 Dense Graphs . . . . . . . . . . . . 1 1.1.2 Stably Dense Graph . . . . . . . . . . . 1 1.1.3 Minimal Stably Dense Subgraph Problem . . . . . 2 1.1.4 Maximal Stably Dense Subgraph Problem . . . . . 2 1.1.5 Maxim um Stably Dense Subgraph Problem . . . . 3 1.1.6 Minim um Stably Dense Subgraph Problem . . . . 3 1.1.7 Examples of v arious Dense Subgraphs . . . . . . 4 1.1.8 Relationships b et w een v arious Graph Problems . . . 5 1.1.9 Optimal Complete Recursiv e Decomp osition . . . . 5 1.2 Summary of Results . . . . . . . . . . . . . 8 1.3 Related W ork in Algorithms Comm unit y . . . . . . . 8 2 APPLICA TIONS TO GEOMETRIC CONSTRAINT SOL VING . . 10 2.1 Geometric Bac kground . . . . . . . . . . . . 10 2.1.1 Geometric Constrain t Problems . . . . . . . 10 2.1.2 The Main Reason to Decomp ose Constrain t Systems . . 11 2.1.3 Decomp osition Recom bination (DR) Plans . . . . 13 2.1.4 Basic Requiremen ts of a DR-plan . . . . . . . 14 2.1.5 Desirable Characteristics of DR-planners for CAD/CAM 17 2.1.6 F ormal Denition of DR-solv ers using P olynomial Systems 22 2.1.7 F ormal Denition of a DR-planner via Constrain t Graphs 30 2.1.8 Tw o old DR-planners . . . . . . . . . . 45 2.1.9 Constrain t Shap e Recognition (SR) . . . . . . 47 2.1.10 Generalized Maxim um Matc hing (MM) . . . . . 57 2.1.11 Comparison of P erformance of SR and MM . . . . 68 2.1.12 Analysis of Tw o New DR-planners . . . . . . . 69 2.1.13 The DR-planner Condense and its P erformance . . . 73 2.1.14 The DR-planner F ron tier and its P erformance . . . 78 vi

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2.2 Relating Problems of Chapter 1 to some Measures of Chapter 2 85 3 MAXIMAL, MAXIMUM AND MINIMAL ST ABL Y DENSE PR OBLEMS 87 3.1 Finding Maxim um Dense Subgraph . . . . . . . . 87 3.2 Finding a Stably Dense Subgraph . . . . . . . . . 88 3.2.1 Distributing an Edge . . . . . . . . . . 88 3.2.2 Finding Dense and Stably Dense Subgraph . . . . 93 3.2.3 PushOutside() . . . . . . . . . . . . 95 3.3 Structure and Prop erties of F ron tier algorithm . . . . . 100 3.3.1 Informal Description of F ron tier Algorithm . . . . 100 3.3.2 Assumptions . . . . . . . . . . . . . 101 3.3.3 Joining P airs of Clusters . . . . . . . . . 101 3.3.4 Relev an t T ransformation Notation . . . . . . . 101 3.3.5 Recom bination b y F ron tier Algorithm . . . . . . 101 3.3.6 Remo ving Undistributed Edges . . . . . . . . 103 3.3.7 Pseudo co de . . . . . . . . . . . . . 103 3.3.8 Example of actions b y F ron tier algorithm . . . . . 104 3.3.9 Mo difying F ron tier Algorithm for Minimalit y . . . . 105 3.3.10 Complete Decomp osition Prop ert y . . . . . . . 109 3.4 Maxim um Stably Dense Prop ert y . . . . . . . . . 109 3.5 Other Prop erties of F ron tier Algorithm . . . . . . . 109 4 MINIMUM ST ABL Y DENSE SUBGRAPH PR OBLEM . . . . . 112 4.1 Relation of Minim um Stably Dense to other Problems . . . 112 4.1.1 Maxim um Num b er of Edges Problem . . . . . . 112 4.1.2 Decision V ersion of Minim um Dense Subgraph . . . 113 4.1.3 Relationships b et w een t w o Decision Problems . . . . 113 4.1.4 Relationships b et w een Appro ximation Algorithms . . 115 4.2 NP-Completeness of Minim um Stably Dense Subgraph Problem 116 4.3 Sp ecial Cases of Minim um Stably Dense . . . . . . . 119 4.3.1 Flo w-based Solution for No-o v erconstrained Case . . . 119 4.3.2 Prero w-push based Solution for No-o v erconstrained Case 121 4.3.3 Finding Smallest Subgraph of Largest Densit y . . . 122 4.3.4 Case of Bounded Num b er of Ov erconstrain ts . . . . 123 4.3.5 Size Ov erconstrained Graphs . . . . . . . . 125 4.4 Appro ximation Algorithms for Minim um Stably Dense . . . 126 4.4.1 Randomized Appro ximation Algorithms . . . . . 126 4.4.2 Minim um Dense as Minim um Cost Flo ws . . . . . 133 4.4.3 Stating Minim um Dense Problem as IP . . . . . 138 REFERENCES . . . . . . . . . . . . . . . . . 145 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 151 vii

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LIST OF FIGURES Figure page 1{1 Nondense and dense graphs (K=1) . . . . . . . . . 1 1{2 Graph ABCDEF is dense but not stably dense (K=-3, w(v)=2,w(e)=1) 2 1{3 An edge/v ertex w eigh ted graph . . . . . . . . . . 4 1{4 Original graph G and a corresp onding RD-dag . . . . . . 6 1{5 Another p ossible RD-dag . . . . . . . . . . . . 7 1{6 Original graph G and NOT a RD-dag . . . . . . . . . 7 1{7 Original graph G RD-dag and complete RD-dag . . . . . . 7 2{1 A solv able system of equations . . . . . . . . . . . 15 2{2 Step 1 rectangles, Step 3 o v als . . . . . . . . . . 16 2{3 CAD/CAM/CAE master mo del arc hitecture . . . . . . . 20 2{4 A constrain t graph . . . . . . . . . . . . . . 32 2{5 Generically unsolv able system . . . . . . . . . . . 34 2{6 Generically unsolv able system that has a solv able constrain t graph . 35 2{7 Original geometric constrain t graph G 1 and simplied graph G 2 . 37 2{8 Geometric constrain t graph and a DR-plan . . . . . . . 38 2{9 Another p ossible DR-plan . . . . . . . . . . . . 40 2{10 The original, cluster graph and the simplied graphs . . . . . 50 2{11 Action of the simplier on G i and C i during Phase One . . . . 51 2{12 Action of the simplier during Phase Tw o . . . . . . . 52 2{13 This solv able graph w ould not b e recognized as solv able b y SR . . 52 2{14 W eigh t of all v ertices is 2, w eigh t of all edges is 1 . . . . . 53 2{15 Tw o triconnected subgraphs not comp osed of triangles . . . . 55 2{16 Solv able graph consisting of n= 3 solv able triangles . . . . . 55 viii

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2{17 Constrain t graph, in tended and actual decomp ositions . . . . 58 2{18 Original graph and it decomp osition . . . . . . . . . 59 2{19 Mo died bipartite graph and maxim um ro w in this graph . . . 60 2{20 Constrain t graph and net w ork ro w with 3 extra ro w units at AB . 61 2{21 DR-plan dep ends on the initial c hoice . . . . . . . . . 62 2{22 Maxim um ro w and decomp osition giv en the bad initial c hoice . . 63 2{23 Decomp osition giv en the go o d initial c hoice . . . . . . . 63 2{24 Bad b est-c hoice appro ximation . . . . . . . . . . . 66 2{25 Sequen tial extension . . . . . . . . . . . . . 73 2{26 Sequence of simplications from left to righ t . . . . . . . 74 2{27 Original and simplied graphs . . . . . . . . . . . 76 2{28 Bad b est-c hoice appro ximation . . . . . . . . . . . 77 2{29 The simplied graph after three clusters has b een replaced b y edges 79 2{30 Original graph BCDEIJK is dense, new graph MCDEIJK is not . 81 2{31 1 =n w orst-c hoice appro ximation factor of DR-planner F ron tier . . 82 3{1 Before the distribution of edge ED . . . . . . . . . . 90 3{2 After the distribution of edge ED . . . . . . . . . . 90 3{3 Before the distribution of edge AD . . . . . . . . . 91 3{4 Before the distribution of edge BD . . . . . . . . . . 91 3{5 Lo cating dense graph instead of stably dense . . . . . . . 93 3{6 Before the distribution of edge BD . . . . . . . . . . 93 3{7 Dense graph BCD found instead of maximal dense ABCD . . . 95 3{8 Graphs ABCDEF and F GHIJA are maximal stably dense in 3D . 97 3{9 Dense graph ABC found instead of minimal dense BC . . . . 97 3{10 Actions of Minimal() algorithm . . . . . . . . . . 98 3{11 T ransformation b y F ron tier Algorithm . . . . . . . . . 102 3{12 General transformation b y F ron tier Algorithm . . . . . . 102 ix

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3{13 Edge CF will not b e distributed, edge A C will . . . . . . 103 3{14 Initial graph . . . . . . . . . . . . . . . . 105 3{15 After A-C ha v e b een distributed . . . . . . . . . . 105 3{16 After A-G ha v e b een distributed . . . . . . . . . . 106 3{17 After A-P ha v e b een distributed . . . . . . . . . . 106 3{18 Corresp onding complete recursiv e decomp osition . . . . . . 106 3{19 Enforcing cluster minimalit y . . . . . . . . . . . 108 3{20 Example of nding minimal . . . . . . . . . . . 109 3{21 Remo ving AF do es not aect solv abilit y of en tire graph . . . . 110 4{1 Reduction . . . . . . . . . . . . . . . . 114 4{2 Gadgets for reduction to CLIQUE . . . . . . . . . . 117 4{3 H ( v ) represen ting v ertex v . . . . . . . . . . . . 120 4{4 Coun terexample for DRFMA . . . . . . . . . . . 132 4{5 Net w ork . . . . . . . . . . . . . . . . . 133 4{6 Cost functions . . . . . . . . . . . . . . . 134 4{7 Net w ork and a ro w for p=3 . . . . . . . . . . . 135 4{8 Graph corresp onding to Figure 4{7 . . . . . . . . . 135 4{9 Flo w for p=2 . . . . . . . . . . . . . . . 136 x

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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y GRAPH AND COMBINA TORIAL ALGORITHMS F OR GEOMETRIC CONSTRAINT SOL VING By Andrew Lomonoso v Ma y 2004 Chair: Meera Sitharam Ma jor Departmen t: Computer and Information Science and Engineering Geometric constrain ts are at the heart of CAD/CAM applications and also arise in man y geometric mo deling con texts suc h as virtual realit y rob otics, molecular mo deling, teac hing geometry etc. Informally a geometric constrain t problem consists of a nite set of geometric ob jects and a nite set of constrain ts b et w een them. The geometric ob jects are dra wn from a xed set of t yp es suc h as p oin ts, lines, circles and conics in the plane, or p oin ts, lines, planes, cylinders and spheres in 3 dimensions. The constrain ts are spatial and include logical constrain ts suc h as incidence, tangency p erp endicularit y and metric constrain ts suc h as distance, angle, radius. The spatial constrain ts can usually b e written as algebraic equations whose v ariables are the co ordinates of the participating geometric ob jects. A solution of a geometric constrain t problem is a real zero of the corresp onding algebraic system. Curren tly there is a lac k of eectiv e spatial v ariational constrain t solv ers and assem bly constrain t solv ers that scale to large problem sizes and can b e used in teractiv ely b y the designer as conceptual to ols throughout the design pro cess. xi

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The requiremen t is a constrain t solv er that uses geometric domain kno wledge to dev elop a plan for decomp osing the constrain t system in to small subsystems, whose solutions can b e recom bined b y solving other small subsystems. The primary aim of this decomp osition plan is to restrict the use of direct algebraic/n umeric solv ers to subsystems that are as small as p ossible. Hence the optimal or most ecien t decomp osition plan w ould minimize the size of the largest suc h subsystem. An y geometric constrain t solv er should rst solv e the problem of ecien tly nding a close-to-optimal decomp osition-recom bination (DR) plan, b ecause that dictates the usabilit y of the solv er. In this thesis w e state this problem of nding a close-to-optimal solution as a problem that deals with w eigh ted graphs and also iden tify sev eral imp ortan t subproblems. One class of suc h subproblem in v olv es nding dense subgraphs graphs suc h that sum of w eigh ts of its edges is greater than sum of w eigh ts of its v ertices. Dense graphs that presen t in terest for nding a DR-plan are (a) minim um (smallest p ossible dense graphs), (b) minimal (not con taining an y other dense subgraphs), (c) maxim um (largest dense ones), (d) maximal (not con tained in an y other dense subgraph). This thesis presen ts p olynomial time algorithms for problems (b), (c) and (d). Problem (a) is sho wn to b e NP-complete, and v arious appro ximation algorithms are suggested, as w ell as explicit solutions for sp ecial cases that arise from CAD/CAM applications. xii

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CHAPTER 1 INTR ODUCTION 1.1 Problem Denitions 1.1.1 Dense Graphs Motiv ation. Supp ose that w e w an ted to nd a smallest subgraph of a giv en graph that has at least t wice the n um b er of edges as v ertices. Or three times. Or 5 times more edges than v ertices. Giv en: an edge and v ertex w eigh ted graph G = ( V ; E ) and a constan t K W eigh ts of v ertices and edges are denoted b y w ( v ) ; w ( e ) resp ectiv ely A graph G is called dense if P e 2 G w ( e ) P v 2 G w ( v ) K F unction d ( G ) = P e 2 G w ( e ) P v 2 G w ( v ) is called density of G A B C 2 2 2 2 2 2 A B C 2 2 2 1 1 5 Figure 1{1: Nondense and dense graphs (K=1) In man y applications w e are in terested in nding a subgraph whose densit y is \uniform", i.e., not con tributed b y some small o v erly dense part. Densit y of this graph is \stable" or preserv ed ev en when some o v erly dense part is replaced b y a barely dense part. F ollo wing denitions describ e this concept. 1.1.2 Stably Dense Graph A graph A that has d ( A ) > K is over c onstr aine d A graph G suc h that d ( G ) = K and 8 A G; d ( A ) K is wel lc onstr aine d 1

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2 A graph G is stably dense if d ( G ) K and after replacing an y of it o v erconstrained non trivial subgraph of b y w ell-constrained subgraph G remains dense. F or example graph ABCDEF in Figure 1{2 is dense but not stably dense ( K = 3 ; w ( v ) = 2 ; w ( e ) = 1 ; 8 v ; w ) b ecause mo died graph AF G is not dense an ymore. F E D C B A A F G 3 Figure 1{2: Graph ABCDEF is dense but not stably dense (K=-3, w(v)=2,w(e)=1) Man y applications require nding smallest or largest subgraph (un w eigh ted) whic h has t wice the n um b er of edges as v ertices. If w e call graphs that ha v e this prop ert y \dense" (b y setting w eigh ts of v ertices and edges appropriately) then the problem b ecomes that of nding \smallest" or \largest dense subgraph." F ollo wing denitions formalize these notions. 1.1.3 Minimal Stably Dense Subgraph Problem A stably dense graph A is called minimal stably dense if 6 9 B A s.t. B is stably dense. Giv en a graph G a minimal stably dense sub gr aph pr oblem in v olv es lo cating a minimal stably dense subgraph A G if suc h A exists. Note that A is minimal stably dense i A is minimal stably dense. 1.1.4 Maximal Stably Dense Subgraph Problem A stably dense graph A is called maximal stably dense if 6 9 B ; A B G s.t. B is stably dense. Giv en a graph G a maximal stably dense sub gr aph pr oblem in v olv es lo cating a maximal stably dense subgraph A G if suc h A exists.

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3 1.1.5 Maxim um Stably Dense Subgraph Problem A largest (in terms of the n um b er of v ertices) maximal stably dense subgraph is called maximum stably dense Giv en a graph G a maximum stably dense sub gr aph pr oblem in v olv es lo cating a maxim um stably dense subgraph A G if suc h A exists. Note that a graph G can ha v e sev eral maxim um stably dense subgraphs (that ha v e the same size). 1.1.6 Minim um Stably Dense Subgraph Problem A smallest (in terms of the n um b er of v ertices) minimal stably dense subgraph is called minimum stably dense Giv en a graph G a minimum stably dense sub gr aph pr oblem in v olv es lo cating a minim um stably dense subgraph A G if suc h A exists. Note that a graph G can ha v e sev eral minim um stably dense subgraphs (that ha v e the same size). Also note that subgraph A is minim um dense if and only if A is minim um stably dense, and if A is minim um dense, then A is also minimal dense. Minimizing n um b er of edges vs n um b er of v ertices There is a mo dication of Minim um Stably Dense Subgraph Problem, where w e w an t to minimize n um b er of edges (and not v ertices) of stably dense subgraph A F ollo wing results demonstrate that optimal solution of mo died problem will yield a constan t factor appro ximate solution of the original problem. Let A G b e the dense subgraph d ( A ) 1, suc h that j E ( A ) j is minim um o v er all dense subgraphs of G Let B G b e the dense subgraph d ( B ) 1, suc h that j V ( B ) j is minim um o v er all dense subgraphs of G Lemma 1 Numb er of e dges j E ( A ) j is at most 3 j V ( A ) j similarly j E ( B ) j is at most 3 j V ( B ) j

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4 Pro of Let k b e an a v erage degree of A then d ( A ) = j V ( A ) j k = 2 2 j V ( A ) j 1. Since A is minim um dense, then remo v al of a v ertex v 2 A suc h that v has smallest degree in A will result in nondense graph A n v Degree of v could b e at most k so this forces 1 > d ( A n v ) j V ( A ) j ( k = 2 2 1) and therefore k 6, and j E ( A ) j = 3 j V ( A ) j W e can use this lemma to relate sizes of A and B Claim 1 F or A and B dene d ab ove, j V ( A ) j 3 = 2 j V ( B ) j 1 = 2 Pro of 2 j V ( A ) j + 1 j E ( A ) j j E ( B ) j 3 j V ( B ) j First inequalit y is due to densit y of A second inequalit y is due to A ha ving smallest n um b er of edges among all dense subgraphs and third inequalit y is due to Lemma 1 1.1.7 Examples of v arious Dense Subgraphs Consider Figure 1{3 If K = 0 then BCDE is minimal (stably) dense, EDF is minim um (stably) dense, A GH is maximal stably dense and ABCDEF is maxim um stably dense subgraph. F 1 2 3 2 2 2 1 2 2 2 2 2 2 2 G H 2 2 2 2 2 E A B C D Figure 1{3: An edge/v ertex w eigh ted graph

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5 1.1.8 Relationships b et w een v arious Graph Problems Here w e will sho w that the problem of nding minim um (minimal) dense subgraph can b e reduced in p olynomial time to the problem of nding a maxim um (maximal) dense subgraph. Let X b e a bipartite graph, comprised of t w o sets of v ertices X 1 and X 2 Let A b e a subset of X 1 a set of all v ertices in X 2 that are connected to some v ertex in A will b e denoted b y I ( A ). No w let G = ( V ; E ) b e giv en graph. Let B G = ( V B [ E B ; F ) b e the bipartite graph corresp onding to G where V B represen ts v ertices V of G E B represen ts edges E of G and in this graph B G v ertex v 2 V B is connected to v ertex e 2 E B b y an edge (in F ) if and only if v is an endp oin t of e in G Lemma 2 L et A b e a dense sub gr aph of G L et E A B G b e the set c orr esp onding to al l e dges of A Then A = I ( E A ) and ther efor e j E A j 3 I ( E A ) K by denition of density of A Henc e nding lar gest E A such that j E A j 3 I ( E A ) K would nd maximum (numb er of e dgeswise) dense gr aph. Lemma 3 L et A b e a dense sub gr aph of G L et C = V n A Note that I ( C ) = E n E A Henc e d ( G ) = j E A j + I ( C ) 3 j V A j 3 j C j and sinc e d ( A ) K ; d ( G ) I ( C ) + 3 j C j K and 3 j C j I ( C ) K d ( G ) Ther efor e nding smal lest dense sub gr aph, is e quivalent to nding lar gest C such that 3 j C j I ( C ) K d ( G ) Claim 2 F r om pr evious two lemmas it fol lows that the pr oblem of nding minimum (minimal) dense sub gr aph c an b e r e duc e d in p olynomial time to the pr oblem of nding a maximum (maximal) dense sub gr aph. (Note that r everse r e duction wil l not work due to the lack of symmetry in the bip artit gr aph a vertex c an have have any numb er of adjac ent e dges, but every e dge has exactly two endp oints.) 1.1.9 Optimal Complete Recursiv e Decomp osition A r e cursive de c omp osition of a graph G in v olv es constructing a so-called RD-dag of G dened b elo w.

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6 A RD-dag of graph G is directed acyclic graph R = ( R E ; R V ) whic h has follo wing prop erties. The rst prop ert y of R is that O V R V where O V is a cop y of the en tire set of v ertices of G No w let X b e a no de in R then U ( X ) denotes a set of v ertices in M V suc h that there is an orien ted path from v to X for ev ery v 2 U ( X ). F or example in Figure 1{4 U ( S 2 ) = f C ; D ; E g (in all examples in this subsection all edges of G ha v e w eigh t 1, all v ertices ha v e w eigh t 2, constan t K = 3). The second prop ert y of R is that for ev ery sink v ertex X of R corresp onding subgraph U ( X ) is a maximal stably dense subgraph of G The third prop ert y of R is that for ev ery no de S i in R with exception of no des in O V corresp onding subgraph U ( S i ) is stably dense. The nal prop ert y of R is that ev ery S i has a cluster minimality pr op erty i.e., that the set P ( S i ) of all ancestors of S i do es not con tain a prop er non trivial subset P 0 ( S i ) P ( S i ) ; j P 0 S i j 6 = 1 suc h that [ X 2 P 0 ( S i ) U ( X ) is stably dense. F or example in Figure 1{6 no de S 2 do es not ha v e cluster minimalit y prop ert y (for prop er subset P 0 = S 1 [ B ; U ( P 0 ) = B C D E is stably dense, hence Figure 1{6 is not a RD-dag. Note that a graph G can ha v e sev eral dieren t RD-dags, see Figure 1{4 and Figure 1{6 S 1 S 2 A C B S 3 D D E A B C E Figure 1{4: Original graph G and a corresp onding RD-dag A c omplete r e cursive de c omp osition of a graph G in v olv es constructing a so-called c omplete RD-dag of G A complete RD-dag of G is a RD-dag of G with

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7 C E D S 2 S 1 S 3 A C B B D E A Figure 1{5: Another p ossible RD-dag A C B A B C E D S S 1 2 D E Figure 1{6: Original graph G and NOT a RD-dag additional requiremen t that for ev ery maximal stably dense subgraph M of G there is a corresp onding no de M 0 2 R suc h that U ( M 0 ) = M F or example middle dag in Figure 1{6 is a RD-dag but not a complete RD-dag of G while the righ t dag is a complete RD-dag. A B D F E C B A F E D C B A 2 1 F E D C S S 3 2 1 S S S Figure 1{7: Original graph G RD-dag and complete RD-dag Since giv en graph G can ha v e sev eral p ossible RD-dags, an imp ortan t measure of eciency of a RD-dag R is it maximum fan-in dened as a maxim um indegree among all v ertices of R W e will pro vide motiv ation for this criteria in

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8 Section 2.1.1 F or example in Figure 1{4 maxim um fan-in is 1. The problem of nding optimal (c omplete) r e cursive de c omp osition of a graph G in v olv es nding (complete) RD-dag of G that has smallest maxim um fan-in among all other (complete) RD-dags of G Non trivial subgraphs. In practice w e are only in terested in nding maximal/minimal/minim um/maxim um stably dense subgraphs of size 3 or more (i.e., not v ertices or edges). Graphs of size 2 or less will b e referred to as trivial 1.2 Summary of Results Numeric en tries are sections where p olynomial time algorithms (or NPcompleteness) are demonstrated. Problem Bounded P arameter Un b ounded V ersion Minim um (stably) dense NP-complete, 4.2 NP-complete, 4.2 sp ecial cases and appr 4.3 4.4 Maximal stably dense 3.2.3 3.2.3 Minimal (stably) dense 3.2.3 3.2.3 Maxim um stably dense 3.4 3.4 Maxim um dense 3.1 NP-complete, 3.1 Maxim um n um b er of edges, 4.1.1 N/A NP-complete Measures and P erformance for GCS 2.1.6 2.1.14 N/A Complete recursiv e decomp osition 3.3.10 Unkno wn 1.3 Related W ork in Algorithms Comm unit y Finding densest graph. The problem of nding subgraph A G of xed size j A j = k that maximizes P w ( e ( A )) is NP-Complete (could b e sho wn b y reduction from CLIQUE, see Asahiro and Iw ama (1995 )). The problem of nding A G that maximizes P w ( e ( A )) P w ( v ( A )) can b e solv ed in p olynomial time using parametric ro w tec hniques. The problem of nding A G s.t j A j k for a giv en k and A maximizes P w ( e ( A )) j A j is NP-Complete (could b e sho wn b y reduction from CLIQUE).

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9 Section 4.1.1 con tains more details on the relationship b et w een these and our graph problems. Clique-lik e results. Let v and e b e the n um b er of v ertices and edges of graph G resp ectiv ely It w as sho wn b y ( Asahiro and Iw ama, 1995 ) that nding a subgraph of G that has at most a v ertices and at least b edges is NP-Complete for b = a ( a 1) = 2 NP-Complete for a = v = 2 ; b e= 4(1 + O (1 = p v )) NP-Complete for a v = 2 ; b a 1+ ; b e= 4(1 + 0 : 45 + O (1)) can b e done in p olynomial time for a = v = 2 ; b e= 4(1 O (1 =v ) (b y greedy algorithm) Semidenite programming based approac hes. F or a maxim um n um b er of edges problem ( F eige and Seltzer, 1997 ) giv es a semidenite programming relaxation appro ximation algorithm with appro ximation ratio roughly O ( n=k ). ( Go emans, 1996 ) studied a linear relaxation of the problem, getting appro ximation ratio of O ( n 1 = 2 ) when k = n= 2. ( Sriv asta v and W olf, 1999 ) used semidenite programming relaxation to get appro ximation factor within .5 for k = n= 2 and ( Y e and Zhang, 1997 ) w ere able to design .5866 appro ximation factor SDP based algorithm for the case of k = n= 2.

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CHAPTER 2 APPLICA TIONS TO GEOMETRIC CONSTRAINT SOL VING 2.1 Geometric Bac kground 2.1.1 Geometric Constrain t Problems Geometric constrain ts are at the heart of computer aided engineering applications (see, e.g., Homann (1997 ); Homann and Rossignac (1996 )), and also arise in man y geometric mo deling con texts suc h as virtual realit y rob otics, molecular mo deling, teac hing geometry and so on. In this dissertation w e will b e lo oking at geometric constrain t systems primarily within the con text of pro duct design and assem bly Figure 2{3 illustrates those (b oldface) comp onen ts within a standard CAD/CAM/CAE master mo del arc hitecture ( Bronsv o ort and Jansen, 1994 ; Krak er et al., 1997 ; Homann and Joan-Arin y o, 1998 ) where our graph problems are relev an t. Informally a ge ometric c onstr aint pr oblem consists of a nite set of geometric ob jects and a nite set of constrain ts b et w een them. The geometric ob jects are dra wn from a xed set of t yp es suc h as p oin ts, lines, circles and conics in the plane, or p oin ts, lines, planes, cylinders and spheres in 3 dimensions. The constrain ts are spatial and include logical constrain ts suc h as incidence, tangency p erp endicularit y and metric constrain ts suc h as distance, angle, radius. The spatial constrain ts can usually b e written as algebraic equations whose v ariables are the co ordinates of the participating geometric ob jects. A solution of a ge ometric c onstr aint pr oblem is a real zero of the corresp onding algebraic system. In other w ords, a solution is a class of v alid instan tiations of the geometric elemen ts suc h that all constrain ts are satised. Here, it is understo o d 10

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11 that suc h a solution is in a particular geometry for example the Euclidean plane, the sphere, or Euclidean 3 dimensional space. F or recen t reviews of the extensiv e literature on geometric constrain t solving see Homann et al. (1998 ); Kramer (1992 ); F udos (1995 ). 2.1.2 The Main Reason to Decomp ose Constrain t Systems Curren tly there is a lac k of eectiv e spatial v ariational constrain t solv ers and assem bly constrain t solv ers that scale to large problem sizes and can b e used in teractiv ely b y the designer as conceptual to ols throughout the design pro cess. Almost all curren t CAD/CAM systems primarily use a non-v ariational, historybased 3 dimensional constrain t mec hanism. This basic inadequacy in spatial constrain t solving seriously hinders progress in the dev elopmen t of in telligen t and agile CAD/CAM/CAE systems. One go v erning issue is eciency: computing the solution of the nonlinear algebraic system that arises from geometric constrain ts is computationally c hallenging, and except for v ery simple geometric constrain t systems, this problem is not tractable in practice without further mac hinery The so-called constrain t propagation based solv ers (e.g. Gao and Chou (1998a ); Klein (1998 )) generally suer from a dra wbac k that cannot b e easily o v ercome: they nd it dicult to decomp ose cyclically constrained systems, an essen tial feature of v ariational problems. Direct approac hes to algebraically pro cessing the en tire system include the follo wing: 1) standard metho ds for p olynomial ideal mem b ership and lo cating solutions in algebraically closed elds, for example using Gr obner bases or the W u-Ritt metho d; 2) n umerous algorithms and implemen tations for solving o v er the reals based on the metho ds of, for example, Cann y (1993 ); Renegar (1992 ); Collins (1975 ); Lazard (1981 ) and

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12 3) algorithms for decomp osing and solving sparse p olynomial systems based on Cann y and Emiris (1993 ); Sturmfels (1993 ); Kho v anskii (1978 ); Sridhar et al. (1993 1996 ). These direct algebraic solv ers deal with general systems of p olynomial equations: that is, they do not exploit geometric domain kno wledge; partly as a result, they ha v e at least exp onen tial time complexit y and they are slo w in practice as w ell. In addition, they do not tak e in to accoun t design considerations and as a result, cannot assist in the conceptual design pro cess. These dra wbac ks apply to direct n umerical solv ers as w ell, including those that use homotop y con tin uation metho ds, see, for example, Durand (1998 ). The problem w ould b e further comp ounded if w e allo w ed constrain ts that are natural in the design pro cess, but whic h m ust b e expressed as inequalities, suc h as \p oin t P is to the left of the orien ted line L in the plane." Suc h additions w ould necessitate using cylindrical algebraic decomp osition based tec hniques ( Collins, 1975 ), suc h as Grigor'ev and V orob jo v (1988 ); Lazard (1991 ); W ang (1993 ) whic h ha v e a theoretical w orst-case complexit y of O (2 n 2 ), where n is the algebraic size of the problem; or alternativ ely require the use of nonlinear optimization tec hniques, all of whic h are slo w enough in practice that they do not represen t a viable option for large problem sizes. With regard to eciency the follo wing rule of th um b has therefore emerged from y ears of exp erimen tation with geometric, spatial constrain t solv ers in engineering design and assem bly: the use of direct algebraic/n umeric solv ers for solving large subsystems renders a geometric constrain t solv er practically useless: (see Durand (1998 ) for a natural example of a geometric constrain t system with 6 primitiv e geometric ob jects and 15 constrain ts, whic h has rep eatedly deed attempts at tractable solution). The o v erwhelming cost in geometric constrain t solving is directly prop ortional to the size of the largest subsystem that is solv ed using a direct algebraic/n umeric solv er. This size dictates the practical utilit y of

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13 the o v erall constrain t solv er, since the time complexit y of the constrain t solv er is at least exp onen tial in the size of the largest suc h subsystem. 2.1.3 Decomp osition Recom bination (DR) Plans Therefore, the constrain t solv er should use geometric domain kno wledge to dev elop a plan for decomp osing the constrain t system in to small subsystems, whose solutions can b e recom bined b y solving other small subsystems. The primary aim of this decomp osition plan is to restrict the use of direct algebraic/n umeric solv ers to subsystems that are as small as p ossible. Hence the optimal or most ecien t decomp osition plan w ould minimize the size of the largest suc h subsystem. An y geometric constrain t solv er should rst solv e the problem of ecien tly nding a close-to-optimal de c omp osition-r e c ombination (DR) plan b ecause that dictates the usabilit y of the solv er. Finding a DR-plan can b e done as a pre-pro cessing step b y the constrain t solv er: a robust DR-plan w ould remain unc hanged ev en as minor c hanges to n umerical parameters or other suc h on-line p erturbations to the constrain t system are made during the design pro cess. In addition to optimalit y (eciency), other equally imp ortan t (and sometimes comp eting) issues arise from the fact that a DR-plan is a k ey conceptual comp onen t of the CAD mo del and should aid the the o v erall design or assem bly pro cess. These issues will b e discussed under \Desirable c haracteristics" later in this section. A clean and precise form ulation of the DR-planning problem is therefore a fundamen tal necessit y T o our kno wledge, despite its longstanding presence, the DR-problem has not y et b een clearly isolated or precisely form ulated, although there ha v e b een man y prior, sp ecialized DR-planners that utilize geometric domain kno wledge ( Bouma et al., 1995 ; Ow en, 1991 1993 ; Homann and V ermeer, 1994 1995 ; Latham and Middleditc h, 1996 ; F udos and Homann, 1996b 1997 ; Cripp en and Ha v el, 1988 ; Ha v el, 1991 ; Hsu, 1996 ; Ait-Aoudia et al., 1993 ; P ab on, 1993 ; Kramer, 1992 ; Serrano and Gossard, 1986 ; Serrano, 1990 ). See Homann et al.

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14 (1998 ) for an exp osition of t w o primary classes of existing metho ds of decomp osing geometric constrain t systems; represen tativ e algorithms from these t w o classes are extensiv ely analyzed in Section 2.1.8 In the next t w o subsections w e informally describ e b oth the basic requiremen ts of a DR-plan(ner) that dictate its o v erall structure, as w ell as desirable c haracteristics of the DR-plan(ner) that impro v e eciency and assist in the design pro cess. 2.1.4 Basic Requiremen ts of a DR-plan Recall that a DR-plan sp ecies a plan for decomp osing a constrain t system in to small subsystems and recom bining solutions of these subsystems later. Therefore the rst requiremen t of a DR-plan is that the solutions of the small subsystems in the decomp osition can b e recom bined in to a solution of the en tire system. In other w ords, it should b e p ossible to substitute the (set of ) solution(s) of eac h subsystem in to the en tire system in a natural manner, resulting in a simpler system. Secondly w e w ould lik e these in termediate subsystems that are solv ed during the decomp osition and recom bination to b e geometrically meaningful. T ogether, these t w o requiremen ts on the in termediate subsystems translate to a requiremen t that the subsystems b e geometrically rigid. A rigid or solvable subsystem of the constrain t system is one for whic h the set of real-zero es of the corresp onding algebraic equations is discrete (that is, the corresp onding real-algebraic v ariet y is zero dimensional), after the lo cal co ordinate system is xed arbitrarily that is, after an appropriate n um b er of degrees of freedom D are xed arbitrarily The constan t D is usually the n um b er of (translational and rotational) degrees of freedom a v ailable to an y rigid ob ject in the giv en geometry (3 in 2 dimensions, 6 in 3 dimensions, and so on.) and in some cases, D dep ends on other symmetries of the subsystem. An under c onstr aine d system is not solv able, that is, its set of real zero es is not discrete (non-zero-dimensional). A

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15 A B C D: (x1-x2)^2+(y1-y2)^2-A^2=0 E: (x2-x3)^2+(y2-y3)^2-B^2=0 P3 F: (x3-x1)^2+(y3-y1)^2-C^2=0 P1 P2 Figure 2{1: A solv able system of equations wel lc onstr aine d system is a solv able system where remo v al of an y constrain t results in an underconstrained system. An over c onstr aine d system is a solv able system in whic h there is a constrain t whose remo v al still lea v es the system solv able. Solv able systems of equations are therefore w ellconstrained or o v erconstrained. I.e, the constrain ts force a nite n um b er of isolated real solutions, so one solution cannot b e obtained b y an innitesimal p erturbation of another. F or example, Figure 2{1 is a solv able subsystem of three p oin ts and three distances b et w een pairs of p oin ts. A c onsistently o v erconstrained system is one whic h has at least one real zero. Note. It is imp ortan t to distinguish \solv able" from \has a real solution." Although (inconsisten tly) o v erconstrained (or ev en certain w ellconstrained) systems ma y ha v e no real solutions at all, b y our denition, since their set of real zero es is discrete, they w ould still b e considered \solv able." In general, whenev er a subsystem of a constrain t system is detected that has no real solutions, then the solution pro cess w ould ha v e to immediately halt and inform the designer of this fact. } Informally a geometric constrain t solv er whic h solv es a large constrain t system E b y using a DR-planner { to guide a direct algebraic/n umeric solv er capable of solving small subsystems { pro ceeds b y rep eatedly applying the follo wing three steps at eac h iteration i

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16 E E = T ( E ) E = T ( E ) E = T T ( E )n-1 1 1 2 2 3 2 1 1 1S T ( S ) S S T ( S ) T(T(S)) S1 1 1 2 3 2 2 2 1 1 n . .Figure 2{2: Step 1 rectangles, Step 3 o v als 1. Find a small solv able subsystem S i of the (curren t) en tire system E i (at the rst iteration, this is simply the giv en constrain t system E i.e, E 1 = E ). This step is indicated b y a rectangle in Figure 2{2 The subsystem S i could b e also c hosen b y the designer. 2. Solv e S i using the direct algebraic/n umeric solv er. 3. Using the output of the solv er, and p erhaps using the designer's help to eliminate some solutions, replace S i b y an abstr action or simplic ation T i ( S i ) thereb y replacing the en tire system E i b y a simplication T i ( E i ) = E i +1 This step is indicated b y an o v al in Figure 2{2 Some informal requiremen ts on the simpliers T i are the follo wing: w e w ould lik e E i to b e (real algebraically) infer able from E i +1 ; i.e, w e w ould lik e an y real solution of E i +1 to b e a solution of E i as w ell. A constrain t solv er that ts the ab o v e structural description, { whic h w e shall refer to as S in future discussions { is called a de c omp osition-r e c ombination (DR) solver (F ormal denition imp oses further requiremen ts on simplier T i { see the next section). This solv er terminates when the small, solv able subsystem S i found in Step 1 is the en tire p olynomial system E i An optimal DR-plan will minimize the size of the largest S i If the whole system is underconstrained, the solv er should still solv e its maximal solv able subsystems. F or the purp ose of planning a solution sequence apriori, w e w ould lik e to execute Steps 1 and 3 alone without access to the algebraic solv er, and get a DR-plan, without actually solving the subsystems. I.e, w e w ould lik e the constrain t solv er to

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17 lo ok as in Figure 2{3 with the DR-planner driving the direct algebraic/n umeric solv er. T o generate a DR-plan apriori, one w ould ha v e to lo cate a solv able subsystem S i and without actually solving it, nd a suitable abstraction or simplication of it that is substituted in to the larger system E i to obtain an o v erall simpler system E i +1 in Step 3. On the other hand, suc h a DR-plan w ould p ossess the adv an tage of b eing robust, or generically indep enden t of particular n umerical v alues attac hed to the constrain ts, and of the solution to the S i and usually only dep ends on the degrees of freedom of the relev an t geometric ob jects and geometric constrain ts. 2.1.5 Desirable Characteristics of DR-planners for CAD/CAM W e en umerate and informally describ e a set C of natural c haracteristics desirable for a DR-planner. W e b egin with four criteria that directly follo w from the o v erall structural description of a t ypical DR-planner S in the previous subsection. (i) The DR-planner should b e general, i.e, it should not b e articially restricted to a narro w class of decomp osition plans; it should output a DR-plan if there is one; and it should b e able to decomp ose underconstrained systems as w ell. F urthermore, if a DR-plan exists, the planner should run to completion irresp ectiv e of ho w and in what order the solv able subsystems S i are c hosen for the plan (a Ch urc h-Rosser prop ert y). (ii) The planner should p oten tially output a close-to-optimal DR-plan (i.e, where the size of the largest solv able subsystem S i is close-to-minimal). This dictates eciency of solution of the constrain t system. (iii) The DR-planner should b e fast and simple; the time complexit y should b e lo w, the planner should b e fast in practice, easily implemen table, and compatible with existing algebraic solv ers, CAD systems, constrain t mo dels and manipulators.

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19 therefore b e a consisten t extension and/or renemen t of this conceptual design decomp osition. (vi) While the DR-plan is used to guide the algebraic solv er, it should remain unaected or adapt easily to the designer's in terv en tion in the solution pro cess, whic h is v aluable for pruning com binatorial explosion: the designer can thro w out meaningless or undesirable solutions of subsystems at an early stage. Suc h designer in terference is also crucial for a v oiding the use of inequalit y constrain ts: for example, instead of adding on a constrain t that p oin t P is to the left of line L the designer can simply thro w out an y partial solutions that put P to the righ t of line L (vii) The DR-plan for solving a geometric constrain t system should remain meaningful in the presence of other than geometric constrain ts, suc h as equational constrain ts or parametric constrain ts expressing design in ten t. Finally the CAD system and the CAD mo del do not stand alone. In standard clien t-serv er based arc hitectures (see e.g, Bronsv o ort and Jansen (1994 ); Krak er et al. (1997 ); Homann and Joan-Arin y o (1998 )), the CAD mo del is just one clien t's view of the pro duct master mo del ( New ell and Ev ans, 1976 ; Semenk o v, 1976 ), with whic h it has to b e con tin ually co ordinated and made consisten t. The master mo del in turn co ordinates with other do wnstream pro duction clien t systems whic h main tain other consisten t views. These clien ts include geometric dimensioning and tolerancing systems (GD& T), and man ufacturing pro cess planners (MPP) for automatically con trolled mac hining or assem bly Eac h clien t view con tains p oten tially proprietary information that m ust b e k ept secure from the master mo del. Figure 2{3 illustrates this arc hitecture and those parts that w e deal with directly Eac h clien t view con tains its o wn in ternal view of the constrain t mo del and therefore co ordination and consistency c hec ks b et w een the v arious views

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20 modelmodelmodelMasterconstraintmodelMasterView/system GD&T plan View/system Other downstreams GD&T MPP modelMPPmodel constr. constr.Master model serverGD&TCAD View/system Constraint solverConstraint model manipulator Decomp. planner Algebraic solverCAD modelOptional design historyof constraint models (for nonvariational case) CAD constraint model decomposition designer's conceptual consistent with Decomposition Constraint Manuf. proc.Figure 2{3: CAD/CAM/CAE master mo del arc hitecture crucially in v olv e, and vicev ersa ma y aect the DR-plan. This leads to the follo wing additions to the set C of desirable c haracteristics for DR-planners. (viii) The DR-plan should b e as robust as p ossible (see e.g. F ang (1992 )) to on-line c hanges made to the constrain t system, and the DR-planner should b e able to quic kly alter the DR-plan in resp onse to suc h c hanges. In particular, the DR-plan should ideally not c hange at all with n umerical p erturbations (within large ranges) to the constrain ts, thereb y p ermitting the DR-plan to b e computed as a pre-pro cessing step. Addition, deletion, and mo dication of constrain ts and geometric ob jects to the constrain t system o ccurs in n umerous circumstances during the design pro cess. F or example: (a) In the pro cess of solving the system using the DR-plan, a subsystem S i deemed solv able b y a degree of freedom analysis ma y b e found to ha v e no real solutions, or ma y in a degenerate case, turn out to b e underconstrained or ha v e innitely man y solutions, prev en ting a con tin uation of the solution pro cess S ;

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21 (b) In history-based CAD systems, the sp ecication of the pro duct tak es the form of a progressiv e sequence of c hanges b eing made to successiv e partial sp ecications and the asso ciated partial constrain t systems; (c) Inferred or understo o d constrain ts are often added to an incomplete constrain t sp ecication at the discretion of the CAD system; (d) In underconstrained situations, as migh t o ccur in designing rexible or mo ving parts and assem blies, the mapping of the conguration space of the resulting mec hanism w ould require the rep eated addition of constrain ts at the discretion of the constrain t solv er { the solutions to the resulting w ellconstrained systems w ould represen t v arious congurations of the mec hanism; (e) F or a v ariational spatial constrain t solv er to b e eectiv e or ev en usable, it w ould ha v e to rely on extensiv e in teractiv e constrain t c hanges b eing made b y the designer, sometimes in the course of solving; and (f ) Finally when one of the v arious clien ts in Figure 2{3 mak es a c hange to its o wn view of the constrain t mo del, this will result in consistency up dates to the master constrain t mo del whic h will, in turn, result in up dates to the other views of the constrain t mo del. (ix) The DR-plan should isolate o v erconstrained subsystems whic h arise in assem bly problems; furthermore, with m ultiple (p ossibly proprietary) views of the constrain t mo del b eing k ept b y v arious clien ts as in 2{3 constrain t reconciliation often tak es place and the DR-plan should facilitate this pro cess, and vicev ersa, should b e robust against this pro cess. F or a precise description of the constrain t reconciliation problem, see Homann and Joan-Arin y o (1998 ). The problem requires isolation of o v erconstrained subsystems and is comp ounded in the case of non-v ariational, history-based design.

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22 2.1.6 F ormal Denition of DR-solv ers using P olynomial Systems W e will formally state the decomp osition-recom bination (DR) problem for p olynomial systems arising from geometric constrain ts. This requires formalizing the notion of a decomp osition-recom bination solution sequence, and of decomp osition-recom bination solv ers that t the description S giv en in the previous section. In addition, w e dene the p erformance measures that capture some of the desirable prop erties C giv en in the last section for comparing suc h sequences and solv ers. Note. As w as men tioned previously for the sak e of gradual exp osition, w e will rst assume that the constrain t system is b eing solv ed ev en as the decomp osition is b eing generated { i.e, the DR-solv er exactly ts the structural description S giv en in the previous section. F urthermore, w e will dene the DR-solv er in the general con text of p olynomial equations that arise from the geometric constrain ts. In Section 2.1.7 w e will shift our atten tion to the DR-planning problem. The DR-planner generates a decomp osition plan apriori, whic h then driv es the direct algebraic/n umeric solv er { together, they form a DR-solv er. The DR-planner, ho w ev er, will b e dened in the con text of constrain t graphs that incorp orate geometric degrees of freedom. The DR-planner and its p erformance measures will b e analogous to the italics terms dened here. In order to formally dene DR-solv ers and their p erformance measures, w e need to sp ecify the mo del of real-algebraic computation used b y the algebraic/n umeric solv er in Figure 2{3 Ho w ev er, the issue of the mo del or ho w real n um b ers are represen ted is en tirely outside the fo cus of this man uscript for the follo wing reason: our DR-solv ers and p erformance measures are robust in that they are conceptually indep enden t of the algebraic/n umeric solv er b eing used or ho w real n um b ers are represen ted. The denition of DR-solv ers and p erformance measures adapts straigh tforw ardly to other natural mo dels of computation. In other w ords,

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23 our conceptual denition of DR-solv ers and p erformance measures are suc h that if DR-solv er A p erforms b etter than DR-solv er B with resp ect to one of our p erformance measures, then it con tin ues to do irresp ectiv e of the (natural) mo del of computation b eing used b y the algebraic/n umeric solv er. F urther, as p oin ted out in an earlier note, w e only emphasize the b oldface comp onen ts of Figure 2{3 sp ecically the DR-planner, whose op eration dened in Section 2.1.7 will b e seen to b e purely com binatorial. In particular, no emphasis is placed on the non-b oldface comp onen ts, whic h includes the algebraic/n umeric solv er, and the mo del of computation it uses. } Ha ving noted the ab o v e, the mo del of computation w e assume for the the sak e of completeness and formalit y of denitions in this man uscript is the Blum-Sh ubSmale mo del of real algebraic computation Blum et al. (1989 ). Briery in this mo del, real n um b ers are assumed to b e represen table as suc h as en tities, without an y recourse to rational appro ximations and nite precision or in terv al arithmetic. Real arithmetic op erations suc h as m ultiplication and addition and division can b e p erformed in constan t time, and nding unam biguous represen tation of eac h real zero of a univ ariate p olynomial p can also b e ac hiev ed in time p olynomial in the degree of p F urthermore, these unam biguous represen tations of real zero es of univ ariate p olynomials are treated as en tities just lik e an y other real n um b er, and can for instance, b e used as co ecien ts of other p olynomials. A system of e quations E is a pair ( P ; X ) where P is a set of p olynomial equations with real co ecien ts, and X is a set of formal indeterminates. The union of two systems E 1 = ( P 1 ; X 1 ) and E 2 = ( P 2 ; X 2 ) is the system ( P 1 [ P 2 ; X 1 [ X 2 ). An interse ction E 1 \ E 2 is the system ( P 1 \ P 2 ; X 1 \ X 2 ). Within the geometric con text, a solv ed subsystem is a rigid or solv able system where all the v ariables ha v e already b een \solv ed for," i.e, they ha v e t ypically b een expressed explicitly as p olynomials of D free v ariables, where D represen ts the

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24 n um b er of degrees of freedom of the rigid b o dy (within its lo cal co ordinate system) in the prev ailing geometry In some cases, for example when the rigid b o dy is a line in 3 dimensions, the n um b er of free v ariables, including redundan t ones, ma y b e greater than D Let E = ( P ; V ) with V = f y 1 ; : : : ; y D g [ X b e suc h that ev ery equation in P is of t yp e x j = F j ( y 1 ; :::; y D ), for all x j 2 X and F j is a rational function (of the form Q 1 =Q 2 where Q 1 and Q 2 are, as usual, p olynomials with real co ecien ts) that can b e ev aluated in time p olynomial in ( j V j ; j P j ). Suc h a system of equations, E is called a solve d system The v ariables y 1 ; :::; y D 2 V of the solv ed system are fr e e variables of E and the v ariables x 1 ; :::; x k 2 V are explicitly xe d variables Note that all the v ariables in a solv ed system are xed or free, whereas a general solv able system ma y ha v e free, explicitly xed and other, implicitly xe d v ariables. A t ypical DR-solv er that follo ws the o v erall structural description S of Section 2.1.4 obtains a sequence E 1 ; : : : ; E m consisting of successiv ely simpler solv able systems. These are general solv able systems and ha v e successiv ely few er implicitly xed v ariables; E = E 1 ; and E m is a solv ed system. New v ariables y i ma y b e in tro duced at in termediate stages whic h represen t free v ariables within subsystems S i that are solv ed with resp ect to these v ariables. These solv ed subsystems represen t v arious rigid b o dies lo cated and xed with resp ect to their lo cal co ordinate systems. Ho w ev er, these lo cal co ordinate systems are still constrained with resp ect to eac h other, and hence in fact only D of the newly in tro duced v ariables are, in eect, free and the remainder are implicitly xed. Some of these newly in tro duced v ariables ma y b e remo v ed at later stages. Ev en tually in the solv ed system E m all the original v ariables and those newly in tro duced v ariables that migh t remain b ecome explicitly xed with resp ect to D free v ariables as w ell. The set of real solutions to E i +1 should also b e a subset of solutions to E i to ensure that the nal solutions to E m actually represen t solutions to the original system E

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25 W e formally dene real algebraic equiv alence and real algebraic inference of t w o geometric constrain t systems as follo ws. Giv en t w o systems E 1 = ( P 1 ; X [ Y 1 ) and E 2 = ( P 2 ; X [ Y 2 ) where the set X represen t the original v ariables that are curren tly implicitly or explicitly xed, and the sets Y i represen t the newly in tro duced v ariables (that are \free" within the solv ed subsystems S i ), w e sa y that the system E 1 = ( P 1 ; X [ Y 1 ) is r e al algebr aic al ly infer able (in short, inferable) from the system E 2 = ( P 2 ; X [ Y 2 ), if for an y real solution Y 2 = A 2 ; X = B that satises the equations in P 2 there is a corresp onding assignmen t A 1 of real v alues to Y 1 suc h that Y 1 = A 1 ; X = B satises P 1 Tw o systems E 1 and E 2 are r e al algebr aic al ly e quivalent if E 1 is real algebraically inferable from E 2 and E 2 is real algebraically inferable from E 1 No w w e are ready to dene the notion of a DR-solution sequence. Let E b e a system of equations. A DR-solution se quenc e of E is a sequence of systems of equations E 1 ; :::; E m suc h that E = E 1 E m is a solv ed system (so E m has a real solution), ev ery E i is solv able, eac h E i is inferable from E i +1 An y solv able system E whic h has a real solution in fact has a DR-solution sequence. A trivial DR-solution sequence E ; E where E is a solv ed system equiv alen t to E will do. (Note that b y E i w e denote abstract algebraic systems, rather than their computer represen tations that could only ha v e rational co ecien ts and therefore ma y only ha v e appro ximate DR-solution sequences). The DR-pr oblem is the problem of nding a DR-solution sequence of a giv en constrain t system. A DR-solver is a constrain t solv er or algorithm that solv es the DR-problem. A DR-solv er is gener al if it alw a ys outputs a DR-solution sequence when giv en a solv able system as input. Aside from b eing general, w e w ould also lik e a DR-solv er to ha v e the Chur ch-R osser pr op erty i.e, the DR-solv er should terminate irresp ectiv e of the order in whic h the solv able subsystems S i are c hosen.

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26 In other w ords, at eac h step, a solv able subsystem S i can b e c hosen greedily to satisfy the (easily c hec k able) requiremen ts of the algorithm. This prev en ts exhaustiv e searc h. P erformance measures. Next, w e formally dene a set of p erformanc e me asur es for the DR-solution sequences and DR-solv ers. These p erformance measures are designed to capture the c haracteristics C of constrain t solv ers giv en in Section 2.1.5 that are desirable for engineering design and assem bly applications. (Note that man y of these measures are Bo olean, i.e, either a certain desirable condition is met or not.) In particular, w e w ould lik e a DR-solution sequence Q = E 1 ; : : : ; E m of a solv able system E to ha v e sev eral prop erties. T o describ e these prop erties w e formalize a simplifying map from eac h system E i to its successor E i +1 In fact, for generalit y w e c ho ose these maps T i called subsystem simpliers to map the set of subsystems of E i on to the set of subsystems of E i +1 First, in order to rerect the Ch urc h-Rosser prop ert y in (i), and p oin ts (iv) and (vi) of C w e w ould lik e these subsystem simpliers T i to b e natural and w ell-b eha v ed, i.e, to ob ey the follo wing simple and non-restrictiv e rules. (1) If A is a subsystem of B then T i ( A ) is a subsystem of T i ( B ) (2) T i ( A ) [ T i ( B ) = T i ( A [ B ) (3) T i ( A ) \ T i ( B ) = T i ( A \ B ) Second, in the description of the t ypical DR-solv er S giv en in Section 2.1.4 eac h system E i +1 in the DR-solution sequence is t ypically to b e obtained from E i b y replacing a solv able subsystem S i in E i (lo cated during Step 1 of S ), b y a simpler subsystem (during Steps 2 and 3). F or a manipulable DR-solution sequence (again satisfying the p oin ts (i), (iv), (v) and (vi) of C ), w e w ould lik e E i +1 to lo ok exactly lik e E i outside of S i This leads to another set of prop erties desirable for the subsystem simpliers T i

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27 (4) Eac h E i = S i [ R i [ U i 1 i m where S i is solv able, R i is a maximal subsystem suc h that S i and R i do not share an y v ariables, S i ; R i ; U i do not share an y equations, and all v ariables of U i are either v ariables of S i or R i F or an y A R i T i ( A ) = A Thirdly in order to address the p oin ts (iv)-(vi), and (i) of C sim ultaneously i.e, p ermitting generalit y of the subsystem S i that is replaced b y a simpler system during the i th step, while at the same time making it con v enien t for the designer to geometrically follo w and manipulate the decomp osition, w e w ould lik e the subsystem simpliers to satisfy the follo wing prop ert y (5) F or eac h i all the pre-images of S i T 1 j :::T 1 i 1 ( S i ) ; 1 j i 1 are algebraically inferable from S i and furthermore, they are solv able or rigid subsystems for the giv en geometry (recall that the denition of \solv able" dep ends on the geometry). It follo ws from (2) and (3) ab o v e that the inverse T 1 i ( A ) = S B where the union is tak en o v er all B E i suc h that T i ( B ) A The ab o v e prop ert y sa ys that while the subsystem simpliers enjo y a high degree of generalit y and are completely free to map solv able subsystems in to solv able systems, they should nev er map (con v ert) subsystems that are originally not solv able in to one of the c hosen, solv able systems S i at an y stage i In other w ords, in the act of simplifying, the subsystem simpliers should not create one of the solv able subsystems S i out of subsystems that w ere originally not solv able. A DRsolution sequence that satises the ab o v e prop erties is called valid Th us a v alid DR-solution sequence for a geometric constrain t system E is sp ecied as a sequence of E 1 ; : : : ; E m suc h that E 1 = E E m is a solv ed system, ev ery E i is solv able and inferable from E i +1 ev ery E i = S i [ R i [ U i as describ ed ab o v e. The motiv ation giv en for eac h of the prop erties ab o v e mak es it clear that v alid DRsolution sequences encompass highly general but geometrically meaningful solutions

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28 of the original constrain t system E Next w e turn to p oin t (ii) of C i.e, optimalit y whic h also comp etes with generalit y (p oin t (i)). F or optimalit y w e w ould lik e to minimize the size of the largest solv able subsystem S i in the DR-solution sequence. F ormally the size of the DR-sequence Q is equal to max 1 i m j S i j where the size j S i j is equal to the total n um b er of it v ariables less the n um b er of its explicitly xed v ariables. The optimal DR-size of the algebraic system E is a minim um size of Q where the minim um is tak en o v er all p ossible DR-solution sequences Q of E An optimal DR-solution se quenc e Q of E is a solution sequence suc h that the size of Q is equal to the optimal DR-size of E The appr oximation factor of DRsolution sequence Q of the system E is dened as the ratio of the optimal DR-size of E to the size of Q As a general rule for optimalit y it is clear that the larger the c hoice for solv able subsystems S i of E i a v ailable at an y stage i the more lik ely that one can nd a small solv able S i in E i In other w ords, w e w ould lik e to mak e sure that the subsystem simplier do es not destro y solv abilit y of to o man y sub systems starting from the original system E Note that while the denition of DR-solution sequence mak es sure that the en tire system E m is solv able if E 1 is, it do es not require the same for subsystems of E i (In fact, ev en if all the E i in a DR-solution-sequence are algebraically equiv alen t, this w ould still not imply that solv abilit y is preserv ed for the subsystems). In addition, while the denition of a DR-solution sequence ensures that E i +1 has a real solution if E i has one, it do es not ensure the same for subsystems of E i The next t w o denitions capture prop erties of subsystem simpliers that preserv e subsystem solv abilit y (resp. solutions) to v arying degrees, thereb y helping the optimalit y of the DR-solv er, i.e, (ii) of C The DR-solution sequence Q = E 1 ; :::; E m is solvability pr eserving if and only if for all A E i A is solv able (resp. has a real solution) and ( A \ S i = ; or A S i ) ( ) T i ( A ) is solv able (resp. has a real solution). A DR-sequence Q = E 1 ; :::; E m is strictly solvability

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29 pr eserving if and only if for all A E i A is solv able (resp. has a real solution) ( ) T i ( A ) is solv able (resp. has a real solution). Requiring suc h a solv abilit y preserving simplier places a w eak restriction on the class of v alid DR-solution sequences, but on the other hand, this restriction cannot eliminate all optimal DR-solution sequences. F urthermore, solv abilit y preserv ation helps to ensure the Ch urc h-Rosser prop ert y in (i) of C Note. T o b e more precise, \solv abilit y preserv ation" should b e termed \solv abilit y and solution-existence preserv ation," but w e c ho ose the shorter phrase. } In fact, for DR-solution sequences to b e optimal, w e w ould prefer that for all i 1 ; S i do es not con tain an y strictly smaller subsystem, sa y B that is solv able, and has not b een found (during Step 1 of the description S in Section 2.1.4 ) and simplied/replaced at an earlier stage j < i The DR-sequence Q = E 1 ; :::; E m is c omplete if and only if for ev ery non trivial solv able B S i B = T i 1 T i 2 :::T j ( S j ) for some j i 1. While the completeness requiremen t restricts the class of v alid DR-solution sequences, it only eliminates sequences that either ha v e size greater than optimal or the same size as some optimal sequence that do es simplify B In addition to aecting optimalit y i.e, (ii) of C completeness also strongly rerects (ix): completeness prev en ts a DR-solv er from o v erlo oking an o v erconstrained subsystem inside a w ellconstrained subsystem, whic h is also useful for constrain t reconciliation (see Homann and Joan-Arin y o (1998 )). P erformance measures. So far, w e ha v e discussed p erformance measures that measure the desirabilit y of DR-solution sequences. Next w e formally dene directly analogous p erformance measures for DR-solv ers whic h generate DR-sequences. A DR-solv er A is said to b e valid, solvability pr eserving, strictly solvability pr eserving, c omplete if and only if for ev ery input constrain t system E ev ery DR-solution sequence pro duced b y A is v alid, solv abilit y preserving, strictly solv abilit y preserving or complete resp ectiv ely

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30 Note. Purely as a to ol to help analysis and exp osition, often w e assume that DR-solv ers for pro ducing DR-sequences are randomized or nondeterministic in a natural w a y i.e those steps in the algorithm where arbitrary c hoices of equations or v ariables are made (for example, to b e the lo w est n um b ered equation or v ariable) are no w tak en to b e randomized or nondeterministic c hoices. } The next denition formalizes p erformance measures related to the c haracteristic (ii) of C that dieren tiates randomized DR-solv ers for whic h some random c hoice leads to an optimal DR-solution sequence, from inferior DR-solv ers where no c hoice w ould lead to an optimal DR-solution sequence. The worst-choic e appr oximation factor of a DR-solv er A on input system E is the minim um of the appro ximation factors of all DR-solution sequences Q of E obtained b y the algorithm A o v er all p ossible random c hoices. The b est-choic e appr oximation factor of the algorithm A on input E is maxim um of the appro ximation factors of all the DR-solution sequences Q of E obtained b y the algorithm A o v er all p ossible random c hoices. 2.1.7 F ormal Denition of a DR-planner via Constrain t Graphs The DR-solution sequence dened in the previous section em b eds a DR-plan that is in tert wined with the actual solution of the system. Therefore, a DR-solv er that simply outputs a DR-solution sequence (ev en one that is strictly solv abilit y preserving, complete, and so on..), ma y not b e mo dular as sho wn in Figure 2{3 Ho w to construct a DR-solv er that rst generates a DR-plan b efore using it to driv e the general algebraic/n umeric solv er? First notice that rep eatedly applying Steps 1 and 3 of the DR-solv er in the description S in Section 2.1.4 could p otentially generate a DR-plan without actually solving an y subsystem (and without applying Step 2), pro vided that the follo wing can b e done: (a) solv abilit y of a subsystem S i of the system E i in Step 1 can b e determined generically without actually solving it; and

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31 (b) a solv able subsystem S i can b e replaced in Step 3 without actually solving S i i.e, b y a h yp othetical solv ed subsystem to giv e the new system E i +1 F or this w e need a consisten t and adequate abstraction of the systems E i and of their subsystems, as w ell as a w a y to abstract a h yp othetical solv ed subsystem. In other w ords, w e need to adapt the subsystem simplier maps describ ed in the last section, so that the iteration can pro ceed with Step 1 again. By th us mo difying the description S of the DR-solv er, w e obtain a DR-planner whic h can b e emplo y ed as a prepro cessing step to output a DR-plan instead of a DR-solution sequence. This DR-plan can thereafter b e used to direct a series of applications of Step 2, leading to a complete DR-solution sequence. This mo dularit y helps, for example, to w ards maneuv erabilit y b y the designer (p oin t (vi)), and to w ards compatibilit y of the DR-solv er with existing solv ers (p oin t (v)). In order to formally dene suc h a DR-plan, w e follo w a common practice and view the constrain t system as the constrain t h yp ergraph: this abstraction p ermits us to build the DR-planner on the foundation of generalized degree of freedom analysis whic h is kno wn to w ork w ell in estimating generic solv abilit y of constrain t systems without actually solving them. (This is explained more precisely after the formal graph-theoretic denitions are in place). Hence using constrain t graphs facilitates b oth the tasks (a) and (b) describ ed ab o v e. In other w ords a DR-planner that is based on a generalized degree of freedom analysis is robust in the follo wing sense: c hanging the n umerical v alues of constrain ts is not lik ely to aect the DR-plan. Th us the motiv ation for using constrain t graphs includes all of the desirable c haracteristics (iii) to (vi) of the set C in Section 2.1.5 In addition, geometry is more visibly displa y ed via constrain t graphs than via equations, thereb y helping in teraction with the designer. Note. DR-plans and planners and their p erformance measures could also b e dened directly in terms of the algebraic constrain t systems, just as w e dened

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32 2 2 1 1 1 E F D 2 Figure 2{4: A constrain t graph DR-solv ers. In this thesis, ho w ev er, due to the reasons men tioned ab o v e, w e dene DR-plans and planners en tirely within the con text of constrain t graphs and degree of freedom analysis. } Constrain t graphs and solv abilit y First w e dene the abstraction that con v erts geometric constrain t system in to a w eigh ted graph. Recall that a geometric constrain t problem consists of a set of geometric ob jects and a set of constrain ts b et w een them. A geometric constrain t graph G = ( V ; E ; w ) corresp onding to a geometric constrain t problem is a w eigh ted graph with n v ertices (represen ting geometric ob jects) V and m edges (represen ting constrain ts) E ; w ( v ) is the w eigh t of v ertex v and w ( e ) is the w eigh t of edge e corresp onding to the n um b er of degrees of freedom a v ailable to an ob ject represen ted b y v and n um b er of degrees of freedom remo v ed b y a constrain t represen ted b y e resp ectiv ely F or example, Figure 2{4 is a constrain t graph of a constrain t problem sho wn in Figure 2{1 Note that the constrain t graph could b e a hyp er gr aph eac h h yp eredge in v olving an y n um b er of v ertices. No w w e in tro duce denitions that will help us to relate the notion of solv abilit y of the geometric constrain t system to the corresp onding geometric constrain t graph. A subgraph A G that satises X e 2 A w ( e ) + D X v 2 A w ( v ) (2.1)

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33 is called dense where D is a dimension-dep enden t constan t, to b e describ ed b elo w. The function d ( A ) = P e 2 A w ( e ) P v 2 A w ( v ) is called density of a graph A The constan t D is t ypically d +1 2 where d is the dimension. The constan t D captures the degrees of freedom asso ciated with the cluster of geometric ob jects corresp onding to the dense graph. In general, w e use w ords \subgraph" and \cluster" in terc hangeably F or planar con texts and Euclidean geometry w e exp ect D = 3 and for spatial con texts D = 6, in general. If w e exp ect the cluster to b e xed with resp ect to a global co ordinate system, then D = 0. A dense graph with densit y strictly greater than D is called over c onstr aine d A graph that is dense and all of whose subgraphs (including itself ) ha v e densit y at most D is called wel lc onstr aine d. A graph G is called wel lover c onstr aine d if it satises the follo wing: G is dense, G has at least one o v erconstrained subgraph, and has the prop ert y that on replacing all o v erconstrained subgraphs b y w ellconstrained subgraphs, G remains dense. A graph that is w ellconstrained or w ello v erconstrained is said to b e solvable A dense graph is minimal if it has no prop er dense subgraph. Note that all minimal dense subgraphs are solv able, but the con v erse is not the case. A graph that is not solv able is said to b e under c onstr aine d If a dense graph is not minimal, it could in fact b e an underconstrained graph: the densit y of the graph could b e the result of em b edding a subgraph of densit y greater than D In order to understand ho w solv able constrain t graphs relate to solv able constrain t systems it is imp ortan t to remem b er that a geometric constrain t problem has t w o asp ects com binatorial and geometric. The geometric asp ect deals with actual parameters of the geometric constrain t problem, while the com binatorial asp ect deals with only the abstractions of ob jects and constrain ts. Unfortunately at the momen t it is not kno wn ho w to completely separate the t w o asp ects, except for some sp ecial cases. In this thesis w e will limit ourselv es to the geometric constrain t problems where generally there is a corresp ondence b et w een solv able constrain t

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34 A B C D EF Figure 2{5: Generically unsolv able system systems and solv able constrain t graphs. In order to do that, w e need to in tro duce and formalize the notion of generic solv abilit y of constrain t systems. Informally a constrain t system is generically (un)solv able if it is (un)solv able for most c hoices of co ecien ts of the system. More formally w e use the notion of genericity of e.g, Co x et al. (1998 ). A prop ert y is said to hold generic al ly for p olynomials f 1 ; : : : ; f n if there is a nonzero p olynomial P in the co ecien ts of the f i suc h that this prop ert y holds for all f 1 ; : : : ; f n for whic h P do es not v anish. Th us the constrain t system E is generically (un)solv able if there is a nonzero p olynomial P in the parameters of the constrain t system suc h that E is (un)solv able when P do es not v anish. Consider for example Figure 2{5 Here the ob jects are 6 p oin ts in 2d and the constrain ts are 8 distances b et w een them. This system is unsolv able since the edge B C can b e con tin uously displaced, unless the length of AD (induced b y the lengths of AE ; D E ; D F ; AF and E F ) is equal to the j AB j + j B C j + j C D j Since w e can create an appropriate nonzero p olynomial P ( AB ; B C ; : : : ; E F ), suc h that P () = 0 if and only if j AD j = j AB j + j B C j + j C D j this system is generically unsolv able. While a generically solv able system alw a ys giv es a solv able constrain t graph, the con v erse is not alw a ys the case. In fact, there are solv able, ev en minimal dense graphs whose corresp onding systems are not generically solv able, and are in fact generically not solv able (note that the p osition of the not' c hanges the meaning,

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35 A B C D E F G H 3 3 3 3 3 3 3 3 Figure 2{6: Generically unsolv able system that has a solv able constrain t graph the latter b eing stronger than the former). Consider for example Figure 2{6 The constrain t system is sho wn on the left, it consists of 8 p oin ts in 3d and 18 distances b et w een them. The corresp onding constrain t graph is sho wn on the righ t, w eigh t of all the edges is 1, w eigh t of the v ertices is 3, geometry dep enden t constan t D = 6. Note that this system is generically unsolv able since rigid b o dies AB C D E and AF GH E can rotate ab out the axis passing through AE This example can also b e transformed in to a constrain t problem in 2d, where the ob jects are circles and the constrain ts are angles of in tersections Saliola and Whiteley (1999 ). Another example is the graph K 7 ; 6 that in 4 dimensions represen ts distances b et w een pairs of p oin ts. The constrain t graph is minimal dense but it do es not represen t a generically solv able system. It should b e noted that in 2 dimensions, according to Laman's theorem Laman (1970 ) if all geometric ob jects are p oin ts and all constrain ts are distance constrain ts b et w een these p oin ts then an y minimal dense subgraph represen ts a generically solv able system. Also there exists a purely com binatorial c haracterization of solv able systems in 1 dimension based on connectivit y of the constrain t graphs. Ho w ev er, as examples ab o v e indicate, the generalization of Laman's theorem fails in higher dimensions ev en for the case of p oin ts and distances. There is a matroid based approac h to v erifying whether solv abilit y of the constrain t graph implies solv abilit y of the constrain t system. It b egins b y c hec king whether a submo dular function f ( E ) = 2 j V j 3, dened on sets of edges of the constrain t graph creates a matroid Whiteley (1992 1997 ), i.e c hec king whether

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36 the created function is p ositiv e on single edges. Thereafter this approac h c hec ks whether the corresp onding geometric structure is generically rigid. Some matroid based approac hes to determining generic solv abilit y are fast in practice, for example those to b e discussed later under \generalized maxim um matc hing". Ho w ev er, so far no matc h has b een established (for all cases) b et w een the created matroid and generic rigidit y of the particular geometric problem. There are sev eral attempts at c haracterization of generic solv abilit y in dimension 3 and higher for the case of p oin ts and distances T a y (1999 ); Gra v er et al. (1993 ). One suc h c haracterization, the so called Henneb erg construction, c hec ks whether a giv en constrain t graph can b e constructed from the initial basic graph b y applying a sequence of standard replacemen ts. A c haracterization due to Dress c hec ks whether the constrain t graph satises a certain inclusion-exclusion t yp e rule. A c haracterization due to Crap o examines whether the constrain t graph can b e decomp osed in to a union of certain edge-disjoin t trees. All of these c haracterizations though in teresting and useful, are so far unpro v en conjectures. Note. Due to the ab o v e discussion, w e restrict ourselv es to the class of constrain t systems where solv abilit y of the constrain t graph in fact implies the generic solv abilit y of the constrain t system. (As p oin ted out earlier, the con v erse is alw a ys true, with no assumptions on the constrain t system). As w as indicated ab o v e, this class is far from empt y it con tains all constrain t problems in v olving p oin ts and distances in 2d, problems resulting from Cauc h y triangulations of the p olyhedra in 3d as w ell as b o dy-and-hinge structures in 3d. Moreo v er, it should b e emphasized that while existing applications stop at nding subgraphs represen ting solv able constrain t systems, w e are in terested in the en tire problem of decomp osition and recom bination, optimizing the size of the largest subsystem to b e solv ed, i.e w e are in terested in nding an optimal DR-plan. Also note that already for the class of generically solv able constrain t systems and

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37 S1A B C G H I D E F V G I H E D FFigure 2{7: Original geometric constrain t graph G 1 and simplied graph G 2 corresp onding graphs, the DR-planning problem is, in general, dicult (see later Note ab out NP-hardness, after the denition of the optimal DR-plan). } F ormal denition of DR-plans using constrain t graphs. Informally stated in terms of constrain t graphs, the DR-planning problem in v olv es nding a sequence of graphs G i a DR-plan suc h that the original constrain t graph G = G 1 and ev ery G i con tains a minimal solv able subgraph S i whic h is simplied or abstracted in to a simpler subgraph T i ( S i ) and substituted in to G i to giv e an o v erall simpler graph G i +1 = T i ( G i ). (While T i ( S i ) should b e simpler than S i it should also b e someho w equiv alen t to S i for example b y ha ving same densit y v alue.) If the original graph G 1 is w ellconstrained, then the pro cess terminates when G m = S m (If not, the pro cess terminates with the decomp osition of G m in to a maximal set of minimal solv able subgraphs). Consider for example Figure 2{7 whic h sho ws one simplication step. On the left is the constrain t graph G 1 the w eigh t of all v ertices is 2, w eigh t of all edges is 1, geometry dep enden t constan t D = 3. Then S 1 = f A; B ; C g is a solv able subgraph. On the righ t is the simplied graph T 1 ( G 1 ) = G 2 after subgraph S 1 is replaced b y a v ertex f V g = T 1 ( S 1 ). Since densit y of the subgraph S 1 is -3, the w eigh t of the v ertex f V g could b e set to 3. A sequence of simplication steps is sho wn in Figure 2{8 The top part depicts a geometric constrain t graph G where the w eigh t of eac h edge is 1, the w eigh t of eac h v ertex is 2, and the dimension dep enden t constan t D is equal to 3 (this

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38S 4 S 2 S 1 S 3 S 5 S 6S S S S S S S S 1 2 3 4 8 7 5 67 S S 8 B A C D E F G HI J K L M N O A B C D E F G H I J K L M N OFigure 2{8: Geometric constrain t graph and a DR-plan

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39 corresp onds to p oin ts and distances in 2d). One of the plans for decomp osing and recom bining G (and the geometric constrain t system that G represen ts) in to small solv able subgraphs (represen ting solv able subsystems), is to decomp ose G in to dense subgraphs S 1 = f A; B ; C g ; S 2 = f D ; E ; F g ; S 3 = f G; H ; I g ; S 5 = f J ; K ; L g ; S 6 = f M ; N ; O g represen t their solutions appropriately in a simplied graph so that they can b e recom bined, one p ossibilit y is to represen t them as v ertices P ; Q; R ; S; T of w eigh t 3 eac h; recursiv ely decomp ose the simplied graph in to S 4 = f P ; Q; R g ; S 7 = f S; T g ; represen t and recom bine their solutions as v ertices U; W of w eigh t 3; and so on un til the en tire graph is represen ted and recom bined as a single v ertex. A corresp onding DR-plan is sho wn at the b ottom part of Figure 2{8 Note that there could b e more than one DR-plan for a giv en constrain t graph. F or example, another p ossible DR-plan for a constrain t graph describ ed ab o v e is sho wn in Figure 2{9 An optimal DR-plan will minimize the size of the largest solv able subgraph S i found during the pro cess. I.e, it will minimize the maxim um fan-in of the v ertices in the DR-tree sho wn in Figure 2{8 and Figure 2{9 where b y fan-in of a v ertex w e mean the n um b er of immediate descendan ts of the v ertex. With this description, it should b e clear, that DR-plans obtained using the w eigh ted, constrain t graph mo del are generically robust with resp ect to the c hanges made to the geometric constrain ts; as long as the n um b er of degrees of freedom attac hed to the ob jects and destro y ed b y the constrain ts remains the same, the same DR-plan will w ork for the c hanged constrain t system as w ell. Th us, suc h DR-plans satisfy the initial robustness requiremen ts of the c haracteristic (viii) of the set C describ ed in Section 2.1.5 Next w e formally dene a DR-plan for constrain t graphs, and the v arious p erformance measures that capture desirable prop erties of DR-plans and DR-planners.

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40 S2S1S3S5S4 S S S S S S S S 8 6 7 5 4 1 2 3 S6S7S8A B C D E F G H I J K L M N OB A C D E F G HI J KL M N OFigure 2{9: Another p ossible DR-plan

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41 The dev elopmen t is parallel to that in Section 2.1.6 where a DR-solution sequence and v arious p erformance measures for these sequences and for DR-solv ers w ere motiv ated and dened in terms of p olynomial systems. Note. Due to the strong analogy of DR-planners to DR-solv ers dened in the previous section, the discussion here is more terse. The follo wing are useful generic corresp ondences to k eep in mind after reading the Note at the end of Section 2.1.7 and the paragraphs preceding it: Solv able subgraphs n solv able subsystems; DR-plans n DR-solution sequences; DR-planner n DR-solv er; subgraph simplier n subsystem simplier. } Let G b e a solv able constrain t graph. A DR-plan Q for G is a sequence Q of graphs G 1 ; :::; G m suc h that G 1 = G G m is a minimal solv able graph, and ev ery G i is solv able. An algorithm is a gener al DR-planner if it outputs a DR-plan when giv en a solv able constrain t graph as input. The union (r esp. interse ction) of two sub gr aphs A and B is the graph induced b y the union (resp. in tersection) of sets of v ertices of A and B All subgraphs are understo o d to b e v ertex induced. The mapping from the graph G i to G i +1 is called a sub gr aph simplier and is denoted b y T i This mapping should ha v e the follo wing prop erties. (1) If A is a subgraph of B then T i ( A ) is a subgraph of T i ( B ). (2) T i ( A ) [ T i ( B ) is the same as the graph T i ( A [ B ). (3) T i ( A ) \ T i ( B ) is the same as T i ( A \ B ). As in the case of DR-solution sequences, assume that ev ery constrain t graph G i in the DR-plan can b e written as S i [ R i [ U i where S i is minimal solv able, R i is a maximal subgraph suc h that S i and R i do not ha v e common v ertices, S i ; R i ; U i do not ha v e common edges and all v ertices of U i are either v ertices of S i of v ertices of

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42 R i Analogous to prop erties (4) and (5) of the subsystem simplier in the previous section, w e w ould lik e the subgraph simplier to ha v e the follo wing additional prop erties. (4) F or ev ery A R i T i ( A ) = A (5) All the pre-images of S i i.e T 1 j T 1 j +1 :::T 1 i 1 ( S i ) for all 1 j < i 1, are solv able. A DR-plan that satises the ab o v e rules is called valid The size of the DR-plan Q of G is the maxim um of the sizes of S i The size of an arbitrary subgraph A G i is computed as follo ws. S iz e ( A ) = 0 F or 1 j i 1 B = T i 1 T i 2 :::T j ( S j ) If A \ B 6 = 0 then S iz e ( A ) = S iz e ( A ) + D ; A = A n B end if end for S iz e ( A ) = S iz e ( A ) + P v 2 A w ( v ) In other w ords, the image of an y of the S j con tributes D to the size of A where D is geometry dep enden t constan t, and the v ertices of A that are not in an y suc h image con tribute their original w eigh t. The optimal size of the constrain t graph G is the minim um size of Q where the minim um is tak en o v er all p ossible DR-plans of G An optimal DR-plan of G is the DR-plan that has size equal to the optimal size of G The appr oximation factor of DR-plan Q of the graph G is dened as the ratio of the optimal size of G to the size of Q Note. The problem of nding the optimal DR-plan for a constrain t graph with un b ounded v ertex w eigh ts is NP-hard. This follo ws from a result in the authors' pap er Homann et al. (1997 ) sho wing that the problem of nding a minimum

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43 dense subgraph is NP-hard, b y reducing this problem to the CLIQUE. The CLIQUE problem is extremely hard to appro ximate Hastad (1996 ), i.e, nding a clique of size within a n 1 factor of the size of the maxim um clique cannot b e ac hiev ed in time p olynomial in n for an y constan t (unless P = NP ). Ho w ev er our reduction of CLIQUE to the optimal DR-planning problem is not a so-called gap-preserving reduction (or L reduction); th us ho w w ell this problem could b e appro ximated is still an op en question. } The denition of solv abilit y preserv ation is largely analogous to the case of DRsolv ers, but is additionally motiv ated b y the follo wing. In the case of DR-solv ers, one condition on a solv abilit y preserving simplier is that it preserv es the existence of real solutions for certain subsystems. Here, in the case of DR-plans, a natural c hoice is to corresp ondingly require that the simplier do es not map w ellconstrained subgraphs or o v erconstrained subgraphs to underconstrained and vicev ersa. The DR-plan Q of G is solvability pr eserving if and only if for all A G i A is solv able and ( A \ S i = ; or A S i ) ( ) T i ( A ) is solv able. The DR-plan Q of G is strictly solvability pr eserving if and only if for all A G i A is solv able T i ( A ) is solv able. The DR-plan Q of G is c omplete if and only if for all non trivial solv able B S i B = T i 1 T i 2 :::T j ( S j ) for some j i 1. Next, w e formally dene DR-planners and their p erformance measures. An algorithm is said to b e a DR-planner if it outputs a DR-plan when giv en a solv able constrain t graph as input. As b efore, w e consider DR-planners to b e randomized algorithms. A randomized DR-planner A is said to b e valid, solvability pr eserving, strictly solvability pr eserving, c omplete if and only if for ev ery G ev ery DR-plan pro duced b y A is v alid, solv abilit y preserving, strictly solv abilit y preserving or complete accordingly The worst-choic e appr oximation factor of a DR-planner A on input graph G is the minim um of the appro ximation factors of all DR-plans Q of G obtained b y the DR-planner A o v er all p ossible random c hoices. The b est-choic e

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44 appr oximation factor of the algorithm A on input graph G is the maxim um of the appro ximation factors of all the DR-plans Q of G obtained b y the DR-planner A o v er all p ossible random c hoices. In addition to the ab o v e p erformance measures, w e dene three others that rerect the Ch urc h-Rosser prop ert y the abilit y to deal with underconstrained subsystems, as w ell as the abilit y to incorp orate an input, design decomp osition pro vided b y the designer.A DR-planner is said to ha v e the Chur ch-R osser prop ert y if the DR-planner terminates with a DR-plan irresp ectiv e of the order in whic h the dense subgraphs S i are c hosen. A DR-planner A adapts to under c onstr aine d c onstr aint gr aphs G if ev ery (partial) DR-plan pro duced b y A terminates with a set of solv able subgraphs Q i suc h that eac h solv able subgraph Q i has no sup ergraph that is solv able, and moreo v er, no subgraph of G that is disjoin t from all of the Q i 's is solv able. A c onc eptual design de c omp osition P is a set of solv able subgraphs P i whic h are partially ordered with resp ect to the subgraph relation. A DR-planner A is said to inc orp or ate a design de c omp osition P if for ev ery DR-plan Q pro duced b y A the corresp onding sequence of solv able subgraphs S i em b eds a top ological ordering of P as a subsequence recall that a top ological ordering is one that is consisten t with the natural partial order giv en b y the subgraph relation on P When a DR-plan incorp orates a design decomp osition P the lev el of a cluster P i in the partial ordering of P can no w b e view ed as a priorit y rating whic h sp ecies whic h comp onen t of the design decomp osition has most inruence o v er a giv en geometric ob ject. In other w ords, a giv en geometric ob ject K is rst xed/manipulated with resp ect to the lo cal co ordinate system of the lo w est lev el cluster P i 2 P con taining K Thereafter, the en tire cluster P i can b e treated as a

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45 unit and can b e indep enden tly xed/manipulated in the lo cal co ordinate system of the next lev el cluster P j con taining (the simplication of ) P i and so on. Finally w e summarize ho w the ab o v e formal p erformance measures capture the informally describ ed c haracteristics in the set C giv en in the Section 2.1.5 The prop ert y of b eing a general DR-planner refers to whether the metho d successfully terminates with a DR-plan in the general case, and rerects c haracteristic (i); since w e use constrain t graphs whic h yield robust DR-plans that can b e obtained ecien tly the prop ert y of b eing a general DR-planner also rerects (iii) and (viii); dealing with underconstrained graphs also rerects (i); incorp orating input, design decomp ositions rerects (v); v alidit y inruences the Ch urc h-Rosser prop ert y and rerects (i) as w ell as (iv),(v), (vi); solv abilit y preserv ation, strict solv abilit y preserv ation inruence the Ch urc h-Rosser prop ert y and rerect (i), (ii), (iv), (v); completeness is based on the criteria (ii) and (ix); w orst and b est c hoice appro ximation factors are based on (ii) and complexit y directly rerects (iii). 2.1.8 Tw o old DR-planners W e concen trate on t w o primary t yp es of prior algorithms for constructing DR-plans using constrain t graphs and geometric degrees of freedom. Note. Due to reasons discussed in Section 2.1.7 w e lea v e out those graph rigidit y based metho ds for distance constrain ts in dimensions 3 or greater suc h as T a y and Whiteley (1985 ); Hendric kson (1992 ) as w ell as metho ds suc h as Cripp en and Ha v el (1988 ); Ha v el (1991 ); Hsu (1996 ) since they are nondeterministic or rely on sym b olic computation, or they are randomized or generally exp onen tial and based on exhaustiv e searc h. Graph rigidit y and matroid based metho ds for more sp ecic constrain t graphs are discussed in Section 2.1.10 under the so-called Maxim um Matc hing based algorithms. Also, for reasons discussed in the in tro duction, w e lea v e out metho ds based on decomp osing sparse p olynomial systems. }

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46 The rst t yp e of algorithms, whic h w e call SR for c onstr aint Shap e R e c o gnition (e.g. Bouma et al. (1995 ); Ow en (1991 1993 ); Bouma et al. (1995 ), Homann and V ermeer (1994 1995 ); F udos and Homann (1996b 1997 )), concen trates on recognizing sp ecic solv able subgraphs of kno wn shap e, most commonly patterns suc h as triangles. The second t yp e, whic h w e call MM for gener alize d Maximum Matching (e.g. Ait-Aoudia et al. (1993 ); P ab on (1993 ); Latham and Middleditc h (1996 ); Kramer (1992 )), is based on rst isolating certain solv able subgraphs b y transforming the constrain t graph in to a bipartite graph and nding a maxim um generalized matc hing, follo w ed b y a connectivit y analysis to obtain the DR-plan. In this section, w e giv e a formal p erformance analysis of SR, and MM based algorithms { c ho osing a represen tativ e algorithm (t ypically the b est p erformer) in eac h class { using the p erformance measures dened in the previous section. Informally one ma jor dra wbac k of the SR and MM algorithms is their inabilit y to p erform a generalized degree of freedom analysis. F or example, SR w ould require an innite rep ertoire of patterns. In the case of spatial constrain ts, some elemen tary patterns ha v e b een iden tied Homann and V ermeer (1994 1995 ). In the case of extending the scop e of planar constrain t solv ers, adding free-form curv es or conic sections requires additional patterns as w ell Homann and P eters (1995 ); F udos and Homann (1996a ). Similarly a decomp osition of underconstrained constrain t graphs in to w ellconstrained comp onen ts is p ossible for SR algorithms but only sub ject to the pattern limitations. In man y cases, MM algorithms will output DR-plans with larger, nonminimal subgraphs S i that ma y con tain smaller solv able subgraphs. This aects the appro ximation factors adv ersely This inabilit y to nd general minimal dense subgraphs also aects their abilit y to deal with o v erconstrained subgraphs that arise in assem blies, whic h is in turn needed to p erform constrain t reconciliation Homann and Joan-Arin y o (1998 ).

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47 2.1.9 Constrain t Shap e Recognition (SR) Consider the algorithm of F udos and Homann (1996b 1997 ) whic h relies on a follo wing strong assumption (SR1) : all geometric ob jects in 2 dimensions ha v e 2 degrees of freedom and all constrain ts b et w een them are binary and destro y exactly one degree of freedom. Th us, in the corresp onding constrain t graph, the w eigh t of all the edges is 1 and of all the v ertices is 2. Because of this assumption, the SR algorithm ignores the degrees of freedom and relies only on the top ology of the constrain t graph. Description of the algorithm. W e giv e a terse description of the algorithm { the reader is referred to F udos and Homann (1996b 1997 ) for details. Our description is mean t only to put the algorithm in to an appropriate DR-planner framew ork that is suited for the p erformance analysis. The algorithm consists of t w o phases. During Phase One, SR uses the b ottomup iterativ e tec hnique of Itai and Ro deh (1978 ): in the curren t graph G i (where G 1 = G ), sp ecic solv able graphs (clusters) are found that can b e represen ted as a union of three previously found clusters that pairwise share a common v ertex. Suc h congurations of three clusters are called triangles The v ertices of G i represen t clusters and edges of G i represen t constrain ts b et w een clusters (initially due to SR1, ev ery v ertex and ev ery edge of G is a cluster. More sp ecically once a triangle formed b y three clusters has b een found, the three v ertices in G i corresp onding to these three clusters are simplied in to one new v ertex in the simplied graph G i +1 The edges of G i +1 are induced b y the edges of the three old v ertices. This is rep eated for k steps un til there are no more triangles left. If there is only one cluster left in G k then the algorithm terminates. Otherwise G k serv es as an input to Phase Tw o of the algorithm. Before w e describ e Phase Tw o, w e note that a so-called cluster gr aph C i corresp onding to G i is used as an auxiliary structure for the purp ose of nding

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48 triangles. Hence the partial DR-plan pro duced during Phase One of the algorithm is of the form: ( G 1 ; C 1 ) ; : : : ; ( G k ; C k ). The v ertices of the cluster graph C i corresp ond to {v ertices of the original graph G 1 ; {cluster-v ertices in the graph G i ; and {edges in G 1 whic h ha v e not b een included in to one of the cluster-v ertices in the graphs G i 1 ; : : : ; G 1 In particular, the rst cluster graph C 1 con tains one v ertex for ev ery v ertex and ev ery edge of G 1 The edges of the cluster graph C i connect the v ertices of G i that represen t clusters to the original v ertices from G 1 that are con tained in these clusters. Due to the structure of cluster graphs, triangles in G i are found b y lo oking for sp ecic 6-cycles in the corresp onding cluster graph C i These 6-cycles consist of 3 cluster-v ertices and 3 original v ertices. The new cluster graph C i +1 is constructed from C i b y adding a new v ertex c i represen ting the newly found cluster S i and connecting it b y edges to the original v ertices from G 1 that are in the cluster S i W e also remo v e the three old cluster-v ertices in C i whic h together formed the new cluster S i W e note that this is the only w a y in whic h clusterv ertices are remo v ed from cluster graphs. In particular, situations ma y arise where t w o clusters (that share a single v ertex) are represen ted b y the same v ertex in the graph G i but they are represen ted b y distinct v ertices in the cluster graph C i During Phase Tw o, SR constructs the remainder of the DR-plan ( G k ; C k ) ; : : : ; ( G m ; C m ). First SR uses a global top-do wn tec hnique of Hop croft and T arjan (1973 ) to divide G k in to a collection of triconnected subgraphs. These subgraphs are found b y recursiv ely splitting the original graph using separators of size at most 2. Th us the triconnected comp onen ts can b e view ed as the lea v es of a binary tree T eac h of whose in ternal no des corresp onds to a v ertex separator of size at most 2.

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49 F or ev ery triconnected subgraph S in G k a new cluster v ertex is created in the cluster graph C k : similar to Phase One, this v ertex replaces all v ertices that represen t existing clusters con tained in S The next pair ( G k +1 ; C k +1 ) in the DR-plan is found as in Phase One b y nding a triangle in G k or eectiv ely lo cating a 6-cycle in C k Ho w ev er, triangles in G k ha v e already b een lo cated in the course of constructing the tree T these triangles w ere originally split b y the v ertex separators at the in ternal v ertices. Th us, if the original constrain t graph G is solv able, the remainder of the DR-plan ( G k +1 ; C k +1 ) ; : : : ; ( G m ; C m ) is formed b y rep eated simplications along a b ottom up tra v ersal of the tree T If the original graph w as underconstrained, then it is still p ossible to construct a DR-plan of its maximal w ellconstrained subgraphs, b y in tro ducing additional constrain ts and making the original graph solv able. I.e, in order to complete the b ottom up tra v ersal of the tree T in Phase Tw o, additional constrain ts need to b e in tro duced, to mak e the whole graph solv able. F or details, see F udos and Homann (1996b 1997 ). Example. Consider Figure 2{10 During Phase One, the triangles AD E ; AB E ; B C E and C E F will b e disco v ered and simplied as P During Phase Tw o, the remainder of the graph will b e divided in to triconnected subgraphs P ; P GK I LN and O K H M J P then P GK I LN and O K H M J P are simplied and nally the union of P ; P GK I LN and O K H M J P is simplied. Dening the Simplier Map. The same simplier is used throughout Phase One and Tw o: replace a subgraph S i consisting of a triangle of clusters in G i (or triconnected subgraphs during Phase Tw o), b y one v ertex in G i +1 represen ting this triangle. The subgraph S i is found as a 6-cycle in the cluster graph C i The cluster graph C i +1 is constructed as describ ed in Phase One.

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50 5 1 5 1 G G C C M O F G D LN OM FO DG P AD AD DGAEFOOM GFO A B C D E F G H I J K L M N O O N M L K J I H G P Figure 2{10: The original, cluster graph and the simplied graphs More formally recalling the denitions in Section 2.1.7 : let G b e a constrain t graph; the rst graph G 1 in the DR-plan is the original graph G Let G i = ( V ; E ) b e the curren t graph and let S i b e a cluster found. Let A G i ; A = ( V A ; E A ). Then the image of A under the subgraph simplier T i is T i ( A ) = A if the in tersection of A and S i is empt y; otherwise T i ( A ) = ( V T i ( A ) ; E T i ( A ) ) where V T i ( A ) is the set of all v ertices of A that are not v ertices of S i plus a v ertex c i that replaces the cluster S i The set of edges E T i ( A ) is a set of all edges of A that are not edges of S i ; the edges of E A that ha v e exactly one endp oin t in S i will ha v e this endp oin t replaced b y the v ertex c i ; and the edges of A that ha v e b oth endp oin ts in S i are remo v ed. Since the cluster S i in G i is lo cated b y nding a 6-cycle in C i w e need to describ e ho w C i +1 is constructed from C i i.e, the eect that T i has on C i (formally T i is a map from ( G i ; C i ) to ( G i +1 ; C i +1 )). T o obtain C i +1 w e start with C i and

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51 T 1 T 3 C C 2 FE FC CE CB 2 C C B E D A 2 C 1 C D A AB EB DE AD AE C 1 ADEBC EBABCBCE G 2 G 1 G 3 C 1 C 2 E B J O N K M L I H G F E D C B A B C F G H I J K L M N O C F G H I J K L M N O Figure 2{11: Action of the simplier on G i and C i during Phase One rst add a new v ertex c i represen ting the cluster S i whic h is connected b y edges to all the original v ertices that it con tains. Finally v ertices in C i that represen t clusters en tirely con tained in S i are remo v ed, and edges adjacen t to these v ertices are also remo v ed. Figures 2{11 and 2{12 illustrate the action of the subgraph simplier T i on b oth G i and C i A F urther Concession for SR. Observ e that the SR algorithm is not a general DR-planner when input geometric constrain t graphs do not comply with assumption SR1. F or example, for the graph sho wn in Figure 2{13 during Phase One, SR w ould not nd an y triangles and during Phase Tw o, it w ould conclude that the graph is underconstrained, ev en though the graph is w ellconstrained. SR implemen tations ma y remedy some cases similar to this one. F or instance, w eigh t 2 edges to w eigh t 2 v ertices often arise from incidence constrain ts b et w een geometric elemen ts of same t yp e (t w o coinciden t p oin ts or t w o coinciden t lines), and suc h cases are easily accoun ted for b y w orking with equiv alence classes of suc h v ertices. Moreo v er, most planners will ev aluate the densit y of a graph b efore

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52 C 8 G G 5 T 6 T 7 T 8 T 5 7 C 6 C 9 F K D R Q P P IL LN G KLN A J O F K L D A DGDIDNIK OF I D P QKHKOKJJF G I P O N M L K J H Figure 2{12: Action of the simplier during Phase Tw o 3 2 22 22 2 1 1 2 2 22 1 1 Figure 2{13: This solv able graph w ould not b e recognized as solv able b y SR announcing under or o v erconstrained situations. Th us, an implemen tation of the SR algorithms of F udos and Homann (1996b 1997 ) ma y or ma y not construct a DR-plan for the graph of Figure 2{13 dep ending on the original problem statemen t. It is clear that suc h heuristics enlarge the class of solv able graphs, but fall short of decomp osing all solv able constrain t graphs. Ho w ev er, ev en when input graphs do comply with SR1, and SR c hec ks o v erall densit y of a graph, SR could still mistak e a graph that is not solv able for solv able. Figure 2{14 sho ws an example whic h the SR algorithm ma y pro cess incorrectly: since the graph con tains no solv able triangles, SR pro ceeds immediately to Phase

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53 Figure 2{14: W eigh t of all v ertices is 2, w eigh t of all edges is 1 Tw o. During Phase Tw o the triconnectivit y of the en tire graph could b e erroneously construed to mean that the graph is solv able. In fact, the graph do es ha v e densit y 3, whic h sup ercially seems to supp ort suc h a conclusion. Ho w ev er, the graph is certainly not w ellconstrained: the eigh t v ertices on the righ t form a subgraph of densit y -2, that is, an o v erconstrained subgraph. Moreo v er, if this is o v erconstrained subgraph is replaced a subgraph of densit y -3, the resulting graph has densit y 4 unco v ering that it is not w ello v erconstrained either, and therefore not solv able. Figure 2{14 demonstrates that the densit y calculation heuristics is insucien t to determine the existence of minimal dense subgraphs of triconnected constrain t graphs. What is needed is a general algorithm for nding minimal dense subgraphs. In order to giv e a p erformance analysis of the SR algorithm for those classes of constrain t graphs where it do es pro duce a satisfactory DR-plan, w e mak e a str ong c onc ession (SR2) that: only \triangular" structures are \acceptable" for the remainder of this section. I.e, w e mo dify the denitions of v alidit y solv abilit y preserving, strictly solv abilit y preserving and completeness b y replacing the w ords \solv able subgraph" b y the follo wing recursiv e denition: \either a subgraph that can b e simplied in to a single v ertex b y a sequence of consecutiv e simplications of triangles of solv able subgraphs, (using the simplier describ ed in Section 2.1.9 ), or a v ertex, edge or triconnected subgraph'.

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54 P erformance Analysis. In this section, w e analyze the SR algorithm with resp ect to the v arious p erformance measures dened in Section 2.1.7 Claim 3 Under the c onc ession SR2, the SR algorithm is valid. Pro of W e will sho w that ev ery pre-image of the \solv able" cluster S i found at the i th iteration of the SR algorithm is also \solv able," using the stricter SR2 denition of \solv able". It can b e easily c hec k ed that the other requiremen ts necessary for v alidit y from Section 2.1.7 also clearly hold. Let S i b e a cluster that has b een found in G i b y the SR algorithm (b y lo cating a 6-cycle in C i ). Then from the description of the algorithm, and b y assertion SR2, there is a sequence of simplications, sa y T i +1 ; : : : ; T m whose comp osition maps S i to a single v ertex. F or ev ery pre-image A G j A = T 1 j : : : T 1 i 1 ( S i ), for j i 1, the comp osition T m : : : T i T i 1 : : : T j will map A to a single v ertex, hence A is \solv able". Claim 4 The SR algorithm is strictly solvability pr eserving under the c onc ession SR2, and ther efor e is solvability pr eserving as wel l. Pro of Let A b e a \solv able" subgraph, i.e there is a sequence of simplications A 1 ; :::; A m suc h that A = A 1 and A m consists of only one v ertex. If the in tersection of A and the curren tly found cluster S i is empt y then T i ( A ) = A and it remains \solv able". If this in tersection is not empt y then a subgraph B = A \ S i is simplied in to a new cluster v ertex c i = T i ( S i ). W e need to sho w that in this case T i ( A ) remains \solv able" as w ell. Supp ose that some subgraph of B is a v ertex of a 6-cycle in one of the cluster graphs formed during the simplication A 1 ; :::; A m Then clearly the new cluster v ertex c i could p erform the same function: i.e, it could also b e a v ertex in that 6-cycle of one of the cluster graphs formed during the simplication A 1 ; :::; A m Also if A w as triconnected, then so is T i ( A ). This pro v es that there is a sequence (essen tially a sequence A 1 ; : : : ; A m mo died b y replacing

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55 A B Figure 2{15: Tw o triconnected subgraphs not comp osed of triangles Figure 2{16: Solv able graph consisting of n= 3 solv able triangles B b y c i ) whic h terminates in a single v ertex, th us demonstrating the \densit y" of T i ( A ). Claim 5 Even under c onc ession SR2, SR is not c omplete. Pro of W e will sho w that there are cases when the SR pic ks large, nonminimal \solv able" subgraphs S i to simplify ignoring smaller \solv able" subgraphs of S i Consider Figure 2{15 If subgraphs A and B are triconnected but not comp osed of triangles, then they are \solv able," but since the en tire graph is triconnected neither A nor B will b e simplied b y SR. Ho w ev er, since the whole graph A [ B is triconnected, it will c hosen b y SR as S 1 during Phase Tw o. Claim 6 The (worst and) b est-choic e appr oximation factor of SR under c onc ession SR2 is at most O ( 1 n ) Pro of Consider Figure 2{16 The en tire graph consists of n 3 triangles. During Phase One SR will successfully lo cate and simplify eac h one of them. Ho w ev er, during Phase Tw o SR will not b e able to decomp ose the en tire solv able graph in to smaller solv able ones (since en tire graph is triconnected) so the size of the corresp onding DR-plan is O ( n ). On the other hand, the optimal DR-plan w ould

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56 simplify neigh b oring pairs of triangles, one pair at a time, th us the optimal size is a small constan t. Next, w e consider three other p erformance measures discussed in Section 2.1.7 Claim 7 Under the c onc ession SR2, the algorithm SR adapts to under c onstr aine d gr aphs. Pro of Supp ose that the graph G is underconstrained. Let A b e a \solv able" subgraph that is not con tained in an y other \solv able" graph. Since Phase Tw o of the SR algorithm is top-do wn, either SR will nd A and simplify it as one of the S i 's, or A \ S i 6 = ; for one of the S i In either case, SR adapts to G Observ ation 1 Under the c onc ession SR2, the SR algorithm has the Chur chR osser pr op erty, sinc e the new gr aph G i +1 r emains \solvable" if G i is \solvable", r e gar d less of the choic e of the \solvable" S i that is simplie d at the i th stage. Claim 8 Under the c onc ession SR2, the SR algorithm c an inc orp or ate design de c omp ositions P if and only if P full ls the fol lowing r e quir ement: any p air of \solvable" sub gr aphs P k and P t in P satisfy P k P t or P t P k or P k \ P t c ontains no e dges. Pro of F or the if part w e consider the most natural mo dication of the original SR algorithm, and nd a top ological ordering O of the giv en design decomp osition P whic h is a set of \solv able" subgraphs of the input graph G partially ordered under the subgraph relation suc h that O is em b edded as a subplan of the nal DR-plan generated b y this mo died SR algorithm; i.e, O forms a subsequence of the sequence of \solv able" subgraphs S i whose (sequen tial) simplication giv es the DR-plan. W e tak e an y top ological ordering O of the giv en design decomp osition P and create a DR-plan for the rst \solv able" subgraph P 1 in P I.e, while constructing the individual DR-plan for P 1 w e \ignore" the rest of the graph. This individual DR-plan induces the rst part of the DR-plan for the whole graph G In particular,

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57 the last graph in this partial DR-plan is obtained b y simplifying P 1 using the simplier describ ed in 2.1.9 (and treating P 1 exactly as SR w ould treat a cluster S j found at some stage j ). Let G i b e the last graph in the DR-plan for G created th us far. No w, w e consider the next subgraph P 2 in the ordering O and nd an individual DR-plan for it, treating it not as a subgraph of the original graph G but as subgraphs of the simplied graph G i This individual DR-plan is added on as the next part of the DR-plan of the whole graph G The crucial p oin t is that the simplication of an y subgraph, sa y P k will not aect an y of the unrelated subgraphs P t ; t k unless P k P t This is b ecause, b y the requiremen t on P P k and P t share no edges. Therefore, when the cluster v ertex for P k is created, none of the clusters inside P t is remo v ed. The pro cess of constructing individual DR-plans for subgraphs in the decomp osition P and concatenating them to the curren t partial DR-plan is con tin ued un til a partial DR-plan for the input graph G has b een pro duced, whic h completely includes top ological ordering O of the decomp osition P as a subplan. Let G p b e the last graph in this partial DR-plan. The rest of the DR-plan of G is found b y running the original SR algorithm on G p and the corresp onding cluster graph C p F or the only if part, consider Figure 2{17 Let P = f P 0 ; P 1 ; P 2 g where P 0 = AB D ; P 1 = B C D ; P 2 = AB C D Then SR cannot pro duce an y DR-plan that w ould incorp orate P as subplan. 2.1.10 Generalized Maxim um Matc hing (MM) Consider the algorithms of Ait-Aoudia et al. (1993 ); P ab on (1993 ); Kramer (1992 ); Serrano and Gossard (1986 ); Serrano (1990 ) as w ell as graph rigidit y and matroid based metho ds for distance constrain ts in 2 dimensions Hendric kson (1992 ), Gab o w and W estermann (1988 ), Imai (1985 ) as w ell as more general constrain ts Sugihara (1985 ) all of whic h more or less use (generalized) maxim um

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58 A C D B P 0 P 2 P 1 A B D B C D P 2 P 0 A B D C Figure 2{17: Constrain t graph, in tended and actual decomp ositions matc hing (or equiv alen t maxim um net w ork ro w) for nding solv able subgraphs in sp ecialized geometric constrain t graphs. These metho ds either assume that the constrain t graph has zero densit y or they reduce the w eigh t of an arbitrarily selected set of v ertices in order to turn the graph in to one of zero densit y In this thesis w e will analyze what w e consider to b e the most general algorithm of this kind P ab on (1993 ) (it generalizes the algorithm of Ait-Aoudia et al. (1993 ), although the latter pro vides a more complete analysis), supplemen ted (b y us, as suggested b y a review er) with a metho d from Hendric kson (1992 ). Note that while the algorithm of P ab on (1993 ) b oth lo cates solv able subgraphs and describ es ho w to construct a corresp onding DR-plan, Sugihara (1985 ), Hendric kson (1992 ), Gab o w and W estermann (1988 ) only describ e algorithms that allo w to v erify whether a giv en graph is solv able, but do not explicitly state ho w to use these algorithms for successiv ely decomp osing in to small solv able subgraphs, i.e, for constructing DR-plans. While neither of the previously kno wn algorithms analyzed in this thesis p erform w ell according to our previously dened set of criteria, later, w e will describ e our net w ork ro w based F r ontier A lgorithm that impro v es the p erformance in sev eral k ey areas. Description of the Algorithm. As in the case of SR, w e giv e a terse description of the MM algorithm { the reader is referred to Ait-Aoudia et al. (1993 );

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59 A B C D A B C D Figure 2{18: Original graph and it decomp osition P ab on (1993 ) for details. Our description is mean t only to put the algorithm in to an appropriate DR-planner framew ork that is suited for the p erformance analysis. By using maxim um ro w, the input constrain t graph is decomp osed in to a collection of subgraphs that are strongly connected. The ro w information also pro vides a partial ordering of the strongly connected subgraphs represen ting the order in whic h these subgraphs should b e simplied. This ordering in turn sp ecies the DR-plan. It is imp ortan t to note that these strongly connected comp onen ts represen t a sequence of solv able subgraphs of the original constrain t graph, only if the input constrain t graph is w ellconstrained. Consider Figure 2{18 All the v ertices ha v e w eigh t 2, all the edges ha v e w eigh t 1, and the geometry is assumed to b e in 2 dimensions (i.e geometry dep enden t constan t D = 3). The output of the MM algorithm is: {a set of v ertices whose total w eigh t is reduced b y 3 units, sa y w eigh t of v ertices A B and C to b e reduced b y one unit eac h, (this corresp onds to xing 3 degrees of freedom the n um b er of degrees of freedom of a rigid b o dy in 2 dimensions); {t w o strongly connected comp onen ts AB C and D ; {the DR-plan, i.e, the information that the subsystem represen ted b y AB C should b e simplied/solv ed rst and then its union with D should b e simplied/solv ed.

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60 AC ABBC CDBD A BCD Sink 11111 2 111 AC ABBC CDBD A BCD Sink 11111 1112 Figure 2{19: Mo died bipartite graph and maxim um ro w in this graph In order to pro duce suc h an output, the MM algorithm rst constructs the net w ork X = ( V X ; E X ) corresp onding to the original w eigh ted constrain t graph G = ( V ; E ) (after the w eigh ts of A; B ; C w ere reduced b y one unit eac h). The set of v ertices V X is the union of V X 1 and V X 2 where the v ertices in V X 1 corresp ond to the v ertices in V v ertices in V X 2 corresp ond to the edges in E An edge ex 2 E X w ould b e created b et w een v x 1 2 V X 1 and v x 2 2 V X 2 if the v ertex in V corresp onding to the v x 1 is an endp oin t of an edge in E corresp onding to v x 2 The edge ex has innite capacit y All the v ertices in V X 1 are connected to the sp ecial v ertex called Sink The capacit y of connecting edges is equal to the w eigh t of the corresp onding v ertices in V F or example, the left half of Figure 2{19 sho ws the bipartite graph corresp onding to the constrain t graph of Figure 2{18 Next the maxim um ro w in the net w ork X is found, with v ertices in V X 2 b eing source v ertices of capacit y equal to the w eigh ts of the corresp onding edges in E See righ t half of Figure 2{19 (thic k edges ha v e nonzero ro w). The maxim um ro w found in X induces a partition of the original graph G in to a partially ordered set of strongly connected comp onen ts { giving a sequence of solv able subgraphs of G pro vided G is w ellconstrained { as in Figure 2{18 according to the follo wing rules: if the ro w in X from the v ertex z 2 V X 2 that is connected to the v ertices x; y 2 V X 1 is sen t to w ard x then in the graph G the edge corresp onding to z (b et w een v ertices x and y ) b ecomes an orien ted edge directed from y to x If the ro w from z is sen t to w ard b oth x and y then the edge

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61 A C D B 1 Sink 1111 22 ADBC 2 2 4 ABADBDBC DC * 2 Figure 2{20: Constrain t graph and net w ork ro w with 3 extra ro w units at AB is bidirected to w ard b oth x and y (Recall that a strongly connected comp onen t S in this directed graph is a subgraph suc h that for an y t w o v ertices a; b 2 S there is an orien ted path from a to b .) The partial ordering of comp onen ts is induced as follo ws. Let K and L b e t w o strongly connected comp onen ts in the orien ted v ersion of G If all the edges b et w een v ertices of K and L are p oin ting to w ard L then L should b e simplied after K Dening the Simplier Map. W e capture the transformations p erformed b y the MM DR-planner describ ed ab o v e, b y describing its simplier maps (recall the denitions in Section 2.1.7 ). Let G = ( V ; E ) b e the geometric constrain t graph. Denote b y G 1 = ( V 1 ; E 1 = E ) the directed graph after w eigh ts of some v ertices ha v e b een reduced as describ ed ab o v e and a partial ordering of strongly connected comp onen ts has b een found. First S 1 G 1 is lo cated suc h that S 1 is strongly connected. Then S 1 is simplied in to the v ertex v 1 of w eigh t zero. The other v ertices of G 1 remain unc hanged. Edges of G 1 that had b oth endp oin ts outside of S 1 are unc hanged, edges that had b oth endp oin ts in S 1 are remo v ed, edges that had exactly one endp oin t in S 1 ha v e this endp oin t replaced b y v 1 In the next step, C 2 the next strongly connected comp onen t in a top ological ordering of the comp onen ts in G 2 = T 1 ( G 1 ) is lo cated. The subgraph S 2 is set to b e f v 1 g [ C 2 Then S 2 is simplied in to the v ertex v 2

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62 22 1 2 2 A B C D E Figure 2{21: DR-plan dep ends on the initial c hoice of w eigh t zero. This pro cess is con tin ued un til G k consists of one v ertex. The simplier maps are formally dened as follo ws. { T i ( S i ) = v i where v i is the v ertex in G i +1 of w eigh t 0. {If B G i ; B \ S i = ; then T i ( B ) = B {If B G i ; B \ S i 6 = ; then the image of B T i ( B ) = f v i g [ ( B n S i ). Redening Solv abilit y Claim 9 The MM algorithm is a not a gener al DR-planner. Pro of While the MM algorithm can correctly classify as solv able and decomp ose w ellconstrained and w ello v erconstrained graphs and correctly classify underconstrained graphs that ha v e no o v erconstrained subgraphs as b eing unsolv able, it is unable to correctly classify an underconstrained graph that has an o v erconstrained subgraph. Consider Figure 2{21 the w eigh t of all the edges is 1, of the v ertices as indicated. Graph AB C D E has densit y -3, it con tains o v erconstrained subgraphs AC ; B C ; C D ; C E and remo v al of sa y edge AC will result in AB C D E b ecoming non-dense, hence AB C D E is not w ello v erconstrained and not solv able. The MM algorithm ma y or ma y not detect this, dep ending on the initial c hoice of v ertices whose w eigh ts are to b e reduced. Supp ose that the w eigh t of the v ertex E w as reduced b y 1 and the w eigh t of the v ertex D b y 2. Then the corresp onding maxim um p ossible ro w f and the corresp onding decomp osition in to strongly comp onen ts are sho wn in Figure 2{22 Note that it is imp ossible to simplify the strongly connected comp onen t A and th us there can b e no DR-plan

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63 AC ABBC A BCD Sink 11111 1 CDCEED E 1 22 1 0 1 0 2 1 A B C D E 2 Figure 2{22: Maxim um ro w and decomp osition giv en the bad initial c hoice 1 0 2 1 A B C D E 2 Figure 2{23: Decomp osition giv en the go o d initial c hoice for this initial c hoice of v ertices for reducing w eigh ts. On the other hand, if the w eigh t of A had b een reduced b y 2 and w eigh t of D b y 1 then the maxim um ro w is larger than f (in fact, no c hoice of v ertices for w eigh t reduction can giv e a larger ro w) and it can b e c hec k ed that MM w ould yield a DR-plan, see Figure 2{23 ( S 1 = A; S 2 = AC ; S 3 = AB C ; S 4 = AB C D ; S 5 = AB C D E ). Th us MM runs in to problems in the presence of o v erconstrained subgraphs, unless G happ ens to b e w ello v erconstrained, whic h cannot b e apriori c hec k ed without relying on an algorithm for detecting o v erconstrained subgraphs and replacing them b y w ellconstrained ones. Therefore it is necessary and sucien t to lo cate o v erconstrained subgraphs and replace them b y w ellconstrained ones, in order to guaran tee that MM will w ork.

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64 The author of P ab on (1993 ) do es not sp ecify ho w to do this. The follo wing mo dication similar to that of Hendric kson (1992 ) could b e used, as w as suggested b y a review er, and completed here. First a maxim um ro w in the unmo died net w ork X (i.e where v ertex w eigh ts are not reduced) is found. After that, for ev ery source v ertex v 2 V X 2 except one v ertex v m the follo wing t w o steps are rep eated. {Three extra units of ro w are sen t from v p ossibly rearranging existing ro ws in X {These 3 units of ro w are remo v ed, without restoring original ro ws. {F or v ertex v m 3 units of ro w are sen t but not remo v ed. F or example, consider Figure 2{20 The constrain t graph is sho wn on the left, w eigh ts of all v ertices is 2, the w eigh ts of all edges is 1. The resulting net w ork ro w is sho wn on the righ t, assuming that AB w as the last v ertex v m The extra 3 units of ro w and their destination v ertices A and B are mark ed b y asterisks. This particular ro w induces a w eigh t reduction of A b y 1 unit and of B b y 2 units. This mo dication iden ties o v erconstrained subgraphs since it is imp ossible to send all 3 extra units from at least one edge of an o v erconstrained graph. These o v erconstrained graphs ha v e to b e mo died b y the designer to b ecome w ellconstrained, and the ro w algorithm is run again This is imp ortan t b ecause otherwise the DR-algorithm cannot pro ceed, since in underconstrained graphs with o v erconstrained subgraphs, strongly connected comp onen ts do not necessarily corresp ond to a sequence of solv able subgraphs. T o rerect the mo dication ab o v e, w e mak e the follo wing c onc ession for MM, (MM1) : the input constrain t graph G has no o v erconstrained subgraphs in G A subgraph A G is \solv able" if it has zero densit y after the v ertices for w eigh t reduction b y D are c hosen (these can b e c hosen arbitrarily pro vided there are no o v erconstrained subgraphs).

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65 P erformance Analysis. In this section, w e analyze the MM algorithm with resp ect to the v arious p erformance measures dened in Section 2.1.7 Note that despite the concession MM1, the MM algorithm has the follo wing dra wbac ks: the DR-plan is uniquely determined, once v ertices whose w eigh ts are reduced are c hosen. F or example, in Figure 2{20 after the w eigh t of A is reduced b y 1 and the w eigh t of B b y 2 units, solv able subgraphs to b e simplied are S 1 = f A; B g ; S 2 = f T 1 ( S 1 ) ; D g ; S 3 = f T 2 ( S 2 ) ; C g While the subgraph B C D is also solv able (if sa y initially w eigh ts of v ertices B and C w ere reduced instead of A and B ), it cannot b e c hosen as one of the S i after w eigh ts of A and B are reduced. This causes MM to ha v e bad w orst-c hoice and b est-c hoice appro ximation factors and to b e unable to incorp orate general designer decomp osition. The follo wing is straigh tforw ard from the description of the simpliers. Claim 10 Under the c onc ession MM1, the MM algorithm is a valid DR-planner. Claim 11 Under the c onc ession MM1, the MM algorithm is strictly solvability pr eserving (and ther efor e solvability pr eserving). Pro of Supp ose that a subgraph A of the input graph G is \solv able," i.e d ( A ) = 0. Let S i b e the \solv able" subgraph to b e simplied at the curren t stage. Let B = A \ S i ; C = A n B Since w e assume that G do es not con tain an y o v er-constrained subgraphs, d ( B ) 0 and d ( A [ S i ) 0 ) d ( A [ S i ) = d ( A ) + d ( S i ) d ( B ) 0 ) 0 + 0 d ( B ) 0 ) d ( B ) = 0. Th us d ( T i ( A )) = d ( T i ( C )) + d ( T i ( B )) = d ( C ) + 0 = d ( A ) d ( B ) = 0, therefore T i ( A ) is also solv able. Claim 12 Under the c onc ession MM1, the MM algorithm is c omplete. Pro of Let A b e a prop er solv able subgraph of the S i ; i 2. Since A is solv able, there cannot b e an y edges outside of A p oin ting to w ard A (this is b ecause there is no ro om for ro ws of \outside" edges to w ard the v ertices of A ). Recall that S i is the union of C i [ f v i 1 g where C i is the rst strongly connected comp onen t

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66 2 2 2 3 3 3 3 3 333 C R R R R R n/2 L L L L L 1 2 3 n/2-1 n/2 n/2-1321 Figure 2{24: Bad b est-c hoice appro ximation in the top ological ordering of the comp onen ts at the stage i and v i 1 is the simplication of S i 1 Ho w ev er, unless A = v i 1 the rst strongly connected comp onen t at this stage w ould ha v e b een A n f v i 1 g not C i whic h con tradicts the c hoice of C i at stage i Claim 13 Under the c onc ession MM1, the b est-choic e (and worst-choic e) appr oximation factor of MM is at most O ( 1 n ) Pro of T o pro v e the b ound on the b est-c hoice appro ximation factor consider Figure 2{28 The left and righ t columns con tain n= 2 v ertices eac h. The w eigh ts of all the v ertical edges are 2, the w eigh ts of all other edges are 1, the w eigh ts of the v ertices are as indicated, and the geometry dep enden t constan t D = 3. Note that all solv able subgraphs in Figure 2{28 could b e divided in to 3 classes. The rst class consists of the subgraphs C L 1 L 2 ; C L 1 L 2 L 3 ; : : : ; C L 1 L 2 : : : L n= 2 1 L n= 2 The second class consists of the subgraphs C R 1 R 2 ; C R 1 R 2 R 3 ; : : : ; C R 1 R 2 : : : R n= 2 1 R n= 2 The third class con tains the solv able subgraphs that con tain b oth L and R v ertices. There is only one elemen t in this class the en tire graph C L 1 L 2 : : : L n= 2 R 1 R 2 : : : R n= 2 There is an optimal DR-plan of constan t size that tak es S 1 = C L 1 L 2 ; S 2 = S 1 [ L 3 ; : : : ; S n= 2 1 = S n= 2 2 [ L n= 2 After that it tak es S n= 2 = C R 1 R 2 ; S n= 2+1 = S n= 2 [ R 3 ; : : : S n = S n 1 [ R n= 2 Finally it tak es S n +1 = S n= 2 1 [ S n

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67 Ho w ev er all DR-plans found b y MM will ha v e size O ( n ). The reason for this is that MM is unable to simplify solv able subgraphs on the left of the Figure 2{28 indep enden tly from the solv able subgraphs on the righ t. More formally let S 1 b e the rst subgraph simplied b y MM under some DR-plan Q If S 1 b elongs to the third class of solv able subgraphs then size of Q is O ( n ). Otherwise, without loss of generalit y w e can assume that S 1 b elongs to the rst class. According to the denition of MM, the simplication of S 1 is a v ertex v 1 of w eigh t 0. After this simplication an y strongly connected comp onen t that con tains some R i should also con tain all of the R 1 : : : R n= 2 Hence there is an S i in Q suc h that R 1 R 2 : : : R n= 2 S i Hence the size of Q is O ( n ). Next, w e consider three other p erformance measures discussed in Section 2.1.7 Claim 14 Under the c onc ession MM1, a simple mo dic ation of the MM algorithm is able to adapt to under c onstr aine d gr aphs. This mo dic ation, however, incr e ases the c omplexity by a factor of n Pro of Supp ose that the graph G is underconstrained. Consider the maxim um ro w found b y MM in the net w ork X corresp onding to G as describ ed in subsection 2.1.10 There are t w o cases. The rst case is when the last v ertex v m in X corresp onds to an edge of some solv able subgraph A 1 Then all v ertices v i in X corresp onding to v ertices of A 1 ha v e their capacities completely lled, since A 1 is solv able. On the other hand, at least one v ertex v of G will not ha v e its capacit y lled completely since G is underconstrained. Let W b e the set of suc h v ertices v The new mo dication of MM w ould pro ceed b y remo ving all v ertices of X corresp onding to v ertices of W and edges adjacen t to W as w ell as ro ws originating at suc h edges. Once this is done, a new set W is recomputed and remo v ed, un til all remaining v ertices of X that represen t v ertices of G ha v e their capacities lled completely These v ertices comprise a solv able subgraph A 1 suc h that no sup ergraph of A 1 is solv able. Once

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68 the subgraph A 1 (and its DR-plan Q 1 ) is found, it could b e remo v ed from G and the pro cess applied recursiv ely to G n A 1 to nd a solv able subgraph A 2 its DR-plan Q 2 and so on. The second case is when the last v ertex v m in X is not con tained in an y solvable subgraph. Then constructing and remo ving sets W as describ ed ab o v e will completely exhaust G without nding an y solv able graphs. The new mo dication of MM w ould pro ceed b y nding another maxim um ro w in net w ork X corresp onding to G suc h that the last v ertex v i m is dieren t from v m Again t w o cases are considered for v ertex v i m If all v i m 2 G are not con tained in an y solv able subgraph, then G do es not con tain an y solv able subgraph and the pro cess terminates. Note that this mo dication ohand increases the complexit y of MM b y a factor of n and there do es not seem to b e an y ob vious w a y to prev en t this factor. This mo dication of MM outputs a DR-plan Q = Q 1 ; : : : ; Q k for set of solv able subgraphs A 1 ; : : : ; A k suc h for an y i no sup ergraph of A i is solv able and there are no solv able subgraphs B G suc h that B \ A j = ; ; 8 j Th us this mo dication of MM is able to adapt to underconstrained graphs. Observ ation 2 (i) Under the c onc ession MM1, MM has the Chur ch-R osser pr op erty sinc e simplifying any S i that is solvable at the curr ent stage pr eserves the density of the whole gr aph G i +1 (i.e G i +1 is solvable if and only if G i is). (ii) Under the c onc ession MM1, MM is able to inc orp or ate a design de c omp osition P sp e cie d by the designer if and only if for every P k ; P t 2 P such that P k \ P t 6 = ; either P k P t or P t P k Pro of is similar to the corresp onding pro of for SR algorithm in Claim 8 2.1.11 Comparison of P erformance of SR and MM Next, w e giv e a table comparing the SR and MM DR-planners with resp ect to the p erformance measures of Section 3. \Underconstr" refers to the abilit y to

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69 deal with underconstrained graphs, \Design decomp osition" refers to the abilit y to incorp orate design decomp ositions sp ecied b y the designer. The sup erscript *' refers to a narro w class of DR-plans: those that require the solv able subsystems S i to b e based on triangles or a xed rep ertoire of patterns. The sup erscript  y refers to results that w ere not true for the original MM algorithm dev elop ed b y ( Ait-Aoudia et al. (1993 ); P ab on (1993 )) and pro v ed in this thesis through a mo dication of MM describ ed in Section 2.1.10 The sup erscript } refers to a restricted class of graphs in whic h there are no o v erconstrained subgraphs. It also refers to a further mo dication of MM describ ed in Claim 14 whic h ho w ev er, increases the complexit y b y a factor of n The sup erscript  refers to strong restrictions on the design decomp ositions that can b e incorp orated in to DR-plans b y SR and MM. 2.1.12 Analysis of Tw o New DR-planners W e presen t t w o new Decomp osition-Recom bination (DR) planning algorithms or DR-planners. The new planners follo w the o v erall structural description of a t ypical DR-planner, based on the DR-solv er S giv en in previous sections. F urthermore, the new DR-planners adopt and adapt features of older decomp osition metho ds suc h as SR (shap e recognition) and MM (generalized maxim um matc hing) that w ere analyzed and compared previously In particular, those metho ds as w ell P erformance measure SR MM Generalit y No Y es y Underconstr. No(Y es ) Y es y ; } Design decomp osition No(Y es ; ) No(Y es y ; ) V alidit y No(Y es ) Y es y Solv abilit y No(Y es ) Y es y Strict solv abilit y No(Y es ) Y es y Complete No(No ) Y es y W orst appro ximation factor 0 ( O ( 1 n ) ) O ( 1 n ) y Best appro ximation factor 0 ( O ( 1 n ) ) O ( 1 n ) y Ch urc h-Rosser No(Y es ) Y es y Complexit y O (( m + n ) 2 ) O ( n ( m + n )) y

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70 as the new planners are based on de gr e e of fr e e dom analysis of geometric constrain t hyp er gr aphs It should b e noted that the SR and MM based algorithms Bouma et al. (1995 ); Ow en (1991 1993 ); Bouma et al. (1995 ); Homann and V ermeer (1994 ), Homann and V ermeer (1995 ); Latham and Middleditc h (1996 ); F udos and Homann (1996b 1997 ), Ait-Aoudia et al. (1993 ); P ab on (1993 ); Kramer (1992 ); Serrano and Gossard (1986 ); Serrano (1990 ), w ere b eing dev elop ed ev en as the issue { of ecien t decomp osition of constrain t systems for capturing design in ten t in CAD/CAM { w as still in the pro cess of crystallization. In con trast, our dev elopmen t of the new DR-planners is systematically guided b y the new p erformance measures, to closely rerect sev eral desirable c haracteristics C of DR-planners for CAD/CAM applications. An imp ortan t building blo c k of b oth the new DR-planners is the routine used to isolate the solv able subsystems S i at eac h step i In b oth new DR-planners, the solv able subsystem/subgraph S i is isolated using an algorithm dev elop ed b y the authors based on a subtle mo dication of incremen tal net w ork ro w: this algorithm, called \Algorithm Dense," rst isolates a dense subgraph, and then nds a minimal dense subgraph inside it, whic h ensures its solv abilit y The in terested reader is referred to earlier pap ers b y the authors: Homann et al. (1997 ) for a description as w ell as implemen tation results, and Homann et al. (1998 ) for a comparison with prior algorithms for isolating solv able/dense subgraphs. Here, w e shall only note sev eral desirable features of Algorithm Dense. (a) A useful prop ert y of the dense subgraph G 0 found b y the Algorithm Dense (in time O ( n ( m + n ))) is that the densities of all prop er subgraphs are strictly smaller than the densit y of G 0 Therefore, when G 0 corresp onds to a w ellconstrained subsystem, then G 0 is in fact minimal, and hence it is unnecessary

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71 to run the additional steps to obtain minimalit y Ensuring minimalit y crucially aects completeness of the new DR-planners. (b) A second adv an tage of Algorithm Dense is that it is m uc h simpler to implemen t and faster than standard max-ro w algorithms, This w as b orne out b y our C++ implemen tation of Algorithm Dense b oth for nding dense and minimal dense subgraphs. By making constantD a parameter of the algorithm, our metho d can b e applied uniformly to planar or spatial geometry constrain t graphs. F urthermore, the new algorithm handles not only binary but also ternary and higher-order constrain ts whic h can b e represen ted as h yp eredges (c) A third sp ecic adv an tage of our ro w-based algorithm for isolating dense subgraphs (solv able subsystems) is that w e can run the algorithm on-line. That is, the constrain t graph and its edges can b e input con tin uously to the algorithm, and for eac h new v ertex or edge the ro w can b e up dated accordingly This promises a go o d t with in teractiv e, geometric constrain t solving applications, i.e, the criterion (viii) of C As will b e seen b elo w the main dierence b et w een the t w o new planners, ho w ev er, lies in their simplier maps T i i.e, the w a y in whic h they abstract or simplify a solv able subgraph S i once it has b een found. Comparison. As a prelude to the analysis of the new DR-planners Condense and F ron tier w e giv e a table b elo w whic h extends the comparison b et w een the SR and MM to the new DR-planners Condense and F ron tier. The \complexit y" en tries for the 2 new DR-planners are directly based on the complexit y of a building blo c k Algorithm Dense (briery discussed ab o v e) for isolating minimal dense subgraphs S i \Under-const" refers to the abilit y to deal with underconstrained systems, \Design-dec" refers to the abilit y to incorp orate design decomp ositions sp ecied b y the designer, \Solv." and \Strict solv." refer

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72 to (strict) solv abilit y preserv ation, \W orst and Best appro x" refer to the w orst and b est c hoice appro ximation factor. P erf. meas. SR MM Condense(new) F ron tier Generalit y No Y es y Y es Y es Under-const No(Y es ) No Y es Y es Design-dec No(Y es ; ) No No(Y es ) Y es V alidit y No(Y es ) Y es + Y es Y es Solv. No(Y es ) Y es + Y es Y es Strict solv. No(Y es ) Y es + No Y es Complete No(No ) No Y es Y es W orst appro x. 0 ( O ( 1 n ) ) O ( 1 n ) O ( 1 n ) O ( 1 n ) Best appro x. 0 ( O ( 1 n ) ) O ( 1 n ) y O ( 1 n ) O ( 1 2 ) Ch urc h-Rosser No(Y es ) Y es y Y es Y es Complexit y O ( s 2 ) O ( n D +1 s ) y O ( n 3 s ) O ( n 3 s ) Note. The v ariable s in the complexit y expressions denotes the n um b er of v ertices, n plus the n um b er of edges, m of the constrain t graph. Recall that D refers to the n um b er of degrees of freedom of a rigid ob ject in the input geometry (in practice, this could b e treated as a constan t). } The sup erscripts *' and +' refer to narro w classes of DR-plans: those that require the solv able subsystems S i to b e based on triangles or a xed rep ertoire of patterns, or to represen t rigid ob jects that xed or grounded with resp ect to a single co ordinate system. The sup erscript  y refers to results that w ere left unpro v en b y the dev elop ers of the MM based algorithms ( Ait-Aoudia et al. (1993 ), P ab on (1993 )) and pro v ed in this pap er through a crucial mo dication of MM describ ed previously The mo dication also results in the impro v emen t of the complexit y of the b est MM algorithm to O ( n ( s = n + m )).

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73 a f e d c b Figure 2{25: Sequen tial extension The sup erscript  refers to a strong restriction on the design decomp ositions that can b e incorp orated in to DR-plans b y SR and the new DR-planner Condense. In fact, the other new DR-planner F ron tier also places a (ho w ev er, m uc h w eak er) restriction on the design decomp ositions that it can incorp orate, as will b e sho wn later.2.1.13 The DR-planner Condense and its P erformance This DR-planner w as sk etc hed b y the authors already in Homann et al. (1998 ). The authors' ro w-based algorithm discussed ab o v e Homann et al. (1997 ) is applied rep eatedly to constrain t graphs to nd minimal dense subgraphs or clusters (whic h w e kno w to b e generically solv able con taining more than one v ertex. DR-planner Condense consists of t w o conceptual steps. A minimal dense cluster can b e se quential ly extende d under certain circumstances b y adding more geometric ob jects one at a time, whic h are rigidly xed with resp ect to the cluster. After a cluster has b een th us extended, it is then simplied in to a single geometric ob ject, and the rest of the constrain t graph is searc hed for another minimal dense subgraph. The follo wing example illustrates sequen tial extension. Consider the constrain t graph G of Figure 2{25 W e assume that all v ertices ha v e w eigh t 2 and all edges ha v e w eigh t 1. The geometry-dep enden t constan t D = 3. The v ertex set f a; b g induces a minimal dense subgraph of G whic h could b e c hosen b y DRplanner Condense as the initial minimal dense cluster, whic h could b e extended sequen tially b y the v ertices c; d; e; f one v ertex at a time, un til it cannot b e

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74 3 3 3 2 3 3 3 2 2 2 Figure 2{26: Sequence of simplications from left to righ t extended an y further. The resulting subgraph is called an extende d dense sub gr aph or cluster The simplication of an extended cluster is tak en to b e a single geometric ob ject with D degrees of freedom. This is done as follo ws: an extended cluster A is replaced b y a v ertex u of w eigh t D ; all edges from v ertices in A to a v ertex w outside A are com bined in to one edge ( u; w ), and the w eigh t of this induced edge is the sum of the w eigh ts of the com bined edges. After the simplication, another solv able subgraph is found, and the pro cess is con tin ued un til the en tire graph is simplied in to a single v ertex. This is illustrated b y the sequence of simplications of Figure 2{26 Initially all v ertices ha v e w eigh t 2, all edges ha v e w eigh t 1. The v ertices connected b y the hea vy edges constitute minimal or sequen tially extended clusters. After four simplications the original graph is replaced b y one v ertex. Dening Subsystem Simpliers. W e capture the transformations p erformed b y the DR-planner Condense b y describing simplier maps. Let G b e the input constrain t graph; the rst graph G 1 in the DR-plan is the original graph G Let G i = ( V ; E ) b e the curren t graph and let S i b e a cluster found at the curren t stage. Let A b e an y subgraph of G i Then T i ( A ) is dened as follo ws. If A \ S i = ; then T i ( A ) = A If A \ S i 6 = ; then T i ( A ) = ( V T A ; E T A ) where V T A is the set of all v ertices of A that are not v ertices of S i plus one v ertex c i of w eigh t D whic h represen ts

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75 the simplication of the cluster S i The set of edges E T A is formed b y remo ving edges with all endp oin ts in S i and com bining edges with at least one endp oin t outside S i (as w ell as their w eigh ts) as describ ed earlier in this section. P erformance Analysis. In this section, w e analyze the DR-planner Condense with resp ect to the v arious p erformance measures that w ere dened in preceeding sections. Claim 15 Condense is a valid DR-planner with the Chur ch-R osser pr op erty. In addition, DR-planner Condense nds DR-plans for the maximal solvable sub gr aphs of under c onstr aine d gr aphs. Pro of If the graph G is not underconstrained, then it will remain so after the replacemen t of an y solv able subgraph b y a v ertex of w eigh t D i.e, after a simplication step b y DR-planner Condense. Th us, if G = G 1 is w ellconstrained, it follo ws that all of the G i are w ellconstrained. Moreo v er, w e kno w that if the original graph is solv able, then at eac h step, DR-planner Condense will in fact nd a minimal dense cluster S i that consists of more than one v ertex, and therefore G i +1 is strictly smaller than G i for all i Th us the pro cess will terminate at stage k when G k is a single v ertex. This is indep enden t of whic h solv able subgraph S i is c hosen to b e simplied at the i th stage, sho wing that DR-planner Condense has the Ch urc h-Rosser prop ert y On the other hand, if G is underconstrained, since the subgraphs S i c hosen to b e simplied are guaran teed to b e dense/solv able, the pro cess will not terminate with one v ertex, but rather with a set of v ertices represen ting the simplication of a set of maximal solv able subgraphs (suc h that no com bination of them is solv able). This completes the pro of that DR-planner Condense is a DR-planner that can adapt to underconstrained graphs.

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76 C A B D E F G H I J K L M O N S E F H I J K L M G Figure 2{27: Original and simplied graphs The pro of of v alidit y follo ws straigh tforw ardly from the prop erties of the simplier map. Claim 16 DR-planner Condense is solvability pr eserving. Pro of The simplier maps T i do not aect subgraphs outside of S i Claim 17 DR-planner Condense is not strictly solvability pr eserving. Pro of Consider the constrain t graph of Figure 2{27 The v ertex w eigh ts are 2, the edge w eigh ts are 1, and the geometry-dep enden t constan t D = 3. The graphs AB C D N O ; E F H GI D and N M K LI J are all dense/solv able. Supp ose that rst the cluster S 1 = AB C D N O w as found and simplied in to one v ertex S of w eigh t 3. No w the graph S E F H GI = T 1 ( E F H GI D ) is not dense/solv able an ymore. In tuitiv ely the reason wh y DR-planner Condense is not strictly solv abilit y preserving is that the remo v al of the v ertices D and N loses v aluable information ab out the structure of the solv able graph. Claim 18 DR-planner Condense is c omplete. Pro of This is b ecause DR-planner Condense nds minimal dense subgraphs at eac h stage. Claim 19 Best-choic e (and worst-choic e) appr oximation factor of DR-planner Condense is at most O (1 =n )

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77 2 2 2 3 3 3 3 3 333 C R R R R R n/2 L L L L L 1 2 3 n/2-1 n/2 n/2-1321 Figure 2{28: Bad b est-c hoice appro ximation Note. This pro of mimics the MM appro ximation factor pro of, except that no w the subgraph S i is not a strongly connected comp onen t. } Pro of T o pro v e the b ound on the b est-c hoice appro ximation factor consider Figure 2{28 The left and righ t columns con tain n= 2 v ertices eac h. The w eigh ts of all the v ertical edges are 2, the w eigh ts of all other edges are 1, the w eigh ts of the v ertices are as indicated, and the geometry dep enden t constan t D = 3. Note that all solv able subgraphs in Figure 2{28 could b e divided in to 3 classes. The rst class consists of the subgraphs C L 1 L 2 ; C L 1 L 2 L 3 ; : : : ; C L 1 L 2 : : : L n= 2 1 L n= 2 The second class consists of the subgraphs C R 1 R 2 ; C R 1 R 2 R 3 ; : : : ; C R 1 R 2 : : : R n= 2 1 R n= 2 The third class con tains the solv able subgraphs that con tain b oth L and R v ertices. There is only one elemen t in this class the en tire graph C L 1 L 2 : : : L n= 2 R 1 R 2 : : : R n= 2 There is an optimal DR-plan of constan t size that tak es S 1 = C L 1 L 2 ; S 2 = S 1 [ L 3 ; : : : ; S n= 2 1 = S n= 2 2 [ L n= 2 After that it tak es S n= 2 = C R 1 R 2 ; S n= 2+1 = S n= 2 [ R 3 ; : : : S n = S n 1 [ R n= 2 Finally it tak es S n +1 = S n= 2 1 [ S n Ho w ev er all DR-plans found b y DR-planner Condense will ha v e size O ( n ). The reason for this is that DR-planner Condense is unable to simplify solv able subgraphs on the left of the Figure 2{28 indep enden tly from the solv able subgraphs

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78 on the righ t. More formally let S 1 b e the rst subgraph simplied b y DR-planner Condense under some DR-plan Q If S 1 b elongs to the third class of solv able subgraphs then the size of Q is O ( n ). Otherwise, without loss of generalit y w e can assume that S 1 b elongs to the rst class. According to the denition of DRplanner Condense, T 1 ( S 1 ) is a single v ertex of w eigh t 3 that replaces sev eral v ertices including v ertex C No w for an y graph A that b elongs to the second class, T 1 ( A ) is not solv able an ymore (it has densit y -4). Hence there is an S i in Q suc h that R 1 R 2 : : : R n= 2 S i Hence the size of Q is O ( n ). Next, w e consider the last p erformance measure incorp orating input decomp osition.Observ ation 3 In gener al, DR-planner Condense c annot inc orp or ate a design de c omp osition of the input gr aph for r e asons similar to those given for the SR algorithm. It c an inc orp or ate a design de c omp osition, only if every p air A and B of sub gr aphs in the de c omp osition ar e either c ompletely disjoint or A B or B A 2.1.14 The DR-planner F ron tier and its P erformance In tuitiv ely the reason wh y DR-planner Condense is not strictly solv abilit y preserving is that the simplication of a minimal or extended dense cluster in to a single v ertex loses v aluable information ab out the structure of the cluster. The algorithms describ ed in this section preserv e this information at least partially b y designing a simplier that k eeps the structure of the fr ontier vertic es of the cluster, i.e, those v ertices that are connected to v ertices outside of the cluster. Ho w ev er, the w a y in whic h the minimal dense clusters and their sequen tial extensions are found is iden tical to that of DR-planner Condense { i.e, b y using the authors' Algorithm Dense from Homann et al. (1997 ). Informally under DR-planner F ron tier, all in ternal (i.e not fron tier) v ertices of the solv able subgraph S i found at the i th stage are simplied in to one v ertex called the c or e c i The core v ertex is connected to eac h fron tier v ertex v b y an edge whose

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79 I D N Figure 2{29: The simplied graph after three clusters has b een replaced b y edges w eigh t is equal to that of v All other edges, except ones b et w een fron tier v ertices, of S i are remo v ed. This is rep eated un til the solv able subgraph S m found is the en tire remaining graph G m If the solv able subgraph S i has only t w o fron tier v ertices, then all in ternal v ertices of S i should b e remo v ed and no new core v ertex created. Instead the t w o fron tier v ertices of S i should b e connected b y an edge whose w eigh t w is c hosen so that the sum of the w eigh ts of the t w o fron tier v ertices less w is equal to the constan t D This ensures that the graphs G i b ecome steadily smaller as i increases. F or instance, Figure 2{27 is suc h a sp ecial case, where ev ery cluster has only 2 fron tier v ertices. Hence after three iterations it w ould b e simplied b y the DR-planner F ron tier in to Figure 2{29 Dening the Subsystem Simplier. W e capture the transformations p erformed b y the F ron tier DR-planner b y describing simplier maps. Let S i b e the solv able subgraph of G i found at stage i S I b e the set of inner v ertices of S i F I b e the set of fron tier v ertices of S i and A b e an y subgraph of G i Then the simplier map T i ( A ) is dened as follo ws { T i ( S i ) = c i where the w eigh t of c i is equal to the geometry-dep enden t constan t D {If A \ S i = ; then T i ( A ) = A {If A \ S i 6 = ; and A \ S I = ; then the image of A under the map T i is A min us those edges that A shares with S i

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80 {If A \ S I 6 = ; then T i ( A ) = ( V T A ; E T A ), where V T A is the set of all v ertices of A that are not v ertices of S I plus the v ertex c i represen ting the simplication of S i The set of edges E T A is formed as describ ed earlier in this section. P erformance Analysis. In this section, w e analyze the DR-planner F ron tier with resp ect to the v arious p erformance measures denied in previous sections. Claim 20 DR-planner F r ontier is a valid DR-planner with the Chur ch-R osser pr op erty. In addition, DR-planner F r ontier nds DR-plans for the maximal solvable sub gr aphs of under c onstr aine d gr aphs. Pro of The pro of is iden tical to the one used for DR-planner Condense. Before w e discuss the solv abilit y preserving prop ert y of DR-planner F ron tier, w e w ould lik e to consider the follo wing example sho wn in Figure 2{30 All edges ha v e w eigh t 1, v ertices as indicated. Initially the graph B C D E I J K is solv able. The graph AB C D E F GH is also solv able, v ertices A and B are its inner v ertices, v ertices C ; D ; E ; F ; G and H are its fron tier v ertices. After AB C D E F GH has b een simplied in to the graph M C D E F GH the new graph M C D E I J K is no longer dense (edges M C ; M D ; M E ; M H and M F ha v e w eigh t 2 no w). Ho w ev er, note that according to the denition of the DR-planner F ron tier simplier map, the image of B C D E I J K is not the graph M C D E I J K but the graph M C D E F GH I J K whic h is dense. This graph M C D E F GH I J K is w ello v erconstrained, since it has densit y -1 and it could b e made w ellconstrained b y sa y remo ving constrain ts F G and F H Th us the image of AB C D E F GH is also solv able. In general, the follo wing claim holds. Claim 21 L et A and B b e solvable sub gr aphs such that A \ B 6 = ; and A \ B 6 = v wher e v is a single vertex of weight less than ge ometry dep endent c onstant D then A [ B is solvable.

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81 2 2 E F C D 3 B 3 I 2 J 2 3 3 4 2 2 A G H K L 7 2 2 E F C D 3 3 I 2 J 3 3 2 2 G H K L 7 6 M Figure 2{30: Original graph BCDEIJK is dense, new graph MCDEIJK is not Pro of Since A is solv able, the densit y of A \ B d ( A \ B ) D (a solv able graph cannot con tain an o v erconstrained subgraph, unless it is a w ello v erconstrained graph in whic h case it can b e replaced b y an equiv alen t w ellconstrained graph). Hence the densit y of A [ B d ( A [ B ) = d ( B ) + d ( A n B ) = D + d ( A ) d ( A \ B ) D Equalit y o ccurs when d ( A \ B ) = D otherwise A [ B is o v erconstrained. If A [ B is o v erconstrained it is w ello v erconstrained, since it could b e con v erted in to w ellconstrained b y reducing w eigh ts of some edges of B n A or A n B This prop ert y can b e used to sho w that Claim 22 DR-planner F r ontier is solvability pr eserving as wel l as strictly solvability pr eserving. Pro of Let B b e a solv able graph, and supp ose that the solv able graph S i w as simplied b y DR-planner F ron tier. Then B w ould only b e aected b y this simplication if B con tains at least one in ternal v ertex of S i (recall that fron tier v ertices of S i remain unc hanged). But then, b y the denition of the DR-planner F ron tier simplier, T i ( B ) = T i ( B [ S i ). Since T i ( B [ S i ) is obtained b y replacing S i b y solv able T i ( S i ), and according to the previous claim, the union of t w o solv able graphs is solv able, th us T i ( B ) = T i ( B [ S i ) is also solv able. Claim 23 DR-planner F r ontier is c omplete

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82 Pro of This is b ecause DR-planner F ron tier (just as Condense) nds minimal dense subgraphs at eac h stage. Claim 24 DR-planner F r ontier has worst-choic e appr oximation factor O (1 =n ) Pro of Consider Figure 2{31 the solv able constrain t graph G Initially DR-planner F ron tier will lo cate the minimal dense subgraph AB C (since this is the only minimal dense subgraph of G ). It will not b e able to lo cate an y dense subgraphs disjoin t from AB C or including only fron tier v ertices of AB C If it attempts to lo cate a dense subgraph that includes the en tire (simplied) cluster AB C and do es so b y insp ecting the other v ertices in the sequence A; B ; C ; H ; I : : : ; F ; E (i.e going coun terclo c kwise), then DR-planner F ron tier w ould not encoun ter an y dense subgraphs after AB C un til the last v ertex of the sequence E is reac hed. The minimal dense subgraph found at this stage is the entire graph G Th us the size of the DR-plan corresp onding to this c hoice of v ertices is prop ortional to n On other hand, there is a DR-plan of constan t size. This DR-plan w ould rst lo cate the minimal dense subgraph S 1 = AB C and simplify it. After that it w ould simplify S 2 = AB C E = S 1 [ f E g after that S 3 = S 2 [ f F g etc going clo c kwise un til the v ertex H is reac hed. A t ev ery stage i the size of S i is constan t, hence the size of this DR-plan is constan t. I H 1 F E C B A 2 2 2 22 2 2 5 5 5 4 5 5 5 5 5 3 3 3 3 3 3 3 5 Figure 2{31: 1 =n w orst-c hoice appro ximation factor of DR-planner F ron tier Claim 25 The b est-choic e appr oximation factor of DR-planner F r ontier is at le ast 1 2

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83 Pro of Let G b e the w eigh ted constrain t graph. Let P b e an optimal DR-plan of G let p b e the size of P (i.e the size of ev ery cluster S i simplied under the optimal DR-plan is less than p + 1). W e will sho w that there is a DR-plan P 0 that is \close to" P Complete resem blance ( P = P 0 ) ma y not b e p ossible, since the in ternal v ertices of the cluster S 0 i found b y DR-planner F ron tier at the i th stage, are simplied in to one core v ertex, thereb y losing some information ab out the structure of the graph. Ho w ev er w e will sho w that there is a w a y of k eeping the size of P 0 within the constan t D of the size of P Supp ose that DR-planner F ron tier is able to follo w the optimal DR-plan up to the stage i i.e S i = S 0 i Supp ose that there is a cluster S j in the DR-plan P suc h that i < j and S j con tains some in ternal v ertices of S i Therefore the simplication of S j b y DR-planner F ron tier ma y b e dieren t from simplication of S j b y P Ho w ev er, since the union of S i and S j is solv able, DR-planner F ron tier could use S 0 j = T 0 i ( S i ) [ S j instead of S j The size of S 0 j diers from the size of S j b y at most D units, where D is the constan t dep ending on the geometry of the problem. Hence the size of P 0 is at most p + D and since p is at least D, the result follo ws. Next, w e consider the last p erformance measure abilit y to incorp orate design decomp osition. Observ ation 4 DR-planner F r ontier c an inc orp or ate a design de c omp osition of the input gr aph if and only if al l p airs of sub gr aphs A and B in the given design de c omp osition satisfy: the vertic es in A \ B ar e not among the internal vertic es of either A or B Note. This condition on the design decomp osition puts no restriction on the sizes of the in tersections of the subgraphs in the decomp osition, and is far less restrictiv e than the corresp onding conditions for SR and Condense. }

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84 Pro of The pro of is similar to the case of the DR-planner SR. F or the if part, w e nd a top ological ordering O of the giv en design decomp osition P whic h is a set of solv able subgraphs of the input graph G partially ordered under the subgraph relation suc h that O is em b edded as a subplan of the nal DR-plan generated b y DR-planner F ron tier; i.e, O forms a subsequence of the sequence of solv able subgraphs S i whose (sequen tial) simplication giv es the DR-plan. W e tak e an y top ological ordering of the giv en design decomp osition P and create a DR-plan for the rst solv able subgraph A in P I.e, while constructing the individual DR-plan for A w e \ignore" the rest of the graph. This individual DR-plan induces the rst part of the DR-plan for the whole graph G In particular, the last graph in this partial DR-plan is obtained b y simplifying A using the simplier describ ed in Section 2.1.14 (and treating eac h A exactly as DR-planner F ron tier w ould treat a cluster S j found at some stage j ). Let G i b e the last graph in the DR-plan for G created th us far. Next, w e consider the next subgraph in the ordering O and nd an individual DR-plan for it, treating it not as a subgraph of the original graph G but as a subgraph of the simplied graph G i This individual DR-plan is added on as the next part of the DR-plan of the whole graph G The crucial p oin t is that the simplication of an y subgraph, sa y A will not aect an y of the unrelated subgraphs B in P This is b ecause b y the requiremen t on the decomp osition P A and B share at most fron tier v ertices. As a result, b y the functioning of the DR-planner F ron tier, when the core v ertex for A is created, none of the solv able subgraphs inside B are aected. The pro cess of constructing individual DR-plans for subgraphs in the decomp osition P and concatenating them to the curren t partial DR-plan is con tin ued un til a partial DR-plan for the input graph G has b een pro duced, whic h completely includes some top ological ordering of the decomp osition P as a subplan.

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85 Again, let G k b e the last graph in this partial DR-plan. The rest of the DR-plan of G is found b y running the original DR-planner F ron tier on G k F or the only if part, w e consider a DR-plan Q pro duced b y DR-planner F ron tier. W e rst observ e that the sequence of (original) solv able subgraphs whose sequen tial simplication giv es Q can nev er con tain t w o subgraphs A and B suc h that A \ B con tains b oth in ternal v ertices of A and in ternal v ertices of B This is b ecause if, for instance, A is simplied b efore B then B (on its o wn) cannot b e simplied at a later stage (although A [ B could), since an in ternal v ertex of B that is also an in ternal v ertex of A will disapp ear from the graph (will b e replaced b y a core v ertex represen ting ev erything in ternal to A ), the momen t A has b een simplied. Next, w e consider the remaining case where A \ B con tains some in ternal v ertices of A but only fron tier v ertices of B In this case, p oten tially B could b e simplied b efore A and A will not b e aected, since the fron tier v ertices of B are unc hanged b y the simplication. Ho w ev er, since a giv en design decomp osition P could con tain an arbitrary n um b er of o v erlapping subgraphs, w e can c ho ose decomp ositions P suc h that all top ological orderings of P are infeasible i.e, no DRplan can incorp orate them as a subsequence. F or instance, if there are 3 subgraphs A B and C in P suc h that A \ B con tains only fron tier v ertices of B but some in ternal v ertices of A ; B \ C con tains only fron tier v ertices of C but some in ternal v ertices of B ; and C \ A con tains only fron tier v ertices of A but some in ternal v ertices of C This forces the DR-plan to simplify B b efore A C b efore B and A b efore C whic h is imp ossible. 2.2 Relating Problems of Chapter 1 to some Measures of Chapter 2 DR-planning problem of geometric constrain t solving comm unit y directly corresp onds to our RD-dag problem. Underlying w eigh ted graph G is the same for b oth problems, n um b er of geometric degrees of freedom D = K densit y function

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86 d () is the same etc. Stably dense graphs corresp ond to solv able graphs/systems. Solving optimal RD-dag problem is equiv alen t to nding optimal DR solution sequence.Maximal stably dense subgraphs corresp ond to the ro ots of tree in DR-plans. Clearly size of the optimal DR solution sequence is greater or equal to the size of the minim um (stably) dense subgraph of G and to the size of the minim um (stably) dense subgraph of T 1 ( G ) and and to the size of the minim um (stably) dense subgraph of T 2 ( G ) etc. Hence an ecien t solution of minim um dense subgraph problem w ould b e helpful for nding optimal DR solution sequence. Since it desirable that ev ery S i w ere cluster minimal (see Section 1.1 ) a minimal dense subgraph problem is also imp ortan t for obtaining optimal DR solution sequence.

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CHAPTER 3 MAXIMAL, MAXIMUM AND MINIMAL ST ABL Y DENSE PR OBLEMS 3.1 Finding Maxim um Dense Subgraph Un b ounded case. Claim 26 Pr oblem of nding maximum dense sub gr aphs when weights ar e al lowe d to b e unb ounde d is NP-Complete Pro of By straigh tforw ard reduction from CLIQUE. Bounded case. Supp ose that the w eigh t of all v ertices is 3, of all edges 1 (it imp ortan t that w eigh ts b e b ounded, b ecause un b ounded maxim um dense subgraph problem is NP-complete). Constan t K = 0. Consider a follo wing LP: max x i s.t y ij 3 x i 0 x i y ij ; 8 i; j 0 x i ; y ij 1 ; 8 i; j 87

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88 Lemma 4 L et S ( G ) b e a solution of the LP ab ove, for a given gr aph G Then vertic es v i s.t x i = 1 form the lar gest dense sub gr aph A of G (dimension dep endent c onstant K = 0 ). (If ther e ar e mor e than 1 dense sub gr aphs of size j A j then the one that has highest density is forme d). Pro of Let A b e the (densest) maxim um dense subgraph. Let B b e a subgraph constructed as describ ed ab o v e. Supp ose that A 6 = B Consider an x p that has the smallest p ositiv e v alue in B If x p = 1 then B = A con tradiction. Supp ose that 0 < x p < 1. Consider edges y pj suc h that 0 < y pj = x p < 1. If there are 3 or more suc h edges than b oth x p and y pj could b e increased b y some p ositiv e resulting in a b etter LP solution than B con tradiction. Therefore there are at most 2 suc h edges. Remo v e x p and its adjacen t edges from B and set some v ertex v l 2 A n B equal to x p This creates another optimal LP solution B 0 Consider an x 0p that has the smallest p ositiv e v alue in B 0 and rep eat. Ev en tually the v alue of x 0p will reac h 1 and b y construction j A n B j = j B n A j and densit y of A is equal to the densit y of B Note that since maxim um dense subgraph is also maximal, tec hnique ab o v e lo cates maximal dense subgraph as w ell. 3.2 Finding a Stably Dense Subgraph 3.2.1 Distributing an Edge In Homann et al. (1997 ) w e ha v e describ ed an algorithm Distribute( e i ; A; K ) that \distributes" w eigh t of an edge e i to its endp oin ts ( v i R ; v i L ) (or in to graph A ), assuming that all other edges in A has already b een \distributed". F ormally edges of A f e j g = A are distributed if there is a mapping (distribution) of edge-w eigh ts w ( e j ) in to t w o parts f j R ; f j L (corresp onding to the endp oin ts of the edge e j ) suc h that (**) 8 e j 2 A b oth f j R and f j L are non-negativ e in tegers and

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89 f j R + f j L = w ( e j ) X j j v j R;L == t f j R;L w ( v t ) ; 1 t n i.e for an y v ertex v t 2 A distribution from all of it adjacen t edges do es not exceed w eigh t of v t Distributing new edge e i in to a graph A edges f e j g has already b een distributed requires nding a distribution of edges f e j g [ e i suc h that all conditions (**) hold for A [ e i except that the second condition for edge e i is more strict (***) f i R + f i L = w ( e i ) K + 1 W e will giv e examples of distribution for K = 0 and b elo w, Homann et al. (1997 ) con tains relev an t implemen tation details that in v olv e nding net w ork ro ws in bipartite graphs. In the examples b elo w all edges ha v e w eigh t 1, all v ertices ha v e w eigh t 2. In Figure 3{1 curren t graph A = f AE ; AB ; AC ; B E ; B C ; E C ; C D g these edges has already b een distributed, distribution is indicated b y the arro ws (so f A AE = 1 ; f E AE = 0 ; f C B C = 1 ; f B B C = 0 ; : : : ). Curren tly w e w an t to distribute edge ED, but for ED w e should use condition (***) i.e w e ha v e to distribute 1 0 + 1 = 2 units. W e can distribute one unit to w ard D without violating condition 3 of (**) but in order to distribute second unit w e need to c hange curren t distribution (since curren tly there is no ro om in either E or D). One w a y to rearrange the distribution

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90 B A C D E Figure 3{1: Before the distribution of edge ED is b y rev ersing ro w in BE from E to B, this will free up one slot in E, so w e can send remaining unit of ED in to E. Resulting distribution is sho wn in Figure 3{2 B A C D E Figure 3{2: After the distribution of edge ED Note that b efore w e will distribute next edge, sa y AD, distribution of ED should comply with second condition of (**) and not (***) so instead of distributing 2 units on ED w e distribute only one, hence w e should remo v e either 1 unit going to w ard E or 1 unit going to w ard D. Resulting distribution is sho wn in Figure 3{3

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91 B A C D E Figure 3{3: Before the distribution of edge AD Wh y do w e add and than remo v e 1 extra unit, and what is the en tire purp ose of algorithm Distribute( e i ; A; K ) ? Consider Figure 3{4 where edge BD is ab out to b e distributed. B A C D E Figure 3{4: Before the distribution of edge BD W e can send 1 unit of BD to D (rerouting ED to E) but there is no ro om an ywhere for 1 extra unit. Note also that the graph A that w ould result from adding edge BD is dense for K = 0 (it has 5 v ertices of w eigh t 2 and 10 edges of w eigh t 1). In general inabilit y to distribute an extra unit indicates a presence of a dense subgraph. Not also that non of the prop er subgraphs of A are dense for K = 0, and that w e w ere able to distribute all the edges up un til no w.

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92 In general it is true if for some sequence f e i g ; i p no p ossible distribution of edge e p exists (and all previous edges e i ; i < p w ere successfully distributed) this means that there is a dense subgraph B induced b y a subset of edges f e i g and e p 2 B Con v ersely if there is a dense subgraph B and all edges of B form a subsequence of f e i g and the last edge of B in f e i g is e p then this edge e p cannot b e distributed. Prop erties of Distribute(). Prop ert y 1 If D istr ibute ( e; A; K ) 6 = ; then d ( D istr ibute ( e; A; K )) K Prop ert y 2 Supp ose that e dge e 2 B and sub gr aph B A is dense. Then D istr ibute ( e; A; K ) 6 = ; Prop ert y 3 Supp ose that e dge e 2 B and sub gr aph B A is unique dense sub gr aph of A c ontaining e Then D istr ibute ( e; A; K ) = B Prop ert y 4 If D istr ibute ( e; A; K ) 6 = ; then at le ast one endp oint of e 2 D istr ibute ( e; A; K ) Prop ert y 5 L et e b e an e dge of a gr aph A Supp ose that gr aph A n e do es not c ontain any over c onstr aine d sub gr aphs. L et B = D istr ibute ( e; A; K ) and B 6 = ; then B is stably dense. Pro of By Prop ert y 1 d ( B ) K Supp ose that B is not stably dense, i.e d ( B ) K and there is C B suc h that d ( C ) > K and when C is replaced in B b y subgraph C 0 ; d ( C 0 ) = K then resulting graph B 0 is not dense, d ( B 0 ) < K Therefore d ( B n C ) < 0. Also e 2 C (otherwise C A n e con tradicting condition that A n e do es not con tain an y o v erconstrained subgraphs). Since d ( B n C ) < 0, there is a v ertex v 2 B n C suc h that sum of ro ws to v is less than w eigh t of v (for example in Figure 3{6 v ertex v = G and edge e = B D ). Ho w ev er this v ertex v w ould not b e lab eled b y D istr ibute ( e; A; K ), b ecause if v w ere lab eled then there w ould b e a sequence of ro ws from v to e and this sequence could b e rev ersed therefore successfully distributing edge e This con tradicts the

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93 assumption that e could not b e distributed. Hence v 62 D istr ibute ( e; A; K ) and therefore D istr ibute ( e; A; K ) B con tradicting denition of B Hence our original assumption that B is not stably dense w as wrong. Note that the condition that A do es not con tain an y o v erconstrained subgraphs is necessary consider for example Figure 3{5 depicting p oin ts and distances in 2D. Let e = H B ; A = B C D E F then H B C D E F = D istr ibute ( e; A; K ) but H B C D E F is not stably dense, only dense. H E C D F B Figure 3{5: Lo cating dense graph instead of stably dense 3.2.2 Finding Dense and Stably Dense Subgraph Algorithm Distribute(e,A) also pro vides a mec hanism for determining a dense subgraph B that con tains undistributable edge e 2 B W e will illustrate this algorithm on an example Figure 3{6 B A C D E F G Figure 3{6: Before the distribution of edge BD

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94 Supp ose that all edges except BD has already b een distributed, arro ws indicating actual distributions. While attempting to distribute 2 units from edge BD w e will rst lab el endp oin ts of BD. I.e lab eled set L = f B ; D g W e will place one unit of BD in to D. Then w e will try to nd a place for the one remaining unit of ED. Using standard augmen ting path tec hniques w e will lab el and examine all suc h v ertices v suc h that there is a ro w distributed from v to at least one curren tly lab eled v ertex. V ertices A, E and C t the description of v Since none of them has ro om for one unit of ED w e up date L = f A; B ; C ; D ; E g Since there are no more v ertices v suc h that there is ro w from v to some v ertex in L algorithm outputs a found dense subgraph L and terminates. A follo wing is pseudo co de description of the algorithm just demonstrated. Algorithm Dense(G) 1. A = ; ; B = ; 2. for ev ery v ertex v do 3. for ev ery edge e inciden t to v and to C do 4. B=Distribute(e,A,K) 5. if B 6 = ; 6. goto Step 11 7. endif 8. endfor 9. add v ertex v to A 10. endfor 11. return B Prop erties of Algorithm Dense(). Prop ert y 6 A lgorithm Dense() nds a stable dense sub gr aph. Pro of F ollo ws from Prop ert y 5 Prop ert y 7 L et G b e a dense gr aph. Then D ense ( G ) 6 = ;

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95 3.2.3 PushOutside() Note that Prop ert y 5 only guaran tees that some stably dense subgraph con taining curren tly distributed edge is found. In general w e are in terested in nding maximal stably dense subgraphs, whic h ma y not b e found b y Distribute. Consider for example Figure 3{7 (constan t K = 0). Supp ose that algorithm D ense () distributed v ertices A,B,C and D in that order. Then D ense () will return BCD instead of ABCD, b ecause ro ws on edges A C and AB will b e directed to w ard v ertex A, hence A will nev er b e lab eled. 1 1 2 2 2 2 2 2 2 A B D C Figure 3{7: Dense graph BCD found instead of maximal dense ABCD Assume that graph A n e do es not con tain an y o v erconstrained subgraphs and supp ose that algorithm D istr ibute ( e; A; K ) has returned stably dense subgraph B W e w ould lik e to lo cate stably dense C A; e 2 C suc h that there is no stably dense D A suc h that C D In other w ords C is stably dense dense subgraph of A that con tains edge e T o construct C w e will start with B = C and w e will c hec k whether C can b e enlarged b y using tec hnique of Pushoutside(B,A) Algorithm Pushoutside(B,e,A) 1. C = B 2. for ev ery v ertex v 2 C do lab el(v)=1 end for 3. for ev ery edge e adjacen t to C do C = C [ D istr ibute ( e; A; K ) end for 4. return C

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96 Prop ert y 8 Assume that gr aph A n e do es not c ontain any over c onstr aine d sub gr aphs. L et B b e a stably dense sub gr aph r eturne d by D istr ibute ( e; A; K ) In 2D c ase if j B j 2 then C = P ushoutside ( B ; e; A ) is a maximal stably dense sub gr aph of G that c ontains e Similar r esult holds in 3D if j B j 3 Pro of F or all maximal stably dense C G suc h that B C there is an edge e 2 C n B Execution of D istr ibute ( e; A; K ) will lo cate suc h C that will b e enlarged accordingly F or a 3D case when found stably dense subgraph B is an edge there migh t b e sev eral maximal stably dense subgraphs of A con taining B as in Figure 3{8 W e can use follo wing algorithm in order to nd all of them. Algorithm Pushoutsidedge(B,e,A) 1. i = 1 2. C i = B ; C 0 = ; 3. while C i 1 6 = B 4. for ev ery v ertex v 2 C do lab el(v)=1 end for 5. for ev ery edge e adjacen t to C i and not in C j ; j < i 6. do C i = C i [ D istr ibute ( e; A; K ) 7. end for 6. return C i 7. end while Prop erties of Pushoutsideedge(). Prop ert y 9 Assume that gr aph A n e do es not c ontain any over c onstr aine d sub gr aphs. L et B b e a stably dense sub gr aph r eturne d by D istr ibute ( e; A; K ) in 3D c ase. If B is an e dge = e then P ushoutsideedg e ( B ; e; A ) r eturns al l maximal stably dense sub gr aphs of A that c ontain e Finding Minimal Dense Subgraphs. Note that Prop ert y 2 do es not guaran tee that subgraph found is minimal dense. Consider for example Figure 3{9 (constan t K = 0). If algorithm D ense () distributed edges in order A C,AB and

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97 A B C D E F G H I J Figure 3{8: Graphs ABCDEF and F GHIJA are maximal stably dense in 3D BC then nonminimal dense graph ABC w ould b e found instead of minimal dense subgraph BC. 1 2 2 1 1 2 A C B Figure 3{9: Dense graph ABC found instead of minimal dense BC W e will x this problem using tec hnique b elo w. Once a dense subgraph B has b een found the follo wing algorithm Minimal(B) can b e used to lo cate minimal dense subgraph C of B Algorithm Minimal(B) 1. A = B 2. for ev ery v ertex v 2 A do lab el(v)=0 3. for ev ery unlab eled v ertex v 2 A do

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98 4. C = A n v 5. D = Dense(C) 6. if D 6 = ; then A = D 7. l abel ( v ) = 1 8. endfor 9. return A Ev en tually out graphs will ha v e clusters as comp onen ts so an appropriate mo dication is needed. b egin ndMinimal(S,C,F) A = S for all clusters S j of S A = A n S j X = distr ibuteC l uster ( C ; A ) if X 6 = ; then A = X else A = A [ S j end if end for return A end ndMinimal v D A B Figure 3{10: Actions of Minimal() algorithm

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99 Prop erties of Minimal(). Prop ert y 10 L et B = D ense ( A ) and C = M inimal ( B ) then C is minimal stably dense. More ecien t Algorithm for Minimal(). It is p ossible to use minim um cut tec hniques of Karger (1996 ) in order to nd a minimal subgraph. Input: a dense graph G last v ertex found v distributed ro ws f ( e; u ) and undistributed ro ws g ( e ). Output: a w eigh ted graph G 0 and a v ertex v 0 2 G 0 The w eigh t of the edge e 0 2 G is equal to the ro w in corresp onding edge e 2 G to w ard v + g ( e ). Goal: nd a cut in G 0 separating v 0 from at least one v ertex, suc h that the w eigh t of the cut is at most K and if L 0 is the part of G 0 con taining v 0 than there is no M 0 L 0 ; v 0 2 M 0 suc h that the cut separating M 0 from G has w eigh t less than K Claim 27 The gr aph L G c orr esp onding to the L 0 is minimal dense. Pro of Let A b e the the total amoun t of undistributed ro ws on the edges from v 0 to L 0 B b e the total amoun t of undistributed ro ws on the edges from v 0 to G 0 =L 0 and C b e the w eigh t of the cut edges suc h that none of the endp oin ts is v 0 Then A + B = K (b y denition of K ) and B + C < K (since w eigh t of the cut w as less than K ). Th us A > C so the d ( L 0 ) > 0. Th us L 0 is dense and minimal b ecause of a minimalit y condition on the cut. Algorithm for nding a minimal cut of w eigh t less than K. 1. Run the con traction algorithm = 1 + ; = 1 = log n 2. T ak e the smallest one con taining v 0 of An y other cut on the side of v 0 will ha v e size at least (1 + 1 = log n ) m where m is the size of the minim um cut. Con tract all the v ertices on the side not con taining v 0 in to one v ertex, run con traction algorithm again un til the size of the second b est cut is > K The n um b er of iterations is O ((log n )(log K )), and the cuts are found with high probabilit y

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100 3.3 Structure and Prop erties of F ron tier algorithm Note 1 We have describ e d DR-planner F r ontier in Chapter 1, but we did not sp e cify an actual algorithm. Now we wil l describ e an actual implementation. A lso DR-planner F r ontier was designing DR-plan, while F r ontier A lgorithm wil l c onstruct c omplete r e cursive de c omp osition, which is a str onger than just a DR-plan b e c ause it involves nding al l maximal stably dense sub gr aphs, not just some set as in DR-plans. In previous sections w e ha v e describ ed subroutines of F ron tier Algorithm for nding maximal stably dense subgraph that con tain a giv en edge, as w ell as transformation replacing this subgraph once it has b een found. No w w e will describ e the high lev el structure of F ron tier Algorithm and later will demonstrate its prop erties. 3.3.1 Informal Description of F ron tier Algorithm F ron tier Algorithm will pro ceed b y adding edges of graph G one b y one to the initially empt y subgraph A and distributing these edges as w as describ ed in Section 3.2.1 If w e w ere able to distribute all the edges of G then G do es not con tain an y dense subgraph, and algorithm terminates. If w e ha v e encoun tered an edge that w e are unable to distribute then this implies that A con tains a dense subgraph B (see algorithm Dense of Section 3.2.1 ). Once w e ha v e lo cated suc h a dense subgraph B w e will replace graph B b y another graph B 0 In order for F rontier Algorithm to correctly lo cate all maximal dense subgraphs this replacemen t B 0 should satisfy t w o criteria. One criteria is that this replacemen t will allo w us to con tin ue the distribution pro cess. This criteria calls for remo v al of edges that curren tly cannot b e distributed, i.e the edges that cause o v erconstrainedness of B Another criteria is that this replacemen t B 0 should not destro y w ell-constrainedness of an y maximal stably dense subgraph, otherwise w e migh t nev er iden tify it later as a stably dense subgraph.

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101 3.3.2 Assumptions While F ron tier Algorithm can b e applied to arbitrary edge and v ertex w eigh ted graph, w e will for sak e of simplicit y restrict analysis of its prop erties to cases of p oin ts and distances in 2D or 3D. I.e dimension dep enden t constan t K = 3 or 6, w eigh ts of all edges are 1, and w eigh ts of all v ertices are either 2 or 3 resp ectiv ely 3.3.3 Joining P airs of Clusters The previous disclaimer allo ws to ab otain follo wing results that limit the n um b er of maximal stably dense graphs. Claim 28 F or any two distinct stably dense gr aphs A; B in 2D, if j A \ B j > 1 then A \ B is stably dense gr aph. Similarly in 3D, if j A \ B j > 2 then A \ B is stably dense gr aph. A lso in 3D if A \ B = ff v g ; f u gg then A [ B is stably dense. Pro of Let C = A \ B In 2D, if j C j > 1 then either d ( C ) K = 3 or C can b e replaced b y C 0 ; d ( C 0 ) = K without violating solv abilit y of A or B Therefore d ( A [ B ) = d ( A ) + d ( B ) d ( C ) K + K K K Similar pro of holds for 3D case, when K = 6. 3.3.4 Relev an t T ransformation Notation F ron tier Algorithm will b egin with a graph G 1 = G and will c hange it in to G 2 ; G 3 ; : : : ; G k Stably Dense graph or cluster found during i th stage of F ron tier Algorithm will b e denoted S i ; S i G i This cluster's transformation b y F ron tier Algorithm that w as describ ed ab o v e will b e denoted T i ( S i ). In general T i : A G i B G i +1 F or a graph A; A G i ; j A \ S i j 1 the mapping is T i ( A ) = A F or a graph A; A G i ; j A \ S i j 2 the mapping is T i ( A ) = ( A n S i ) [ T ( S i ). F or A G w e will denote T i ( A ) = T i 1 ( T i 2 ( : : : ( T 1 ( A )))). Let X G i an underlying sub gr aph T 1 ( X ) G is a subgraph in G induced b y v ertices that ended up in X at stage i 3.3.5 Recom bination b y F ron tier Algorithm Figure 3{11 illustrate transformation b y F ron tier Algorithm. Left half represen t a graph G b efore transformation. All v ertices ha v e w eigh t 2, all edges ha v e

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102 A B C E GH IJ F D L D F J I H G E Figure 3{11: T ransformation b y F ron tier Algorithm S S S 1 2 3 core core core frontier frontier frontier inner core 4 S Figure 3{12: General transformation b y F ron tier Algorithm w eigh t 1, geometry-dep enden t constan t K = 3. Supp ose that F ron tier Algorithm lo cated a stably dense graph S = AB C D E F It will recom bine this graph in to new graph S 0 as follo ws. All fr ontier v ertices of S that are connected to v ertices outside of S remain unc hanged. I.e v ertices D,E and F remain unc hanged. All edges b et w een fron tier v ertices remain unc hanged. All non-fron tier or inner v ertices of S are com bined in to one new c or e v ertex (i.e v ertex L ). The w eigh t of edges connecting core v ertex L to a fron tier v ertex, sa y v ertex D, is equal to the sum of all edges b et w een D and inner v ertices, i.e 1 in our example. The w eigh t of core v ertex is c hosen so that o v erall densit y of S 0 = K = 3, i.e w eigh t of L is equal to 3. More generally in Figure 3{12 dark no des represen t fron tier v ertices, unlled no des represen t inner v ertices and dashed v ertices represen t core v ertices of previously found stably dense subgraphs S 1 ; : : : ; S 3 F ron tier Algorithm will

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103 transform a stably dense union of S 1 ; : : : ; S 3 in to one new subgraph on the righ t, using rules ab o v e. This replacemen t satises rst criteria ab o v e, since B 0 has densit y exactly K and second criteria as demonstrated in Prop ert y of F ron tier Algorithm 15 b elo w. 3.3.6 Remo ving Undistributed Edges Sometimes while executing line 6 of F ron tier Algorithm w e ma y nd a new stably dense graph while there are still some edges of v remain undistributed. F or example, consider Figure 3{13 Supp ose that C is the last v ertex to b e distributed, and 3 edges remain to b e distributed CE, CF and CA. Once a CE is distributed a cluster CDEF is disco v ered. No w all undistributed edges from C to other v ertices of this cluster (i.e CF) will nev er b e distributed (they just mak e this cluster o v erconstrained, and under denition of stably dense graphs they can b e remo v ed. Their remo v al will not aect nding maximal stably dense subgraphs, as stated in Prop ert y 12 ). All other undistributed edges (i.e CA) will no w b e distributed, so as to enable us to disco v er cluster ABC for example. F E D C B A Figure 3{13: Edge CF will not b e distributed, edge A C will 3.3.7 Pseudo co de F ron tier Algorithm 1. F = ; 2. ClustersInF = ; 3. VList = ff 1 g ; f 2 g ; : : : ; f n gg 4. while VList 6 = ; 5. v = V List:r emov e () ; f l ag = 0

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104 6. for edges e adjacen t to v and F (see Section 3.3.6 ) 8. S = distr ibuteE dg e ( e; F ; K ) ( see Section 3.2 ) 9. S = check F or I nter sections ( S; F ) ( see Claim 28 ) 10. S = pushO utside ( S; F ) ( see Section 3.2.3 ) 11. if S 6 = ; 12. then f l ag = 1 ; S 0 = simpl if y ( S; F ) ( see Section 3.3.5 ,Section 3.3.6 ) 13. C l uster sinF :add ( S 0 ) 15. end for F = F [ v 16. end while end F ron tier Algorithm 1. b egin c hec kF orIn tersections(S,F) 2. for all clusters S i 2 V List () 3. A i = S \ S i 4. if j A i j 2 (in 2D, or 3 in 3D, or A i is a pair of v ertices in 3D) 5. then S = S [ S i 6. V List:r emov e ( S i ) 7. end if 8. end for 9. end c hec kF orIn tersections(S,F) 3.3.8 Example of actions b y F ron tier algorithm W e will demonstrate ho w F ron tier algorithm generates a complete recursiv e decomp ostion for an example constrain t graph G b y describing v arious stages of simplication of the Figure 3{14 Figures 3{15 to 3{17 resulting in a complete recursiv e decomp osition of Figure 3{18

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105 P O N M L K J G F E D C B A Figure 3{14: Initial graph B 2 core1 P O N M L K J G F E D C Figure 3{15: After A-C ha v e b een distributed The w eigh t of all v ertices in G = G 1 is equal to 2, the w eigh t of all the edges is equal to 1, the constan t K is equal to 3. V ertices are distributed in alphab etic order. Circles indicate found non trivial clusters. 3.3.9 Mo difying F ron tier Algorithm for Minimalit y Here w e will sho w ho w to generalize minimalit y-forcing tec hnique of Section 3.2.3 so it w ould guaran tee cluster minimalit y prop ert y Let S b e the stably dense subgraph found b y F ron tier Algorithm. Recall that a sequence ; = S 0 S 1 : : : S k = S has a cluster minimalit y if there is no prop er dense

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106 E F G JK LM NO P 2 core3 Figure 3{16: After A-G ha v e b een distributed E F G JK M P 2 2 core3 core4 Figure 3{17: After A-P ha v e b een distributed A B D F G J K L 1 2 3 4 S S S M N E C S P O Figure 3{18: Corresp onding complete recursiv e decomp osition

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107 subgraph X S j suc h that S j 1 X Consider for example Figure 3{20 The follo wing sequence of dense subgraphs S 1 = f A; B ; C ; D ; E ; F g ; S 2 = f A; G; H ; I ; J ; K g ; S 3 = f K ; L; M ; N ; O ; F g ; S 4 = S 1 [ S 2 [ S 3 [ f P g is not cluster minimal, since X = S 1 [ S 2 [ S 3 is dense and S 1 X S 4 In order to construct cluster-minimal sequences w e will mo dify F ron tier Algorithm, forcing ev ery cluster S j to con tain cluster S j 1 that w as found during the preceeding stage j 1. In other w ords S j is an \extension" of S j 1 and w e apply PushOutside() tec hnique to assure that this extension is minimal. FindSequence(SC,SI,M) Idea of an algorithm W e will b e using algorithm F indS eq uence ( S C ; S I ; X ). Graph S C is the curren t dense subgraph that w e w an t to decomp ose. Graph S I S C is assumed to ha v e b een decomp osed already and a curren t step S j in decomp osition sequence of S C is required to tak e S I as one of comp onen t clusters. Graph X S C n S I is a subset of v ertices an y p ossible S j has to con tain. Initially S C = S; S I = ; ; X = ; ( X can also b e tak en to b e the last v ertex v distributed when S w as found, since b y Prop ert y 4 v ertex v is con tained in ev ery dense subgraph of S ). Figure 3{19 sho ws recursiv e applications of F indS eq uence Pseudo co de 1. While M [ S I do esn't con tain all the v ertices in S C 2. do 3. Let w b e a v ertex not in M 4. While S n f w g con tains no dense subgraph that includes S I 5. (successful pushoutside S I and distribution of all other edges in S n S I ). 6. do 7. add w to M 8. end do

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108 9. Let S N b e the dense subgraph found (that includes M cupS I and excludes w ) 10. Let F S N = F r ontier ( S N ) = F ron tier simplication of S N 11. Let F S C = S C with S N replaced b y F S N 12. Return concatenate( F indseq uence ( S N ; S I ; M ) ; F indseq uence ( F S C ; F S N ; ; )); 13. end do 14. Return the list [SI, SC] M w k1 k1 S k S 2 w 1 3 2 1 S w SN=S S Figure 3{19: Enforcing cluster minimalit y Example of Actions b y Mo died F ron tier Algorithm. Consider Figure 3{20 where S = f A; : : : ; Q g is the stably dense subgraph and S I = M = ; Supp ose v ertices f A; : : : ; F g w ere distributed and minimal dense subgraph S 1 = f A; : : : ; F g w as disco v ered and simplied as usual b y F ron tier Algorithm (if graph is minimal dense then it decomp osition sequence con tains only one step and w e don t ha v e to w orry cluster minimalit y prop ert y). Next all remaining v ertices except K and P w ere distributed i.e. no clusters other than S 1 could b e found so far. Finally v ertex P w as distributed and after that v ertex K w as distributed. No w dense graph S C = f A; : : : ; Q g has b een found, S I = S 1 = f A; : : : ; F g and X = f K g Application of P ushO utside ( S I ; S C ) will return S 2 = f A; : : : ; O g Subsequen t steps will lo cate S 3 = f A; : : : ; P g and S 4 = f A; : : : ; Q g

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109 A Q P O K M N L J I H G F E D C B Figure 3{20: Example of nding minimal Prop ert y 11 F r ontier A lgorithm mo die d as describ e d ab ove has cluster minimality pr op erty. 3.3.10 Complete Decomp osition Prop ert y Theorem 1 Given a gr aph G F r ontier A lgorithm solves r e cursive de c omp osition pr oblem for G Pro of W e already kno w F rom Section 2.1.14 that DR-planner F ron tier constructs a DR-plan (=recursiv e decomp ostion). Using Prop erties 15 14 and 11 w e can see that it in fact constructs complete recursiv e decomp ostion since it nds all maximal stably dense subgraphs. Hence F ron tier algorithm solv es recursiv e decomp osition problem. 3.4 Maxim um Stably Dense Prop ert y Claim 29 Sinc e F r ontier A lgorithm lo c ates al l maximal stably dense sub gr aphs it c an b e use d to nd maximum stably dense sub gr aph as wel l. However the running time is only gur ante e d to b e p olynomial if weights of vertic es and e dges ar e b ounde d. 3.5 Other Prop erties of F ron tier Algorithm Prop ert y 12 F or every i and every maximal stably dense sub gr aph A G i the image T i ( A ) is also stably dense sub gr aph (of G i +1 )

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110 Pro of In 2D, if j S i \ A j 1 then T i ( A ) = A otherwise T i ( A ) = ( A n ( S i \ A )) [ T i ( S i ) whic h is stably dense b y Claim 28 In 3D w e migh t ha v e a sligh t complication when w e remo v e some of the undistributed edges (Section 3.3.6 ). Consider for example Figure 3{21 Supp ose that v ertex A is the last one to b e distributed. Supp ose that there only edges AE and AF remaining and AE is distributed b efore AF. Then cluster ABCDEF will b e disco v ered and edge AF will nev er b e distributed. It seems that graph AF GHIJ that used to b e stably dense no w b ecome unstably dense. Ho w ev er note that the image T i ( AF GH I J ) = T i ( AB C D E F GH I J ) that still remains stably dense, so maximal graph still remains stably dense. This also true in general case since d ( T i ( S i )) = K = 6 is equal to the densit y of a pair of v ertices 3 units eac h. A B C D E F G H I J Figure 3{21: Remo ving AF do es not aect solv abilit y of en tire graph Prop ert y 13 F or every stably dense sub gr aph A G i the gr aph T 1 ( A ) G i 1 is also stably dense. Pro of If j T 1 ( A ) \ S i 1 j 1 then T 1 ( A ) = A and result follo ws. Supp ose that j T 1 ( A ) \ S i 1 j > 1 and supp ose that B = T 1 ( A ) is not stably dense, i.e d ( B ) K and there is C B suc h that d ( C ) > K and when C is replaced in B b y subgraph C 0 ; d ( C 0 ) = K then resulting graph B 0 is not dense, d ( B 0 ) < K Consider C \ T 1 ( c i ), where c i is a core v ertex of cluster S i simplied at stage i If C \ T 1 ( c i ) = ; then T i ( C ) = C con tradicting the fact that A is stably dense. If

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111 C \ T 1 ( c i ) 6 = ; then w e can replace C \ T 1 ( c i ) b y c i and resulting graph ( A ) will b e stably dense, hence (using denition of stably dense graph) graph T 1 ( A ) is also stably dense, con tradiction. Prop ert y 14 F or every cluster S i chosen by F r ontier A lgorithm, sub gr aph T 1 0 ( S i ) G is stably dense. Pro of Prop ert y 5 guaran tees that S i is stably dense and then applying Prop ert y 13 i times insures that T 1 0 ( S i ) G is stably dense as w ell. Prop ert y 15 F or every maximal stably dense sub gr aph A G F r ontier A lgorithm wil l nd cluster S i such that A = T 1 ( S i ) Pro of Due to the Prop ert y 12 T i ( A ) is stably dense for ev ery i A t some stage j the last undistributed edge e 2 T i ( A ) will b e distributed. Because of the Prop ert y 8 and Prop ert y 9 cluster S j suc h that A = T 1 ( S j ) will b e disco v ered.

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CHAPTER 4 MINIMUM ST ABL Y DENSE SUBGRAPH PR OBLEM 4.1 Relation of Minim um Stably Dense to other Problems 4.1.1 Maxim um Num b er of Edges Problem A maximum numb er of e dges (optimization) pr oblem P 1( G; k ) in v olv es nding a subgraph A G suc h that n um b er of v ertices of A j V ( A ) j = k and n um b er of edges of A; j E ( A ) j is maximized. In other w ords the problem is to nd subgraph of a giv en size that has largest n um b er of edges (in some pap ers Kortsarz and P eleg (1993 ) this problem is also referred to as k-dense pr oblem Ho w ev er throughout this pap er w e are using dieren t meaning of term dense as dened b elo w). A corresp onding de cision version of maximum numb er of e dges pr oblem AD 1 is this: giv en ( G; k ; p ), is there A G; j A j = k ; j E ( A ) j j A j p ? Claim 30 AD 1 is NP-c omplete Pro of By reduction from k-CLIQUE. Let Y = ( G; k ) b e an instance of k-CLIQUE, tak e p = ( k 1) = 2 ; G 0 = G; k 0 = k then Z = ( G 0 ; k 0 ; p ) is in AD 1 i Y is in k-CLIQUE. A b ounde d de cision version of maximum numb er of e dges pr oblem AD B 1 is this: giv en ( G; k ; p ), where p const is there A G; j A j = k ; j E ( A ) j j A j p ? Claim 31 AD B 1 is NP-c omplete. Pro of By reduction from k-CLIQUE problem, using the same construction of new graph G 0 as in Theorem 2 of Section 4.2 nding the subgraph A 0 G 0 ; j A 0 j = (( k 1) = 4 1) k ( k 1) + k ( k 1) = 4 that has largest n um b er of edges among all subgraphs of this size and nally c hec king whether the corresp onding subgraph A in the original graph G is a clique of size k Note that p 3 for A 0 112

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113 4.1.2 Decision V ersion of Minim um Dense Subgraph A minimum dense sub gr aph optimization pr oblem P 2( G; X ) for a giv en constan t X and edge/v ertex-w eigh ted graph G in v olv es nding a smallest subgraph A G that is dense i.e X e 2 E ( A ) w ( e ) X v 2 V ( A ) w ( v ) X A corresp onding de cision version of minimum dense sub gr aph pr oblem AD 2 is this: giv en ( G; X ; p ) is there a dense A G; j A j p ? Claim 32 AD 2 is NP-c omplete. Pro of See Lemma 5 in Section 4.2 A b ounde d de cision version of minimum dense sub gr aph pr oblem AD B 2 is this: giv en ( G; X ; p ) where X const is there a dense A G; j A j p ? Claim 33 AD B 2 is NP-c omplete. Pro of See Theorem 2 in Section 4.2 4.1.3 Relationships b et w een t w o Decision Problems Claim 34 Pr oblem P 2 T P 1 Pro of Let ( G; X ) b e an instance of problem P 2. Wlog assume that all w eigh ts w ( e ) = 1 ; 8 e 2 E ; w ( v ) = c; 8 v 2 V W e will nd smallest dense subgraph b y examining subgraphs of size 1,2,3,... that con tain largest n um b er of edges un til dense one is found. I.e tak e k = 1. Let A k b e the graph found b y P 1( G; k ). If A k is dense then return A k as an output of P 2( G; X ). Otherwise set k = k + 1, rep eat. Claim 35 Pr oblem P 1 T P 2 Pro of Let ( G; k ) b e an instance of problem P 1. Construct another graph G 0 = G [ f N g (see Figure 4{1 ). Set w eigh ts of all v ertices w ( v ) = 0, w eigh ts of all edges adjacen t to N to k 2 w eigh ts of all other edges to 1. Let X = k 3 + k ( k 1) 2

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114 Note that if there is a subgraph B G suc h that j V ( B ) j = k ; j E ( B ) j = k ( k 1) 2 then corresp onding subgraph B [ f N g G 0 is dense. Also for an y smaller subgraph C G 0 suc h that j V ( C ) j k < k + 1 = j V ( B ) [ f N gj it holds that C is not dense, since j E ( C ) j ( k 1) k 2 + ( k 1)( k 2) 2 < k 3 + k ( k 1) 2 = X Therefore B [ f N g is minim um dense subgraph of G 0 In general if subgraph B G of size k has largest n um b er of edges T among all subgraphs of size k then when w ( N ) = w ( v i ) = 0 ; w ( v i ; N ) = k 2 ; w ( v i ; v j ) = 1 ; 8 v i ; v j 2 G and X = k 3 + T the corresp onding subgraph B [ f N g G 0 is the smallest dense subgraph of G 0 This suggest the follo wing algorithm for nding subgraph of G of size k that has largest n um b er of edges. W e will set X = k 3 + k ( k 1) 2 and c hec k whether the smallest dense subgraph B [ f N g found b y P 2 has size k + 1. If y es then solution of P 1 is B terminate. If not then set X = X 1, rep eat. N V V V V 1 2 4 5 V 3 V V V V 1 2 4 5 V 3 G G' Figure 4{1: Reduction Claim 36 Pr oblem AD B 2 m AD B 1 Pro of Since problem AD B 2 is NP-Complete, AD B 2 m k C LI QU E And it w as sho wn in Claim 31 that k C LI QU E m AD B 1. Hence AD B 2 m AD B 1. Claim 37 Pr oblem AD B 1 m AD B 2

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115 Pro of Similar to pro of of Claim 36 using comp osition of man y-one reduction from AD B 1 to k C LI QU E and from k C LI QU E to AD B 2 in Theorem 2 in Section 4.2 4.1.4 Relationships b et w een Appro ximation Algorithms Let A 1( G; k ; c ) b e a p olynomial time algorithm that returns a graph B ; j V ( B ) j = k suc h that j E ( B ) =E ( L ) j > c where L is a subgraph of G of size k that has largest n um b er of edges among all the subgraphs of this size. Let A 2( G; X ; c ) b e a p olynomial time algorithm that returns a dense subgraph B suc h that j B j = j L j < c where L is the minim um dense subgraph of G V arious appro ximation algorithms for problem AD 1 has b een prop osed Asahiro and Iw ama (1995 ), F eige and Seltzer (1997 ), Y e and Zhang (1997 ), Kortsarz and P eleg (1993 ), Sriv asta v and W olf (1999 ). Consider follo wing natural algorithm AP 1 for nding appro ximate solution of problem P 1 assuming the existence of algorithm A 2(). (Appro ximating A 1 b y A 2). Consider the follo wing algorithm AP 1. Let ( G; k ; c ) b e an input for A 1(). Construct a graph G 0 as in Claim 35 Let X = k 3 + k ( k 1) = 2. Let B G 0 b e the subgraph found b y A 2. If B = ; then set X = X 1, rep eat. Otherwise if j B j = k + 1 then return B If j B j 6 = k + 1 then remo v e ( j B j =k 1) j B j v ertices of B in greedy fashion using tec hnique of Asahiro and Iw ama (1995 ) (the one that has smallest degree should b e remo v ed rst, then second smallest etc) Conjecture 1 A p olynomial time algorithm AP 1 pr ovides a guar ante e d appr oximation factor solution of pr oblem P 1 On other hand it seems dicult to nd appro ximate solution of problem P 2 (assuming the existence of algorithm A 1()), as example b elo w demonstrate. (Appro ximating A 2 b y A 1). Consider the follo wing appro ximation algorithm AP 2 for nding minim um dense subgraph of G Wlog supp ose that w ( e ) = 1 ; w ( v ) =

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116 c; 8 e 2 E ; v 2 V W e start with k = 1, nd B = A 1( G; k ; c ) and c hec k whether B is dense. If y es then output B otherwise increase k rep eat. Claim 38 F or any d > 0 ; d < j G j = j L j ther e is an instanc e ( G; X ) such that AP 2 r eturns dense sub gr aph B such that j B j = j L j > d wher e L is the minimum dense sub gr aph. Pro of Let f O k g b e the set of minimal dense graphs of densit y exactly X size of eac h j O k j = k ; k n Let f S k g b e the set of underconstrained graphs of densit y exactly X 1, size of eac h j S k j = k ; k < n and S n = O n Let G k = O k [ S k and G = [ G k Supp ose that for ev ery k < n a call to A 1 b y AP 2 returns S k and not O k hence AP 2 will output O n instead of O 1 4.2 NP-Completeness of Minim um Stably Dense Subgraph Problem Notation: ( a; b; c; G ) problem denotes a minim um dense graph problem where all v ertices of graph G ha v e w eigh t a all edges ha v e w eigh t b and constan t K = c Lemma 5 The gener al ( a; b; c; G ) pr oblem is NP-Complete. Pro of This is sho wn b y reduction from CLIQUE OF SIZE P b y con v erting an instance of CLIQUE OF SIZE P ( G; P ) ; G = ( V ; E ) in to an instance of (( P 1) = 2 ; 1 ; 0 ; G = G 0 ) problem (w e assume that P 1 is ev en). Then there is a minim um dense graph A 0 of size P in G 0 if and only if there is a clique A of size P in G since densit y of an y graph A 0 of size x d ( A 0 ) x ( x 1) = 2 x ( P 1) = 2, hence all dense graphs ha v e size x P and clique of size P w ould b e the smallest dense graph Theorem 2 The (2 ; 1 ; 0 ; G ) pr oblem is NP-Complete. Pro of This is sho wn b y con v erting an instance of (( P 1) = 2 ; 1 ; 0 ; G ) problem in to an instance of (2 ; 1 ; 0 ; G 0 ) problem and using previous lemma. Description of the con v ersion. Ev ery v ertex v of G is replaced b y a subgraph H ( v ) = ( V 0 ; E 0 ) in G 0 This subgraph has ( deg ( v ) + 1)( P 1 4 1) + 1 v ertices, where deg ( v ) is degree of the

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117 v ertex v in graph G see Figure 4{3 (w e assume that 4 divides ( P 1)). All v ertices ha v e w eigh t 2, all edges ha v e w eigh t 1. All v ertices of H ( v ) are arranged either in a leftmost column of ( P 1) = 4 reed v ertices r i or a rectangle of deg ( v ) columns and ( P 1) = 4 1 ro ws of k ey v ertices k i;j Set of all reed v ertices is denoted b y R set of all k ey v ertices b y K F or all i; j s.t 0 i ( P 1) = 4 2 ; 1 j deg ( v ) there are edges b et w een k i;j and k i +1 ;j as w ell as b et w een k i;j and r i Also there is an edge b et w een k ( P 1) = 4 1 ;j and r ( P 1) = 4 2 as w ell as b et w een k ( P 1) = 4 1 ;j and r ( P 1) = 4 1 for all j; 1 j deg ( v ). There are deg ( v ) k ey v ertices k 0 ;j that are connected to the v ertices outside of H ( v ). These v ertices corresp ond to the edges in G that are inciden t to v see Figure 4{2 These v ertices are called p ortals their set is denoted b y S Original graph G A B C D AB AC AD DA CA CB BC BA New graph G' H(A) H(B) H(D) H(C) Figure 4{2: Gadgets for reduction to CLIQUE

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118 Lemma 6 Density of H ( v ) is e qual to ( P 1) = 2 Pro of Consider a follo wing mapping F : E 0 V 0 ; F ( k i;j ; r x ) = k i;j ; F ( k i;j ; k i +1 ;j ) = k i;j (essen tialy ev ery edge is mapp ed to its upp er endp oin t). Then for all k i;j there exist exactly 2 edges e 1 and e 2 suc h that F ( e 1 ) = F ( e 2 ) = k i;j hence densit y of H ( v ) ; d ( H ( v )) = j E 0 j 2 j V 0 j = 2 j K j 2 j K j 2 j R j = 2 j R j = ( P 1) = 2. (Similarly densit y of a union of an y n um b er of ro ws of k eys and of all reeds is equal to ( P 1) = 2). Lemma 7 Density of any A H ( v ) ; d ( A ) < 0 Pro of Consider a function L A : V A f 0 ; 1 ; 2 g for w 2 A; L A ( w ) = 2 if there exist 2 edges e 1 and e 2 in A suc h that F ( e 1 ) = F ( e 2 ) = w L A ( w ) = 1 if there exists 1 suc h edge e 1 in A suc h that F ( e 1 ) = w L A ( w ) = 0 otherwise. Let L tA = f w j L A ( w ) = t g Then densit y of A; d ( A ) = 2 j L 0A j j L 1A j Consider B = A \ R L 0A If B 6 = ; then L 0A 6 = ; therefore d ( A ) < 0. If B = ; then consider a k i;j 2 A; i i ; j j ; 8 k i ;j 2 A (lo w er-left k ey in A ). This k i;j 2 L 0A therefore d ( A ) < 0. Lemma 8 L et B 0 G 0 b e a minimal dense sub gr aph. Then A = B 0 \ H ( v ) is either empty or has density ( P 1) = 2 for al l vertic es v 2 G (in other wor ds B has the same density as original vertex v ). Pro of Let S A = A \ S (set of all p ortal v ertices of A ). If A 6 = ; then b y Lemma 7 S A 6 = ; Since B 0 is minimal dense d ( A ) > j S A j otherwise d ( B 0 n A ) 0 but B 0 is minimal. If k 0 ;j 2 S A then k i;j 2 A; 8 i (i.e en tire column of k eys is in A ) otherwise d ( B 0 n ( S i k i;j )) 0 but B 0 is minimal. F or the same reason all suc h k i;j 2 L 2A hence all reeds r i 2 A Therefore d ( A ) = j E A j 2 j V A j = j E A j 2 j L 2A = A \ K j 2 j R j = 2 j R j = ( P 1) = 2. Lemma 9 L et B G b e a clique of size P Then in c orr esp onding H ( B ) = B 0 G 0 ther e exists a dense sub gr aph C 0 B 0 such that j C 0 j = (( P 1) = 4 1) P ( P 1) + P ( P 1) = 4

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119 Pro of Graph C 0 con tains all reed v ertices of H ( B ) as w ell as all columns of k ey v ertices k i;j of H ( v 1 ) i k 0 ;j 2 H ( v 1 ) is connected to k 0 ;j 0 2 H ( v 2 ) and b oth v 1 ; v 2 2 B I.e columns of k ey v ertices k i;j of H ( v ) suc h that p ortal k 0 ;j is connected to H ( w ) ; w 62 B are not included in C 0 This graph C 0 = C 1 [ : : : [ C P where C i = C \ H ( v i ) for P v ertices v i 2 B Densit y of eac h C i ; d ( C i ) = ( P 1) = 2. There are P ( P 1) = 2 edges b et w een C i 's. Therefore C 0 is dense. Lemma 10 L et B G b e a clique of size P L et C 0 b e a minimum dense gr aph in G 0 Then j C 0 j = (( P 1) = 4 1) P ( P 1) + P ( P 1) = 4 and U ( C 0 ) = f v 2 G j H ( v ) \ C 0 6 = ;g is a clique of size P Pro of By Lemma 9 C 0 \ H ( v ) is either empt y or has densit y ( P 1) = 2 for all v ertices v 2 G Let C = C 1 [ : : : [ C t ; C i = C 0 \ H ( v i ). Let I E C 0 b e a n um b er of edges b et w een C i 's. Since C 0 is dense I E C 0 t ( P 1) = 2. Since I E C 0 t ( t 1) = 2 it follo ws that t P Therefore I E C 0 P ( P 1) = 2. Hence the total n um b er of k ey v ertices in C 0 is greater than or equal to 2( P ( P 1) = 2)(( P 1) = 4 1) and the total n um b er of reed v ertices is greater than or equal to P ( P 1) = 4. Since equalit y are reac hed when C 0 deriv ed from H ( B ) is considered, claim follo ws. This completes pro of of Theorem 2 4.3 Sp ecial Cases of Minim um Stably Dense 4.3.1 Flo w-based Solution for No-o v erconstrained Case Assume that all dense subgraphs A of G ha v e the same densit y d ( A ) = K (i.e there are no o v erconstrained subgraphs). In this case the follo wing t w o claims hold. Claim 39 If d ( X ) K ; 8 X G then for any two minimal dense gr aphs A; B G it fol lows that A \ B = ; Pro of Pro of b y con tradiction, supp ose that A \ B = C 6 = ; Since A and B w ere minimal, d ( C ) < K hence d ( A [ B ) = d ( A ) + d ( B ) d ( C ) = 2 K d ( C ) > K

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120 eeds R deg(v) (P1)/4 k k kk k k k kk k k k k k k 01 02 03 04 05 11 12 13 14 1525 21 22 23 24 r r rr Portals K e y s 2 013 Figure 4{3: H ( v ) represen ting v ertex v con tradiction since w e assumed that no subgraph of G has densit y greater than K Th us when there are no o v erconstrained subgraphs graph G can b e decomp osed in to a union of pairwise disjoin t minimal dense graphs [ i A i and remainder subgraph C that do es not con tain an y dense subgraphs. F ormally G = [ i A i [ C ; d ( A i ) = K ; A i \ A j = ; ; 8 B C ; d ( B ) < K F rom this structure it clearly follo ws that

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121 Claim 40 If d ( X ) K ; 8 X G then then N D = numb er of minimal dense sub gr aphs of G is less than or e qual to j G j = n Pro of Let A 1 ; : : : ; A N D b e minimal dense subgraphs of G then [ A i G and since A i \ A j = ; ; 8 i; j it follo ws that j A i j = [j A i j j G j = n Since ev ery j A i j 1 it follo ws that N D n Also since A i are edge-disjoin t (as w ell as v ertex-disjoin t) it follo ws that Claim 41 First dense sub gr aph A of G found by Dense(G) algorithm is minimal dense. Pro of Pro of b y con tradiction, supp ose that A = A 1 [ A p ; p > 1. Let e b e the last edge distribute d in b y routine Distribute(e) Since A 1 ; : : : ; A p are edge-disjoin t it follo ws that e b elongs to only one subgraphs, wlog e 2 A 1 This means that all edges of sa y A 2 w ere distributed b efore e but then dense graph A 2 w ould ha v e b ee found b efore A con tradiction. Prop erties ab o v e yield follo wing algorithm for nding minim um dense subgraphAlgorithm 1 Supp ose that d ( X ) K ; 8 X G Then fol lowing r outine nds al l minimal dense sub gr aphs of G : apply Dense(G) to nd rst minimal dense sub gr aph A 1 T ake G = G n A 1 and r ep e at until no mor e dense sub gr aphs left. By examining the sizes of A i s we lo c ate a minimum dense sub gr aph A p 4.3.2 Prero w-push based Solution for No-o v erconstrained Case Again, supp ose that d ( X ) K ; 8 X G Assume that there is at least one dense subgraph. Assume that w eigh ts of all v ertices are the same (= c ). Then an algorithm of Kortsarz and P eleg (1993 ) can b e used to nd a subgraph A G that maximizes ( A ) = P w ( e ( A )) K + 1 P w ( v ( A ))

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122 Claim 42 Sub gr aph A found ab ove is a minimal dense sub gr aph of smal lest c ar dinality. Pro of Since ( A ) 1 = d ( A ) K + 1 c j A j ; the maxim um of () is reac hed b y suc h an A that n umerator is equal to 1 (th us A is dense) and denominator is as small as p ossible (th us A is minimal and of smallest cardinalit y). Note 2 The algorithm of [GGT] c ould b e adapte d to nding maximum of () sinc e sub gr aph A that maximizes mo die d dual dier enc e P w ( e ( A )) K + 1 ( P w ( v ( A ))) is the same A that maximizes original dual dier enc e P w ( e ( A )) ( P w ( v ( A ))) The only dier enc e arises in the c omputing values of as P w ( e ( A )) K +1 P w ( v ( A )) inste ad of the original values P w ( e ( A )) P w ( v ( A )) If w eigh ts of v ertices are not constan t but b ounded b y some constan t sa y C then algorithm ab o v e yields C appro ximation of a minimal dense subgraph. Ho w ev er there is a b etter w a y of dealing with unequal v ertex w eigh ts describ ed b elo w. Remark 1 It is p ossible to r emove the c ondition ab ove that demands e quality of al l vertex weights. L et M b e the maximum vertex weight of G Then every vertex v of weight L M c ould b e r eplac e d by vertex v 0 of weight M and a lo oping e dge fr om v 0 to v 0 of weight M L Mo die d gr aph G 0 has the same dense sub gr aphs and al l vertic es of G 0 have the same weight M 4.3.3 Finding Smallest Subgraph of Largest Densit y If w e don't ha v e an y restrictions on densit y of subgraphs of G then w e still can use mo dication of [GGT] algorithm describ ed ab o v e to nd subgraph A G

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123 suc h that d ( A ) d ( B ) ; 8 B G and j A j j C j ; 8 C G; d ( C ) = d ( A ). This is done b y trying K = n 2 ; n 2 1 ; : : : un til A has b een found. 4.3.4 Case of Bounded Num b er of Ov erconstrain ts In this subsection w e will wlog supp ose that all edges of G ha v e w eigh t 1 and all v ertices ha v e w eigh t 3, but results could b e generalized for arbitrary in teger w eigh ts. Here w e will relax original condition that graph G has no o v erconstrained subgraphs and instead will assume that constan t n um b er of edges of G can b e remo v ed without violating solvability of an y subgraph of G Recall that dense graph A = ( V ( A ) ; E ( A )) is solvable if either A do es not con tain an y o v erconstrained subgraphs i.e 8 B A; d ( B ) K or there is a set of edges R ( A ) E ( A ) suc h that ( V ( A ) ; E ( A ) n R ( A )) is dense but do es not con tain an y o v erconstrained subgraphs. Claim 43 F or any gr aph G = ( V ; E ) we assume ther e is an e dge set R ( G ) such that for al l solvable sub gr aphs B G a r e duc e d sub gr aph B 0 = ( V ( B ) ; E ( B ) n R ( G )) r emains solvable and for al l sub gr aphs C G 0 = ( V ; E n R ( G )) ; d ( C ) K i.e G 0 has no over c onstr aine d sub gr aphs. Pro of Let A 1 ; : : : ; A k b e the set of maximal solv able subgraphs of G Due to maximalit y of A i s, all j A i \ A j j 1, i.e no t w o subgraphs ha v e an edge in common. Therefore edges of E can b e decomp osed in to m utually disjoin t sets E ( G ) = E ( A 1 ) [ E ( A 2 ) : : : [ E ( A k ) [ S F or ev ery A i there is an edge set R ( A i ) that can b e remo v ed without aecting solv abilit y of A i and of other subgraphs as w ell (due to disjoin tedness). Then R ( G ) = [ R ( A i ) [ S satises the criteria ab o v e. Denition 1 Such a set of e dges R ( G ) describ e d ab ove is c al le d maximal solvability pr eserving r emovable e dge set or MSPRES. Gr aph G that has MSPRES R ; j R j = c wher e is c al le d c-over c onstr aine d.

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124 Claim 44 It is p ossible to nd a minimum dense sub gr aph of c-over c onstr aine d gr aph G = ( V ; E ) in j E j c +3 time. A lso the numb er of wel lc onstr aine d sub gr aphs of G is O ( j E j c +3 ) Pro of This can b e done b y brute force examination of all p ossible com binations of c edges can to b e remo v ed, considering resulting graph G and applying tec hnique from rst subsection to G 0 in order to nd a minim um dense subgraph. Note that at least one of the com binations of c edges will corresp ond to no-o v erconstrained case examined in the rst subsection, hence algorithm will yield correct result. Note also that w e can map an y solv able subgraph to edge com binations up on whose remo v al the resulting subgraph is still solv able and has densit y exactly K A related result b elo w sho ws that a minimal dense subgraph cannot ha v e v ery large densit y Hence if w e are giv en a dense graph G of small densit y w e can use brute force algorithm of Claim 44 to nd minim dense subgraph, and if the densit y of G is large w e kno w that it lik ely to ha v e smaller minimal dense graph of small densit y Claim 45 Wlo g that al l e dges have weight 1, al l vertic es have weight 3. If dimension-dep endent c onstant K = 3 then no minimal dense sub gr aph c an have density mor e than 4. Pro of Pro of b y con tradiction, supp ose that graph A has densit y 5 or more, tak e an y v ertex v of A then the n um b er of edges adjacen t to v is greater than 8 (otherwise A n v is dense). If w e add up this inequailt y o v er all v ertices then 2 j E j > 8 N ; E > 4 N T ak e an y v ertex v It degree is less than or equal to N 1 hence densit y of A n v is greater or equal to E ( N 1) 3( N 1) 4 N 4 N + 4 4, hence A n v is dense and A w as not minimal dense, con tradiction.

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125 4.3.5 Size Ov erconstrained Graphs No w w e will consider a problem of nding minim um dense subgraphs of at least a certain size c + 1, under restriction that all o v erconstrained graphs ha v e size exactly c (w e will not b e making an y restrictions on densit y of suc h subgraphs). Note that without the restriction on size of o v erconstrained subgraph the problem of nding minim um dense subgraph is NP-Complete b y reduction from CLIQUE, so w e cannot generalize it to o m uc h in that direction. Observ ation 5 L et A 1 ; : : : ; A k b e the set of over c onstr aine d sub gr aphs of G; j A i j c; 8 i L et X b e a minimum dense sub gr aph of G; j X j > c Then for al l B G n X ; d ( B ) < 0 (otherwise B [ X would b e over c onstr aine d sub gr aph of size gr e ater than c ). Claim 46 L et e b e an e dge in G such that e 62 A i ; 8 i Then ther e is at most one minimal dense sub gr aph C G such that e 2 C and j C j > c Pro of Supp ose that this edge e 2 C ; D minimal dense subgraphs of size at least c + 1. Consider F = C [ D The d ( F ) = d ( C ) + d ( D ) d ( C \ D ) = K + K d ( C \ D ) K Hence d ( C \ D ) K but j C \ D j > c + 1 (since e 62 A i ; 8 i ), con tradicting minimalit y of C and D The follo wing algorithm lo cates all minimal dense subgraphs of size c + 1 or more. Wlog w e assume that all dense graphs are connected graphs and all o v erconstrained subgraphs ha v e size exactly c Algorithm 2 1. Apply A lgorithm 1 to gr aph F = G n ( A 1 [ : : : A k ) R e c or d the size of the smal lest found sub gr aph. 2. Distribute al l the e dges in F (they wer e r emove d during stage 1). 3. F or al l A i distribute rows within A i 4. F or i = 1 : k ; take F = F [ A i and p erform Pushoutside( A i ) on al l e dges e adjac ent to A i in or der to nd a dense gr aph M c ontaining e and A i Onc e M has b e en found p erform Minimal(M,e) in or der to nd a minimal dense sub gr aph that

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126 c ontains e and A i R e c or d the size of the smal lest minimal sub gr aph found so far, r ep e at. Claim 47 Every minimal dense sub gr aph M of G of size c + 1 or mor e, wil l b e found by A lgorithm 2 Pro of Ev ery edge of M will b e distribute d during execution of Algorithm 2 Consider edge e of M that w as distributed last. Supp ose that e 2 G n ( A 1 [ : : : A k ) Since e 2 M during its distribution some minimal dense graph con taining e w as found during stage 1. According to Claim 46 this graph has to b e M No w supp ose that e has exactly one endp oin t in A i (if it has b oth endp oin ts in A i then it is not going to b e last edge of M to b e distributed, due to connectivit y of M ). While p erforming Pushoutside( A i ) on edge e some dense graph con taining e and A i will b e found and then a minimal dense graph con taining e and A i w as found. Then b y reasoning similar to Claim 46 this graph has to b e M Here is an estimate on running time of this algorithm. Claim 48 The numb er of minimal dense sub gr aphs of size c + 1 or mor e is less than or e qual to ci =0 n i Pro of This is direct consequence of a result in ( F rankl, 1982 ). 4.4 Appro ximation Algorithms for Minim um Stably Dense 4.4.1 Randomized Appro ximation Algorithms In Section 3.2 w e describ e a p olynomial time net w ork-ro w based algorithm Dense that lo cates dense subgraphs in p olynomial time. Giv en a graph G Dense will either lo cate a dense subgraph A G or determine that G do es not con tain an y dense subgraphs. Using algorithm Dense as subroutine a follo wing randomized algorithm R andomize d Dense is prop osed for nding minim um dense subgraphs (of the en tire graph G that is assumed to b e dense throughout this section). This algorithm lo cates minimal dense subgraph, whic h is not guaran teed to b e minim um.

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127 0. A = G unmark all v ertices v 2 G 1. Cho ose an unmark ed v ertex v 2 A at random. If there are no unmark ed v ertices left (in A ) then return A 2. Mark and remo v e v and it adjacen t edges from A apply Dense to A n v 3. If algorithm Dense found dense subgraph B A n v then set A = B goto 1. 4. Else restore v (and edges) in to A goto 1 ( v remains mark ed). W e will sho w that this algorithm pro vides bad appro ximation to the minim um dense problem b oth in general case and in the case when w eigh ts of graph edges/v ertices are b ounded (second coun terexample do es mak e rst (general) counterexample unnecessary but w e do include general coun terexample b elo w since it is easy to analyze). Theorem 3 Consider gener al ( a; b; c; G ) pr oblem (r e c al l that this me ans that weights of e dges/vertic es and dimension dep endent c onstant ar e unb ounde d). Ther e is a gr aph G; j G j = n + 2 y such that algorithm R andomize d Dense wil l lo c ate, with pr ob ability 1 n y a minimal dense sub gr aph of G of size p n wher e y is any p ositive c onstant. On other hand this gr aph G has a minimum dense sub gr aph of size y that algorithm R andomize d Dense wil l nd with (low) pr ob ability n y Ther efor e R andomize d Dense is not FP AS. Pro of Consider a graph G that consists of t w o nono v erlapping dense subgraphs A and B Graph A is minim um dense subgraph, j A j = x where x is a constan t (= 2 y ). W eigh ts of edges and v ertices of A are not imp ortan t, as long as A is minim um dense. Graph B is a clique of size n Ev ery v ertex of B has w eigh t p n 1 2 ev ery edge of B has w eigh t 1, dimension dep enden t constan t c = 0. Therefore ev ery subgraph of G of size p n is minimal dense. Supp ose that at stage t of running R andomize d Dense algorithm w e ha v e reduced G = A [ B to G t = A t [ B t If j B t j p n then B t still con tains dense subgraphs. Hence if j A t j < x and j B t j p n then R andomize d Dense will

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128 output subgraph of size p n Therefore the only w a y subgraph A could b e found b y R andomize d Dense is when (all) rst n p n + 1 v ertices v remo v ed b y R andomize d Dense are lo cated in B (note that new dense subgraph found b y R andomize d Dense in G t n v will b e G t n v itself ). What is the probabilit y of this? Probabilit y that rst remo v ed v ertex w as in B is B A + B = n n + x Probabilit y that second remo v ed v ertex w as also in B is B 1 A + B 1 = n 1 n + x 1 etc. Hence the probabilit y that all n p n + 1 v ertices v remo v ed b y R andomize d Dense are lo cated in B is n n + x n 1 n + x 1 : : : p n + 1 p n + x + 1 p n p n + x After cancelling lik e terms this probabilit y ev aluates to O ( ( p n ) x n x ) = O ( n 1 = 2 x ). T aking x = 2 y claim of Theorem 3 follo ws. (Note that there is nothing magic ab out p n in the pro of ab o v e, man y other functions of n w ould ha v e b een equally go o d). The follo wing tec hnical lemma will allo w us to estimate probabilit y of R andomize d Dense outputting A and will help to analyze b ounded case. Lemma 11 If ther e is a dense gr aph G = A [ B ; j A j << j B j sub gr aph A is minimum dense, al l minimal dense sub gr aphs C of B have size j C j ; j C j >> j A j and every time vertex is r emove d fr om B t in Step 2 of R andomize d Dense the size dier enc e is j B t j j B t +1 j = d then pr ob ability of R andomize d Dense outputting A is O (( j C j j A j + j B j ) j A j =d ) Pro of Probabilit y that rst remo v ed v ertex w as in B is B A + B Probabilit y that second remo v ed v ertex w as also in B is B d 1 A + B d 1 etc. Hence the probabilit y that all ( j B j j C j ) =d + 1 v ertices v remo v ed b y R andomize d Dense are lo cated in B is

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129 B B + A B d 1 B + A d 1 : : : C + d C + A + d C C + A = O (( j C j j A j + j B j ) j A j =d ) The follo wing sho ws that Theorem 3 could b e extended to the b ounded case. Theorem 4 Consider (2 ; 1 ; 0 ; G ) pr oblem (r e c al l that this me ans that weights of e dges/vertic es and dimension dep endent c onstant ar e b ounde d). Ther e is a gr aph G; j G j = ( n log n + log 2 n ) such that algorithm R andomize d Dense with pr ob ability 1 ( log 3 n n ) log n wil l lo c ate a minimal dense sub gr aph of G of size log 3 n On other hand this gr aph G has a minimum dense sub gr aph of size log 2 n that algorithm R andomize d Dense wil l nd with (low) pr ob ability ( log 3 n n ) log n Ther efor e R andomize d Dense is not FP AS. Pro of Consider a graph G that consists of t w o nono v erlapping dense subgraphs A and B Graph A is minim um dense subgraph, j A j = log 2 n W eigh ts of edges and v ertices of A are not imp ortan t, as long as A is minim um dense. Graph B is an image under con v ersion H () describ ed in pro of of Theorem 2 of a clique of size p n V alue P = 2 log n Then size of an image of an y v ertex H ( v ) is (log ( n ) p n ) and since there are p n v ertices in the original clique, size of B is ( n log ( n )). Also size of an y minimal dense subgraph of B is (log 3 n ). Supp ose that at stage t of running R andomize d Dense algorithm w e ha v e reduced G = A [ B to G t = A t [ B t If B t still con tains dense subgraphs and j A t j < log 2 n then algorithm R andomize d Dense will return graph of size (log 3 n ). Therefore in order for R andomize d Dense to return graph A all the v ertices to b e remo v ed (in Step 2) m ust come from B un til B t is empt y Ho w man y v ertices should b e remo v ed? Note that when R andomize d Dense remo v es a v ertex v from B t it ma y nd that B t n v is not dense, but instead some prop er subgraph B t +1 B t n v is dense. Supp ose that for all t un til B = ; the size

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130 dierence j B t j j B t +1 j = log n (Then n um b er of v ertices remo v ed in Step 2, when R andomize d Dense outputs A is n ). If w e w ere able to sho w that for our graph G; j B t j j B t +1 j = log n then Theorem 4 w ould follo w from Lemma 11 Consider Step 2 of R andomize d Dense If v ertex v is a Key (see denitions in con v ersion used in Theorem 2 ) then j B t j j B t +1 j is at most O (log n ) If the v ertex v is a Reed then j B t j j B t +1 j is at most O ( p n log n ) (generally m uc h less b oth for Key and Reed). Probabilit y that randomly c hosen v ertex v is a Reed is 1 p n Hence exp ected v alue of j B t j j B t +1 j is 1 p n p n log n + (1 1 p n ) log n = O (log n ) Using Lemma 11 Theorem 4 follo ws. Com bination of Randomized Dense and Mo died Minimal. In this section w e com bine Randomized Dense algorithm and a mo dication of Minimal Dense algorithm to nd an appro ximate solution of Minim um dense subgraph problem. W e will sho w that there are graph coun terexamples where this com bination do es not yield constan t factor appro ximation. This com bination will b e called Double R andom Flow Minimum A lgorithm (DRFMA) It consists of t w o indep enden t parts. 1. Run Randomized Dense Algorithm, resulting in subgraph A 1 an appro ximation of minim um dense subgraph. 2. Run Glob al Minimal A lgorithm describ ed b elo w, resulting in subgraph A 2 an appro ximation of minim um dense subgraph. Output the smallest of A 1 and A 2 Description of Glob al Minimal A lgorithm

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131 Run algorithm Dense(G). Recall that this algorithm terminates once an edge e cannot b e distributed, indicating a presence of a dense subgraph B Run an algorithm Minimal( B ), nding subgraph A 2 curren t appro ximation of minim um dense subgraph. If it w as the w eigh t of e that cannot b e distributed then remo v e e from G Con tin ue distributing edges of G via algorithm Dense( G ). If another dense subgraph B has b een found then run Minimal( B ), up date A 2 if necessary con tin ue. After all edges of G ha v e either b een distributed or remo v ed, output A 2 Designing coun terexample for DRFMA. W e w ould lik e to construct a coun terexample, where DRFMA do es not yield constan t factor appro ximation. Th us b oth Randomized Dense and Global Minimal Algorithms should fail to yield constan t factor appro ximation to minim um dense subgraph. Supp ose that while constructing coun terexample w e limit ourself to graphs G that has X large minimal dense subgraphs C 1 ; : : : ; C X of size L = j C i j and one small minim um dense subgraph A of size R = j A j where L=R = f ( n ) is a non-constan t function. In tuitiv ely Randomized Dense w ould fail if X is sucien tly large. When will DRFMA fail to lo cate minim um dense subgraph? Consider what happ ens when w e distribute the edges of the last v ertex of A to b e distributed. Since A is dense, at this stage Dense() should return non-empt y dense subgraph B ; A B Hence in order for DRFMA to fail, Minimal(B) should output one of C i instead of A Either one edge of A w as imp ossible to distribute previously (and therefore remo v ed from G ) (and hence Minimal( B ) failed to lo cate A ) or one of the v ertices v of A w as remo v ed at random while running Minimal( B ) and some other minimal dense subgraph C i ; v 62 C i w as lo cated. Probabilit y of latter o ccurrence is similar to probabilit y of Randomized Dense eliminating A so (as will b e sho wn b elo w) it is alone sucien t to guaran tee a failure of Global Minimal with high probabilit y Ho w ev er, the probabilit y of former, that is a failure due to the w a y the algorithm deals with o v erconstrained graphs (b y simply remo ving edges that it

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132 cannot distribute and con tin uing) also o ccurs with high probabilit y as w e will sho w b elo w. Consider a Figure 4{4 Graph ABDEF is minim um dense for dimension dep enden t constan t K = 3. W eigh ts of all edges is 1, except that of AF, BF, DF and EF whic h is equal to 4, w eigh ts of v ertices A,B,D and E is 4, that of F is 5. There are X= 2 cliques C 1 ; : : : ; C X = 2 on the left and X= 2 cliques C X = 2+1 ; : : : ; C X on the righ t. Ev ery clique con tains Z v ertices. All of the Z v ertices in an y clique are connected to eac h other and also either to A and B or D and E. W eigh t of ev ery v ertex in the clique is equal to W, suc h that Z W + 8 = Z ( Z 1) = 2 + 2 Z + 3, i.e the clique plus a pair A,B or D,E form a minimal dense subgraph. Ho w ev er an edge AB (DE) will mak e this subgraph o v erconstrained. First w e will sho w that with high probabilit y algorithm Global minimal will distribute all edges in one of suc h large dense graphs b efore AB or DE, th us AB or DE w ould ha v e to b e remo v ed since they could not b e distributed. C 1 C X/2+1 C X 4 4 44 5 4 4 4 4 C X/2 A B D E F Figure 4{4: Coun terexample for DRFMA

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133 Claim 49 L et W = l og ( n ) ; Z = 2 l og ( n ) 3 then with pr ob ability 1 e n log 2 n al l e dges of some lar ge minimal dense sub gr aph (c ontaining vertic es A and B, but not e dge AB) wil l b e distribute d b efor e e dge AB. Pro of Consider the probabilit y that edge AB will b e distributed b efore some edge edge of all large minimal dense subgraphs con taining A and B. This probabilit y for an y giv en large subgraph of size Z is equal to Z Z +1 Hence for n= 2 Z graphs this probabilit y is ( Z Z +1 ) n= 2 Z that go es to e n 2 Z 2 hence claim follo ws. So with high probabilit y the rst edge (either AB or DE) will b e remo v ed from G Therefore with high probabilit y when the second edge will b e distributed (either DE or AB) the minim um dense subgraph ABDEF will not b e found b y Minimal() algorithm. This pro v es follo wing lemma. Lemma 12 Ther e is a gr aph G such that with pr ob ability 1 e n log 2 n algorithm DRFMA wil l lo c ate a minimal dense sub gr aph that is log ( n ) = 5 gr e ater than minimum one. 4.4.2 Minim um Dense as Minim um Cost Flo ws s t 1+e, c 1,0 1,0 1,0 i j i,j c+(deg(j)-c)e, c 1+e, c 1 2 2 capacity, cost Figure 4{5: Net w ork

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134 C () 1 C () 2 c(1+e) (1+e) 1+e Figure 4{6: Cost functions Let G = ( V ; E ) b e the graph where the minim um dense subgraph is to b e found. Without loss of generalit y assume that all v ertices of G ha v e the same w eigh t w ( v ) = c and constan t K = 0. The net w ork N that corresp onds to G consists of the bipartite graph ( V N ; E N ) and the pair of source and sink v ertices s; t V ertices of V N corresp ond to the v ertices of V v ertices of E N corresp ond to the edges of E The net w ork N has follo wing (orien ted) edges. A v ertex v N 2 V N is connected b y w ( e ) edges ( v N ; e N ) to a v ertex e N 2 E N i corresp onding v ertex v 2 V is an endp oin t of e 2 E Capacit y of ev ery one suc h edge ( v N ; e N ) is 1, cost function is 0. The source v ertex s is connected to ev ery v ertex v N 2 V N b y the edge ( s; v N ). The capacit y of the edge ( s; v N ) is c + ( deg r ee ( v ) c ) where is a small p ositiv e constan t. The cost function of the edge ( s; v N ) is c 1 ( x ), where c 1 ( x ) = 0 if x = 0, c (1 + ) otherwise. Ev ery v ertex e N 2 E N is connected to the sink v ertex t b y w ( e ) edges. The capacit y of an y one suc h edge ( e N ; t ) is 1 + The cost function of the edge ( e N ; t ) is c 2 ( x ), where c 2 ( x ) = 0 if x < 1 + 1 otherwise. See Figure 4{5 and Figure 4{6

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135 AC ABBC t s ABC 1, 0 1,0 1,0 1,0 1+e, C 1+e, C 2+4e, C 2+4e, C 2, C 1 1 1 2 2 2 1+e,C 1,0 1,0 AC ABBC t s ABC 2+3e 2+3e 2 e 1 1 2+2e 2+2e e 1+e 4+4e 1+e Figure 4{7: Net w ork and a ro w for p=3 A B C 2 2 2 2 2 2 Figure 4{8: Graph corresp onding to Figure 4{7 Lemma 13 L et N b e the network that c orr esp onds to the gr aph G c onstructe d as describ e d ab ove. L et b () b e the fol lowing demand ve ctor dene d on the vertic es of N b ( s ) = b ( t ) L; b ( v ) = 0 ; v 6 = s; t wher e L = 2 p (1 + ) Wlo g dimension-dep endent c onstant K = 0 Then ther e is a non-empty dense gr aph A G; j A j p i ther e exists a non-zer o fe asible row in N (i.e it satises the demand ve ctor b () and the c ap acities of al l the e dges of N ) of c ost 0 or less, for some demand ve ctor b () Example. Consider Figure 4{7 On the left is a net w ork N corresp onding to the graph G on the righ t of of Figure 4{8 ( c = 2). On the righ t of Figure 4{7 is a ro w of cost zero corresp onding to the p = 3 i.e the dense subgraph ABC. This ro w is of cost zero b ecause costs on the righ t (AB,BC,A C-t) cancel the costs on the left (s-A,B,C). (Note that costs of edges (AB,t), (A C,t), (BC,t) are negativ e, but they could b e con v erted in to p ositiv e one b y c hanging directions of the arcs and their capacities). The w a y the net w ork N is designed is suc h that an y ro w of cost zero will corresp ond to a feasible dense subgraph and vice v ersa. A Figure 4{9 sho ws a

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136 ro w of cost zero corresp onding to the p = 2 i.e the dense subgraph A C. F or p = 1 there are no ro ws of cost zero since there are no dense subgraphs of size 1. AC ABBC t s ABC 2+2e 2+2e 4+4e 2+2e 2+2e Figure 4{9: Flo w for p=2 The net w ork N giv en ab o v e is just one p ossible reduction of minim um dense graph to minim um cost ro w. F or example other cost functions are p ossible: for an edge ( s; v N ) cost function could b e c 1 ( x ) = 0 if x = 0 otherwise c 1 ( x ) = 2 + ( deg r ee ( v ) 2) x and for an edge ( e N ; t ) cost function could b e c 2 ( x ) = 0 if x = 0 otherwise 1 + x Then there is a non-empt y dense graph A G; j A j p i there exists a non-zero feasible ro w in N (i.e it satises the demand v ector b () and the capacities of all the edges of N ) of cost less than 1, for some demand v ector b (). Con v erting min cost ro w problem in to mixed-in teger problem. The problem of nding a minim um cost ro w in a net w ork describ ed in the previous section can b e stated as a follo wing optimization problem OPT : min n +3 m X i =1 f i ( x i ) s.t Ax = b; 0 x u

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137 where i suc h that 1 i n denotes \left" edges in the bipartite net w ork, i suc h that m + 1 i n + 2 m denotes \middle edges in the bipartite net w ork and i suc h that n + 2 m + 1 i n + 3 m denotes \righ t" edges in the bipartite net w ork. Here n and m is n um b er of v ertices and edges of the original w eigh ted graph (and accordingly n um b er of v ertices on the left and on the righ t of the corresp onding net w ork). F or notational simplicit y w e assume that all edges ha v e w eigh t 1 and all v ertices ha v e w eigh ts c V ariable x i represen ts ro w across an edge i v ector u represen ts capacit y constrain ts describ ed in the previous section. Matrix A and v ector b describ e the usual conserv ation/demand constrain ts. F unction f i ( x i ) = c 1 ( x i ) ; m + 1 i n + 2 m f i ( x i ) = 0 ; m + 1 i n + 2 m f i ( x i ) = c 2 ( x i ) ; n + 2 m + 1 i n + 3 m where c 1 () ; c 2 () are the cost functions describ ed in the previous section. A t this stage function that is to b e optimized in OPT in v olv es step functions. These step functions can b e replaced b y additional v ariables with in tegralit y constrain ts. Let 8 i = 1 ; : : : ; n v ariable y i = 1 if x i > 0 and y i = 0 otherwise. Let 8 i = n + 2 m + 1 ; : : : ; n + 3 m v ariable y i = 1 if x i = 1 + and y i = 0 otherwise.

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138 The OPT can b e restated as min n X i =1 c (1 + ) y 1 + n +3 m X i = n +2 m +1 (1 + ) y i s.t Ax = b 0 x i u i y i ; 8 i = 1 ; : : : ; n 0 x 1 1 ; 8 i = n + 1 ; : : : ; n + 2 m (1 + ) y i x i (1 + ) ; 8 i = n + 2 m + 1 ; : : : ; n + 3 m y i 2 f 0 ; 1 g Op en question: ho w to use this form ulation in order to construct FP AS for the original minim um dense problem? 4.4.3 Stating Minim um Dense Problem as IP In order to simplify the notation w e assume for this section that all the v ertices ha v e w eigh t 3, all edges ha v e w eigh t 1. K densit y constan t, P -size of the graph V ersion 1 of minim um dense problem stated as IP:

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139 min X x i X e ij X 3 x i K x i e ij ; 8 i; j X e ij = 3 P + K x i ; e ij 2 0 ; 1 LP relaxation is not alw a ys in tegral, consider Figure 4.4.3 K=6 P=3 3/5 e=3/5 x= V ersion 2 : max X e ij X x i

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140 X x i = P x i e ij ; 8 i; j x i ; e ij 2 0 ; 1 LP relaxation is not alw a ys in tegral, again consider Figure 4.4.3 Sev eral other IP statemen ts of the minim um dense problem are giv en b elo w, essen tially using densit y condition, size of graph condition and the fact that if the edge is c hosen to b e in a subgraph, then b oth of it endp oin ts should also b e in this subgraph. None of their LP relaxations are guaran teed to b e in tegral. Op en question ho w to state minimalit y condition in IP form? (p erhaps via cuts?). This migh t lead to m uc h b etter IP form ulations. V ersion 3: max X e ij x i e ij ; 8 i; j X x i P

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141 X e ij X 3 x i K V ersion 4: min X e ij X x i P x i e ij ; 8 i; j X e ij X 3 x i K V ersion 5: max X e ij X e ij + X x i E 2 P X x i P V ersion 6:

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142 min X e ij X e ij + 3 X x i 3 E x i + e ij 1 X x i N P V ersion 7: min X e ij X e ij 3 X x i 0 x i e ij X x i P Semidenite approac h. An IP form ulation, sa y

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143 min n X i =1 x i X x i = P X e ij 3 X x i = K x i e ij = 0 can b e easily con v erted in to a semidenite program max 0 : 25(1 + x i 0 + x j 0 + x ij ) X x i 0 = P X x ij = K 2 P x i 0 x ij = 0 x ii = 1

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144 matrix X is symmetric and p ositiv e denite Standard approac h Y e and Zhang (1997 ) is to solv e semidenite programming relaxation ab o v e c ho ose a random unit v ector in a n-dimensional unit sphere and select P v ertices closest to this v ector. Unfortunately it p ossible that this solution corresp onds to a subgraph that is not dense (in Y e and Zhang (1997 ) w e need to nd a subgraph of size P that has largest n um b er of v ertices, so feasibilit y of solution is guaran teed. F or our problem feasibilit y i.e subgraph b eing dense, is not guaran teed).

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149 Ruiz, O. E., F erreira, P M. (1996). Algebraic geometry and group theory in geometric constrain t satisfaction for computer-aided design and assem bly planning. In IIE T r ansactions on Design and Manufacturing 28 pages 281-294. Saliola, F., Whiteley W. (1999). Constrain t congurations in CAD: circles, lines and angles in the plane. Pr eprint Y ork University Semenk o v, O. (1976). An exp erimen tal CAD/CAM system. In 3r d international IFIP/IF A C c onfer enc e on Pr o gr amming L anguages for Machine T o ols NorthHolland, Stirling, Scotland. Serrano, D. (1990). Managing constrain ts in concurren t design: rst steps. Pr o c. Computers in Engine ering MIT press, Boston. Serrano, D., Gossard, D. (1986). Com bining mathematical mo dels with geometric mo dels in cae systems. Pr o c. Computers in Engine ering Univ ersit y of Chicago Press, Chicago. Sridhar, N., Aggarw al, R., Kinzel, G. (1993). Activ e o ccurence matrix based approac h to design decomp osition. In Computer A ide d Design 25 pages 500-512. Sridhar, N., Aggarw al, R., Kinzel, G. (1996). Algorithms for the structural diagnosis and decomp osition of sparse, underconstrained, systems. In Computer A ide d Design 28 pages 237-249. Sriv asta v, A., W olf, K. (1998). Finding dense subgraphs with semidenite programming. In L e ctur e Notes in Computer Scienc e 1444 pages 181{191. Sturmfels, B. (1993). Sparse elimination theory Pr o c. Computational A lgebr aic Ge ometry and Commutative A lgebr a Cam bridge Univ ersit y Press, Cam bridge, UK. Sugihara, K. (1985). Detection of structural inconsistency in systems of equations with degrees of freedom and its applications. In Discr ete Applie d Mathematics 10 pages 297-312. T a y T. (1999). On the generic rigidit y of bar framew orks. In A dvanc es in Applie d Math. 23 pages 14-28. T a y T., Whiteley W. (1985). Generating isostatic framew orks. In T op olo gie Structur ale 11 pages 21-69. W ang, D. (1993). An elimination metho d for p olynomial systems. In J. Symb olic Computation 16 pages 83-114. Whiteley W. (1992). Matroids and rigid structures. Matr oid Applic ations Cam bridge Univ ersit y Press, Cam bridge, UK.

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150 Whiteley W. (1997). Rigidit y and scene analysis. Handb o ok of Discr ete and Computational Ge ometry CR C Press, Bo ca Raton. Y. Y e and J. Zhang (2001). Appro ximation of densen= 2-subgraph and the complemen t of min-bisection. Journal of Glob al Optimization 25 pages 55{73.

PAGE 163

BIOGRAPHICAL SKETCH I w as b orn in Mosco w, Russia. I attended the Mathematics Departmen t of Mosco w State Univ ersit y in 1989-1991. I attended Ken t State Univ ersit y Ohio, as an undergraduate in 1991-1995 and graduated in Spring of 1995 with a bac helor's degree in mathematics and a bac helor's degree in computer science. I con tin ued m y graduate studies in Ken t State Univ ersit y un til 1997 and w ork ed on a join t pro ject in Purdue Univ ersit y in 1997-1998. In 1998 I transferred to the Univ ersit y of Florida where I ha v e b een w orking on m y PhD in computer science. 151

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Full Text

GRAPH AND COMBINATORIAL ALGORITHMS
FOR GEOMETRIC CONSTRAINT SOLVING

By

ANDREW LOMONOSOV

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

2004

by

Andrew Lomonosov

Dedicated to H.

ACKNOWLEDGMENTS

I am mostly indebted to my advisor, Dr. Meera Sitharam, who has been

supervising me with patience and everlasting interest. In the beginning she

offered me a very extensive choice of research problems and was willing to spend

considerable time explaining the nature and background of these problems, so I was

able to choose the area that personally appealed to me. While all these problems

were interesting and challenging, the problem statements were easy to understand,

so one could start thinking about them right away, learning necessary technical

skills along the way. My advisor has given me every opportunity to acquire these

technical skills by spending a lot of her time meeting me, running relevant special

topics seminars, introducing me to experts in the field and sending me to plenty of

conferences and workshops.

While allowing for my independent growth, my advisor was trying to make

the entire research process as efficient as possible. When working on one problem

she always kept another one on "backb, l i i so if the progress on one problem

slowed down somewhat then I could always turn to the second problem. This

topics, enthusiasm that was highly motivating whenever I would feel temporarily

bogged down.

Throughout the research process my advisor was mindful of my future career,

and was steering me toward learning proper mix of theoretical and practical skills.

Also in all the classes where I was her Teaching Assistant, she always took trouble

to explain the intricacies and potential pitfalls of the teaching process.

To some advisors the dissertation is not the only final product, the individual

growth of a student is at least as important. I am very happy that Dr. Meera

I would also like to thank my committee members Drs. Sartaj Sahni, Gerhardt

Ritter, Tim Davis, Sanjay Ranka and Ravi Ahuja for discussions we had and their

helpful fI-, -Ht, regarding my research, especially concerning network flow

related issues and relationships between various problems considered.

I would like to thank Dr. Christopher Hoffman for various assistance he has

provided during our joint work.

I would like to thank Dr. Pardalos as well as Burak and Sandra Eksioglu for

helping me with step-cost function approaches to obtaining approximate solution of

minimum dense problem.

Finally I would like to thank people with whom I have been working on

Geometric Constraint Solving Team: Jianjun Oung, Naganandhini Kohareswaran,

Yong Zhou, Aditee Kumthekar, Ramesh Balasubramaniam and Heping Gao for

helping me with various aspects of this large research area.

page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF FIGURES ................... ......... viii

ABSTRACT .................................. xi

CHAPTER

1 INTRODUCTION .................... ....... 1

1.1 Problem Definitions ........... ............... 1
1.1.1 Dense Graphs ......................... 1
1.1.2 Stably Dense Graph .............. ... ....... 1
1.1.3 Minimal Stably Dense Subgraph Problem .......... 2
1.1.4 Maximal Stably Dense Subgraph Problem .......... 2
1.1.5 Maximum Stably Dense Subgraph Problem ......... 3
1.1.6 Minimum Stably Dense Subgraph Problem ......... 3
1.1.7 Examples of various Dense Subgraphs ............ 4
1.1.8 Relationships between various Graph Problems . 5
1.1.9 Optimal Complete Recursive Decomposition . . 5
1.2 Summary of Results .................. ... 8
1.3 Related Work in Algorithms Community ... . 8

2 APPLICATIONS TO GEOMETRIC CONSTRAINT SOLVING ..... 10

2.1 Geometric Background .................. .... .. 10
2.1.1 Geometric Constraint Problems . . . 10
2.1.2 The Main Reason to Decompose Constraint Systems . 11
2.1.3 Decomposition Recombination (DR) Plans . ... 13
2.1.4 Basic Requirements of a DR-plan . . .. 14
2.1.5 Desirable Characteristics of DR-planners for CAD/CAM 17
2.1.6 Formal Definition of DR-solvers using Polynomial Systems 22
2.1.7 Formal Definition of a DR-planner via Constraint Graphs 30
2.1.8 Two old DR-planners .................. .. 45
2.1.9 Constraint Shape Recognition (SR) . . ..... 47
2.1.10 Generalized Maximum Matching (i) . .. 57
2.1.11 Comparison of Performance of SR and MM . ... 68
2.1.12 Analysis of Two New DR-planners . . ..... 69
2.1.13 The DR-planner Condense and its Performance ...... 73
2.1.14 The DR-planner Frontier and its Performance ...... ..78

2.2 Relating Problems of Chapter 1 to some Measures of Chapter 2 85

3 MAXIMAL, MAXIMUM AND MINIMAL STABLY DENSE PROBLEMS 87

3.1 Finding Maximum Dense Subgraph ................ 87
3.2 Finding a Stably Dense Subgraph ................. 88
3.2.1 Distributing an Edge .................. .. 88
3.2.2 Finding Dense and Stably Dense Subgraph . ... 93
3.2.3 PushOutside() .................. ..... .. 95
3.3 Structure and Properties of Frontier algorithm . . ... 100
3.3.1 Informal Description of Frontier Algorithm . ... 100
3.3.2 Assumptions .................. .. .... .. .. 101
3.3.3 Joining Pairs of Clusters .................. 101
3.3.4 Relevant Transformation Notation . . ..... 101
3.3.5 Recombination by Frontier Algorithm . .... 101
3.3.6 Removing Undistributed Edges ............... ..103
3.3.7 Pseudocode .................. ..... .. 103
3.3.8 Example of actions by Frontier algorithm . ... 104
3.3.9 Modifying Frontier Algorithm for Minimality . ... 105
3.3.10 Complete Decomposition Property . . ..... 109
3.4 Maximum Stably Dense Property ................. ..109
3.5 Other Properties of Frontier Algorithm .............. ..109

4 MINIMUM STABLY DENSE SUBGRAPH PROBLEM . ... 112

4.1 Relation of Minimum Stably Dense to other Problems ...... ..112
4.1.1 Maximum Number of Edges Problem . ..... 112
4.1.2 Decision Version of Minimum Dense Subgraph ...... ..113
4.1.3 Relationships between two Decision Problems . ... 113
4.1.4 Relationships between Approximation Algorithms . 115
4.2 NP-Completeness of Minimum Stably Dense Subgraph Problem .116
4.3 Special Cases of Minimum Stably Dense . . .... 119
4.3.1 Flow-based Solution for No-overconstrained Case . 119
4.3.2 Preflow-push based Solution for No-overconstrained Case .121
4.3.3 Finding Smallest Subgraph of Largest Density ...... ..122
4.3.4 Case of Bounded Number of Overconstraints . ... 123
4.3.5 Size Overconstrained Graphs ................ ..125
4.4 Approximation Algorithms for Minimum Stably Dense . 126
4.4.1 Randomized Approximation Algorithms . . ... 126
4.4.2 Minimum Dense as Minimum Cost Flows . ... 133
4.4.3 Stating Minimum Dense Problem as IP . .... 138

REFERENCES .................. ................ .. 145

BIOGRAPHICAL SKETCH .................. ......... .. 151

LIST OF FIGURES
Figure page

1-1 Nondense and dense graphs (K=1) ................. 1

1-2 Graph ABCDEF is dense but not stably dense (K=-3, w(v)=2,w(e)=l) 2

1-3 An edge/vertex weighted graph .................. 4

1-4 Original graph G and a corresponding RD-dag . . 6

1-5 Another possible RD-dag .................. .... 7

1-6 Original graph G and NOT a RD-dag ................ 7

1-7 Original graph G, RD-dag and complete RD-dag . . ... 7

2-1 A solvable system of equations .................. ..... 15

2-2 Step 1 rectangles, Step 3 ovals ................ 16

2-3 CAD/CAM/CAE master model architecture . ..... 20

2-4 A constraint graph .................. ......... .. 32

2-5 Generically unsolvable system .................. ..... 34

2-6 Generically unsolvable system that has a solvable constraint graph 35

2-7 Original geometric constraint graph G1 and simplified graph G2 37

2-8 Geometric constraint graph and a DR-plan . . ...... 38

2-9 Another possible DR-plan .................. .... .. 40

2-10 The original, cluster graph and the simplified graphs . ... 50

2-11 Action of the simplifier on Gi and C, during Phase One . ... 51

2-12 Action of the simplifier during Phase Two .............. .. 52

2-13 This solvable graph would not be recognized as solvable by SR . 52

2-14 Weight of all vertices is 2, weight of all edges is 1 . ... 53

2-15 Two triconnected subgraphs not composed of triangles . ... 55

2-16 Solvable graph consisting of n/3 solvable triangles . .... 55

2-17 Constraint graph, intended and actual decompositions . ... 58

2-18 Original graph and it decomposition .............. .. 59

2-19 Modified bipartite graph and maximum flow in this graph ..... ..60

2-20 Constraint graph and network flow with 3 extra flow units at AB 61

2-21 DR-plan depends on the initial choice ............. .. 62

2-22 Maximum flow and decomposition given the bad initial choice . 63

2-23 Decomposition given the good initial choice . . 63

2-24 Bad best-choice approximation ................ ...... 66

2-25 Sequential extension ............... ........ .. 73

2-26 Sequence of simplifications from left to right . . ..... 74

2-27 Original and simplified graphs ................ ...... 76

2-28 Bad best-choice approximation ................ ...... 77

2-29 The simplified graph after three clusters has been replaced by edges .79

2-30 Original graph BCDEIJK is dense, new graph MCDEIJK is not 81

2-31 1/n worst-choice approximation factor of DR-planner Frontier . 82

3-1 Before the distribution of edge ED ................ 90

3-2 After the distribution of edge ED ................ 90

3-3 Before the distribution of edge AD .................. .. 91

3-4 Before the distribution of edge BD ................ 91

3-5 Locating dense graph instead of stably dense . . ..... 93

3-6 Before the distribution of edge BD ................ 93

3-7 Dense graph BCD found instead of maximal dense ABCD . 95

3-8 Graphs ABCDEF and FGHIJA are maximal stably dense in 3D 97

3-9 Dense graph ABC found instead of minimal dense BC . ... 97

3-10 Actions of Minimal() algorithm .................. .. 98

3-11 Transformation by Frontier Algorithm . . ...... 102

3-12 General transformation by Frontier Algorithm . . 102

3-13 Edge CF will not be distributed, edge AC will .....

3-14 Initial graph . . . . . .

3-15 After A-C have been distributed .. ...........

3-16 After A-G have been distributed .. ...........

3-17 After A-P have been distributed .. ...........

3-18 Corresponding complete recursive decomposition .

3-19 Enforcing cluster minimality .. ............

3-20 Example of finding minimal .. ............

3-21 Removing AF does not affect solvability of entire graph

4-1 R education . . . . . . .

4-2 Gadgets for reduction to CLIQUE .. ..........

4-3 H(v) representing vertex v .. .............

4-4 Counterexample for DRFMA .. .............

4-5 N network . . . . . . .

4-6 Cost functions . .. .. .. ... .. .. .. .. .

4-7 Network and a flow for p=3 .. ............

4-8 Graph corresponding to Figure 4-7 .. .........

4-9 Flow for p=2 ...........

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

GRAPH AND COMBINATORIAL ALGORITHMS
FOR GEOMETRIC CONSTRAINT SOLVING

By

Andrew Lomonosov

May 2004

Chair: Meera Sitharam
Major Department: Computer and Information Science and Engineering

Geometric constraints are at the heart of CAD/CAM applications and

also arise in many geometric modeling contexts such as virtual reality, robotics,

molecular modeling, teaching geometry, etc.

Informally, a geometric constraint problem consists of a finite set of geometric

objects and a finite set of constraints between them. The geometric objects are

drawn from a fixed set of types such as points, lines, circles and conics in the plane,

or points, lines, planes, cylinders and spheres in 3 dimensions. The constraints are

spatial and include logical constraints such as incidence, tangency, perpendicularity

and metric constraints such as distance, angle, radius. The spatial constraints can

usually be written as algebraic equations whose variables are the coordinates of the

participating geometric objects. A solution of a geometric constraint problem is a

real zero of the corresponding algebraic system.

Currently there is a lack of effective spatial variational constraint solvers

and assembly constraint solvers that scale to large problem sizes and can be used

interactively by the designer as conceptual tools throughout the design process.

The requirement is a constraint solver that uses geometric domain knowledge

to develop a plan for decomposing the constraint system into small -il1 .-i' t' in-

whose solutions can be recombined by solving other small -11.-\--t' iL"- The primary

aim of this decomposition plan is to restrict the use of direct algebraic/numeric

solvers to --~ll.-i-t iu-H that are as small as possible. Hence the optimal or most

efficient decomposition plan would minimize the size of the largest such -il1 .-\-t' iI

Any geometric constraint solver should first solve the problem of efficiently finding

a close-to-optimal decomposition-recombination (DR) plan, because that dictates

the usability of the solver.

In this thesis we state this problem of finding a close-to-optimal solution as

a problem that deals with weighted graphs and also identify several important

subproblems. One class of such subproblem involves finding dense subgraphs -

graphs such that sum of weights of its edges is greater than sum of weights of its

vertices. Dense graphs that present interest for finding a DR-plan are (a) minimum

(smallest possible dense graphs), (b) minimal (not containing any other dense

subgraphs), (c) maximum (largest dense ones), (d) maximal (not contained in any

other dense subgraph).

This thesis presents polynomial time algorithms for problems (b), (c) and (d).

Problem (a) is shown to be NP-complete, and various approximation algorithms are

-11. -.I' 1 as well as explicit solutions for special cases that arise from CAD/CAM

applications.

CHAPTER 1
INTRODUCTION
1.1 Problem Definitions

1.1.1 Dense Graphs
Motivation. Suppose that we wanted to find a smallest subgraph of a given
graph that has at least twice the number of edges as vertices. Or three times. Or 5
times more edges than vertices.
Given: an edge and vertex weighted graph G = (V, E) and a constant K. Weights
of vertices and edges are denoted by w(v), w(e) respectively.
A graph G is called dense if LeC, w(e) C,,, w(v) > K

Function d(G) = LeGo w(e) EcG w(v) is called 1. ,.i,,',, of G.

B 2 B 2

22 1 1

A 2 C A 5
(2 2 )2 2

Figure 1-1: Nondense and dense graphs (K=1)

In many applications we are interested in finding a subgraph whose density is
"uniform", i.e., not contributed by some small overly dense part. Density of this

graph is "stable" or preserved even when some overly dense part is replaced by a
barely dense part. Following definitions describe this concept.
1.1.2 Stably Dense Graph

A graph A that has d(A) > K is overconstrained.
A graph G such that d(G) = K and VA C G, d(A) < K is wellconstrained.

A graph G is .Jll,j dense if d(G) > K and after replacing any of it over-

constrained nontrivial subgraph of by well-constrained subgraph G remains

dense. For example graph ABCDEF in Figure 1-2 is dense but not stably dense

(K = -3, w(v) = 2, w(e) = 1,Vv, w) because modified graph AFG is not dense

anymore.

B

D A
A c 3

G
0E
F F

Figure 1-2: Graph ABCDEF is dense but not stably dense (K=-3, w(v)=2,w(e)=l)

Many applications require finding smallest or largest subgraph unweightedd)

which has twice the number of edges as vertices. If we call graphs that have this

property "dense" (by setting weights of vertices and edges appropriately) then the

problem becomes that of finding "smallest" or "largest dense subgraph." Following

definitions formalize these notions.

1.1.3 Minimal Stably Dense Subgraph Problem

A stably dense graph A is called minimal f,tlll, dense if /EB C A, s.t. B is

stably dense. Given a graph G, a minimal f4,ll,, dense u,,.ii1,.ql, problem involves

locating a minimal stably dense subgraph A C G, if such A exists.

Note that A is minimal stably dense iff A is minimal stably dense.

1.1.4 Maximal Stably Dense Subgraph Problem

A stably dense graph A is called maximal f,tlol,, dense if /EB, A C B C G,

s.t. B is stably dense. Given a graph G, a maximal fo'l,l, dense ub,,Ilqil, problem

involves locating a maximal stably dense subgraph A C G, if such A exists.

1.1.5 Maximum Stably Dense Subgraph Problem

A largest (in terms of the number of vertices) maximal stably dense subgraph

is called maximum ,tf,l'l,, dense. Given a graph G, a maximum ,tflll,, dense

.,,1.lii ,"l, problem involves locating a maximum stably dense subgraph A C G, if

such A exists.

Note that a graph G can have several maximum stably dense subgraphs (that

have the same size).

1.1.6 Minimum Stably Dense Subgraph Problem

A smallest (in terms of the number of vertices) minimal stably dense subgraph

is called minimum tf,lbl,, dense. Given a graph G, a minimum tfb,ll,, dense o,,1l.i.,1./,

problem involves locating a minimum stably dense subgraph A C G, if such A

exists.

Note that a graph G can have several minimum stably dense subgraphs (that

have the same size). Also note that subgraph A is minimum dense if and only if A

is minimum stably dense, and if A is minimum dense, then A is also minimal dense.

Minimizing number of edges vs number of vertices

There is a modification of Minimum Stably Dense Subgraph Problem, where

we want to minimize number of edges (and not vertices) of stably dense subgraph

A. Following results demonstrate that optimal solution of modified problem will

yield a constant factor approximate solution of the original problem.

Let A C G be the dense subgraph d(A) > 1, such that IE(A)| is minimum over

all dense subgraphs of G.

Let B C G be the dense subgraph d(B) > 1, such that |V(B)I is minimum over

all dense subgraphs of G.

Lemma 1 Number of edges |E(A)| is at most 31V(A)|, similarly IE(B)l is at most

31V(B)I.

Proof Let k be an average degree of A, then d(A) = |V(A)I*k/2-2* V(A)| >

1. Since A is minimum dense, then removal of a vertex v E A, such that v has

smallest degree in A, will result in nondense graph A \ v. Degree of v could be at

most k, so this forces 1 > d(A \ v) > IV(A)l(k/2 2 1) and therefore k < 6, and

|E(A) = 3|V(A)|. I
We can use this lemma to relate sizes of A and B

Claim 1 For A and B defined above, IV(A)| < 3/21V(B)| 1/2.

Proof

2|V(A)| +1 < |E(A)I < IE(B)| < 3|V(B)|

First inequality is due to density of A, second inequality is due to A having

smallest number of edges among all dense subgraphs and third inequality is due to

Lemma 1. I

1.1.7 Examples of various Dense Subgraphs

Consider Figure 1-3. If K = 0 then BCDE is minimal stablyy) dense, EDF is

minimum stablyy) dense, AGH is maximal stably dense and ABCDEF is maximum

stably dense subgraph.

B 3 E
2 2
G\ 2
G \2 F
2 2
2C
H2 2 /2
A
2
D

Figure 1-3: An edge/vertex weighted graph

1.1.8 Relationships between various Graph Problems

Here we will show that the problem of finding minimum (minimal) dense

subgraph can be reduced in polynomial time to the problem of finding a maximum

(maximal) dense subgraph.
Let X be a bipartite graph, comprised of two sets of vertices X1 and X2. Let

A be a subset of X1, a set of all vertices in X2 that are connected to some vertex in

A will be denoted by I(A).

Now let G = (V, E) be given graph. Let BG = (VB U EB, F) be the bipartite

graph corresponding to G, where VB represents vertices V of G, EB represents

edges E of G, and in this graph BG, vertex v E VB is connected to vertex e E

by an edge (in F) if and only if v is an endpoint of e in G.

Lemma 2 Let A be a dense o,,io1,il,,i, of G. Let EA C BG be the set corresponding

to all edges of A. Then A = I(EA) and therefore IEA 3 I(EA) > K by definition

of ~ sfli of A. Hence finding 'i,,., tI EA such that IEA 3 I(EA) > K would find

maximum (number of edgeswise) dense iI,,l,

Lemma 3 Let A be a dense "oloil,/, of G. Let C = V \ A. Note that I(C) =

E \ EA. Hence d(G) = IEA + I(C) 3 IVAI 3 ICI, and since d(A) >

K,d(G)-I(C)+3C*C| > K and 3*,1C-I(C) > K-d(G). Therefore finding smallest

dense b,,1-i,,,l, is equivalent to finding largest C such that 3* CI-I(C) > K-d(G).

Claim 2 From previous two lemmas it follows that the problem of finding minimum

(minimal) dense uo,,bp, ,'l, can be reduced in t ,,oj,,;;,il time to the problem of

filit,.i a maximum (maximal) dense ,uo/b",'i/ (Note that reverse reduction will not
work due to the lack of symmetry in the bipartit vinl,, a vertex can have have any

number of adjacent edges, but every edge has ,l, fli two endpoints.)

1.1.9 Optimal Complete Recursive Decomposition

A recursive decomposition of a graph G involves constructing a so-called

RD-dag of G defined below.

A RD-dag of graph G is directed acyclic graph R = (RE, RV) which has

following properties.

The first property of R is that OV C RV, where OV is a copy of the entire

set of vertices of G. Now let X be a node in R, then U(X) denotes a set of vertices

in MV such that there is an oriented path from v to X for every v E U(X). For

example in Figure 1-4 U(S2) = {C, D, E} (in all examples in this subsection all

edges of G have weight 1, all vertices have weight 2, constant K = -3). The second

property of R is that for every sink vertex X of R, corresponding subgraph U(X)

is a maximal stably dense subgraph of G. The third property of R is that for every

node Si in R, with exception of nodes in OV, corresponding subgraph U(S1) is

stably dense. The final property of R is that every Si has a cluster l!;; i;''aliif

property, i.e., that the set P(S1) of all ancestors of Si does not contain a proper

nontrivial subset P'(S,) c P(SJ), |P'\ | 1 such that Uxp',(s,)U(X) is stably dense.

For example in Figure 1-6 node S2 does not have cluster minimality property (for

proper subset P' = S U B, U(P') = BCDE is stably dense, hence Figure 1-6 is not

a RD-dag.

Note that a graph G can have several different RD-dags, see Figure 1-4 and

Figure 1-6.

S3

B E S 2

A C
A B CDE

Figure 1 4: Original graph G and a corresponding RD-dag

A complete recursive decomposition of a graph G involves constructing a

so-called complete RD-dag of G. A complete RD-dag of G is a RD-dag of G with

A C D
A B CDE

Figure 1-5: Another possible RD-dag

B E S

Si

A C D
AB CDE

Figure 1-6: Original graph G and NOT a RD-dag

additional requirement that for every maximal stably dense subgraph M of G,

there is a corresponding node M' E R such that U(M') = M. For example middle

dag in Figure 1-6 is a RD-dag but not a complete RD-dag of G while the right dag

is a complete RD-dag.

E S2

F A B C DE F

S1 S2 S3

B CD
A B CD E F

Figure 1-7: Original graph G, RD-dag and complete RD-dag

Since given graph G can have several possible RD-dags, an important mea-

sure of efficiency of a RD-dag R is it maximum fan-in defined as a maximum

indegree among all vertices of R. We will provide motivation for this criteria in

C D

Section 2.1.1. For example in Figure 1-4 maximum fan-in is 1. The problem of

finding optimal (complete) recursive decomposition of a graph G involves find-

ing (complete) RD-dag of G that has smallest maximum fan-in among all other

(complete) RD-dags of G.

Nontrivial subgraphs. In practice we are only interested in finding

maximal/minimal/minimum/maximum stably dense subgraphs of size 3 or more

(i.e., not vertices or edges). Graphs of size 2 or less will be referred to as trivial.

1.2 Summary of Results

Numeric entries are sections where polynomial time algorithms (or NP-

completeness) are demonstrated.

Problem Bounded Parameter Unbounded Version
Minimum stablyy) dense NP-complete, 4.2 NP-complete, 4.2
special cases and appr 4.3, 4.4
Maximal stably dense 3.2.3 3.2.3
Minimal stablyy) dense 3.2.3 3.2.3
Maximum stably dense 3.4 3.4
Maximum dense 3.1 NP-complete, 3.1
Maximum number of edges, 4.1.1 N/A NP-complete
Measures and Performance for GCS 2.1.6, 2.1.14 N/A
Complete recursive decomposition 3.3.10 Unknown

1.3 Related Work in Algorithms Community

Finding densest graph. The problem of finding subgraph A C G of fixed

size IAI = k that maximizes Ew(e(A)) is NP-Complete (could be shown by

reduction from CLIQUE, see Asahiro and Iwama (1995)).

The problem of finding A C G that maximizes (A))can be solved in polyno-

mial time using parametric flow techniques.

The problem of finding A C G, s.t |A| < k for a given k, and A maximizes EZwi(A))
is NP-Complete (could be shown by reduction from CLIQUE).
is NP-Complete (could be shown by reduction from CLIQUE).

Section 4.1.1 contains more details on the relationship between these and our

graph problems.

Clique-like results. Let v and e be the number of vertices and edges of

graph G respectively.

It was shown by (Asahiro and Iwama, 1995) that finding a subgraph of G that

has at most a vertices and at least b edges is

NP-Complete for b = a(a 1)/2

NP-Complete for a = v/2, b < e/4(1 + 0(1/ ))

NP-Complete for a < v/2, b < al+6, b < e/4(1 + 0.45 + 0(1))

can be done in polynomial time for a = v/2, b < e/4(1 O(1/v) (by greedy

algorithm)

Semidefinite programming based approaches. For a maximum number

of edges problem (Feige and Seltzer, 1997) gives a semidefinite programming

relaxation approximation algorithm with approximation ratio roughly O(n/k).

(Goemans, 1996) studied a linear relaxation of the problem, getting approximation

ratio of O(n-1/2) when k = n/2.

(Srivastav and Wolf, 1999) used semidefinite programming relaxation to get

approximation factor within .5 for k = n/2 and (Ye and Zhang, 1997) were able to

design .5866 approximation factor SDP based algorithm for the case of k = n/2.

CHAPTER 2
APPLICATIONS TO GEOMETRIC CONSTRAINT SOLVING

2.1 Geometric Background

2.1.1 Geometric Constraint Problems

Geometric constraints are at the heart of computer aided engineering applica-

tions (see, e.g., Hoffmann (1997); Hoffmann and Rossignac (1996)), and also arise

in many geometric modeling contexts such as virtual reality, robotics, molecular

modeling, teaching geometry, and so on. In this dissertation we will be looking at

geometric constraint systems primarily within the context of product design and

assembly. Figure 2-3 illustrates those (boldface) components within a standard

CAD/CAM/CAE master model architecture (Bronsvoort and Jansen, 1994; Kraker

et al., 1997; Hoffmann and Joan-Arinyo, 1998) where our graph problems are

relevant.

Informally, a geometric constraint problem consists of a finite set of geometric

objects and a finite set of constraints between them. The geometric objects are

drawn from a fixed set of types such as points, lines, circles and conics in the plane,

or points, lines, planes, cylinders and spheres in 3 dimensions. The constraints are

spatial and include logical constraints such as incidence, tangency, perpendicularity

and metric constraints such as distance, angle, radius. The spatial constraints can

usually be written as algebraic equations whose variables are the coordinates of the

participating geometric objects.

A solution of a geometric constraint problem is a real zero of the corresponding

algebraic system. In other words, a solution is a class of valid instantiations of the

geometric elements such that all constraints are satisfied. Here, it is understood

that such a solution is in a particular geometry, for example the Euclidean plane,

the sphere, or Euclidean 3 dimensional space. For recent reviews of the extensive

literature on geometric constraint solving see Hoffmann et al. (1998); Kramer

(1992); Fudos (1995).

2.1.2 The Main Reason to Decompose Constraint Systems

Currently there is a lack of effective spatial variational constraint solvers

and assembly constraint solvers that scale to large problem sizes and can be used

interactively by the designer as conceptual tools throughout the design process.

Almost all current CAD/CAM systems primarily use a non-variational, history-

based 3 dimensional constraint mechanism. This basic inadequacy in spatial

constraint solving seriously hinders progress in the development of intelligent and

One governing issue is efficiency: computing the solution of the nonlinear

algebraic system that arises from geometric constraints is computationally challeng-

ing, and except for very simple geometric constraint systems, this problem is not

tractable in practice without further machinery. The so-called constraint propaga-

tion based solvers (e.g. Gao and Chou (1998a); Klein (1998)) generally suffer from

a drawback that cannot be easily overcome: they find it difficult to decompose

cyclically constrained systems, an essential feature of variational problems. Direct

approaches to algebraically processing the entire system include the following:

1) standard methods for polynomial ideal membership and locating solutions

in algebraically closed fields, for example using Grobner bases or the Wu-Ritt

method;

2) numerous algorithms and implementations for solving over the reals based

on the methods of, for example, Canny (1993); Renegar (1992); Collins (1975);

Lazard (1981) and

3) algorithms for decomposing and solving sparse polynomial systems based

on Canny and Emiris (1993); Sturmfels (1993); Khovanskii (1978); Sridhar et

al. (1993, 1996). These direct algebraic solvers deal with general systems of

polynomial equations: that is, they do not exploit geometric domain knowledge;

partly as a result, they have at least exponential time complexity and they are slow

in practice as well. In addition, they do not take into account design considerations

and as a result, cannot assist in the conceptual design process. These drawbacks

apply to direct numerical solvers as well, including those that use homotopy

continuation methods, see, for example, Durand (1998). The problem would

be further compounded if we allowed constraints that are natural in the design

process, but which must be expressed as inequalities, such as "point P is to

the left of the oriented line L in the plane." Such additions would necessitate

using cylindrical algebraic decomposition based techniques (Collins, 1975), such

as Grigor'ev and Vorobjov (1988); Lazard (1991); Wang (1993) which have a

theoretical worst-case complexity of 0(2"2), where n is the algebraic size of the

problem; or alternatively require the use of nonlinear optimization techniques, all

of which are slow enough in practice that they do not represent a viable option for

large problem sizes.

With regard to efficiency, the following rule of thumb has therefore emerged

from years of experimentation with geometric, spatial constraint solvers in en-

gineering design and assembly: the use of direct algebraic/numeric solvers for

solving large -i1 .-\--f' t'i i renders a geometric constraint solver practically useless:

(see Durand (1998) for a natural example of a geometric constraint system with

6 primitive geometric objects and 15 constraints, which has repeatedly defied

attempts at tractable solution). The overwhelming cost in geometric constraint

solving is directly proportional to the size of the largest -'i1l .-\-t itl that is solved

using a direct algebraic/numeric solver. This size dictates the practical utility of

the overall constraint solver, since the time complexity of the constraint solver is at

least exponential in the size of the largest such -i1t -\'-f int

2.1.3 Decomposition Recombination (DR) Plans

Therefore, the constraint solver should use geometric domain knowledge to

develop a plan for decomposing the constraint system into small -1 i-\ -t'i it- whose

solutions can be recombined by solving other small -111 -\--t'i i- The primary aim

of this decomposition plan is to restrict the use of direct algebraic/numeric solvers

to -iil. -\'t ii- that are as small as possible. Hence the optimal or most efficient

decomposition plan would minimize the size of the largest such l.-\-t'. iit Any

geometric constraint solver should first solve the problem of efficiently finding a

close-to-optimal decomposition-recombination (DR) plan, because that dictates

the usability of the solver. Finding a DR-plan can be done as a pre-processing

step by the constraint solver: a robust DR-plan would remain unchanged even as

minor changes to numerical parameters or other such on-line perturbations to the

constraint system are made during the design process.

In addition to optimality (efficiency), other equally important (and sometimes

competing) issues arise from the fact that a DR-plan is a key conceptual compo-

nent of the CAD model and should aid the the overall design or assembly process.

These issues will be discussed under "Desirable characteristics" later in this section.

A clean and precise formulation of the DR-planning problem is therefore a

fundamental necessity. To our knowledge, despite its longstanding presence, the

DR-problem has not yet been clearly isolated or precisely formulated, although

there have been many prior, specialized DR-planners that utilize geometric domain

knowledge (Bouma et al., 1995; Owen, 1991, 1993; Hoffmann and Vermeer, 1994,

1995; Latham and Middleditch, 1996; Fudos and Hoffmann, 1996b, 1997; Crippen

and Havel, 1988; Havel, 1991; Hsu, 1996; Ait-Aoudia et al., 1993; Pabon, 1993;

Kramer, 1992; Serrano and Gossard, 1986; Serrano, 1990). See Hoffmann et al.

(1998) for an exposition of two primary classes of existing methods of decomposing

geometric constraint systems; representative algorithms from these two classes are

extensively analyzed in Section 2.1.8.

In the next two subsections we informally describe both the basic require-

ments of a DR-plan(ner) that dictate its overall structure, as well as desirable

characteristics of the DR-plan(ner) that improve efficiency and assist in the design

process.

2.1.4 Basic Requirements of a DR-plan

Recall that a DR-plan specifies a plan for decomposing a constraint system

into small -l l-\-t iivt and recombining solutions of these -l l-\-t it-r later. There-

fore the first requirement of a DR-plan is that the solutions of the small -1i ,-\- ti it-

in the decomposition can be recombined into a solution of the entire system. In

other words, it should be possible to substitute the (set of) solutions) of each sub-

system into the entire system in a natural manner, resulting in a simpler system.

Secondly, we would like these intermediate -1i ,-'-t'i i,-t that are solved during the

decomposition and recombination to be geometrically meaningful.

Together, these two requirements on the intermediate -1i ,-'-t'i i,-t translate

to a requirement that the -iil.-\--t'i i,- be geometrically rigid. A rigid or solvable

-,il -\-t it of the constraint system is one for which the set of real-zeroes of

the corresponding algebraic equations is discrete (that is, the corresponding

real-algebraic variety is zero dimensional), after the local coordinate system is

fixed arbitrarily, that is, after an appropriate number of degrees of freedom D,

are fixed arbitrarily. The constant D is usually the number of translationall

and rotational) degrees of freedom available to any rigid object in the given

geometry (3 in 2 dimensions, 6 in 3 dimensions, and so on.) and in some cases,

D depends on other symmetries of the \1 l-i\t' it An underconstrained system is

not solvable, that is, its set of real zeroes is not discrete (non-zero-dimensional). A

D: (x1-x2)^2+(yl-y2)^2-A^2=0
E: (x2-x3)^2+(y2-y3)^2-B^2=0
P2
F: (x3-x1)^2+(y3-yl)^2-C^2=0

A B

P1 C P3

Figure 2-1: A solvable system of equations

wellconstrained system is a solvable system where removal of any constraint results

in an underconstrained system. An overconstrained system is a solvable system in

which there is a constraint whose removal still leaves the system solvable. Solvable

systems of equations are therefore wellconstrained or overconstrained. I.e, the

constraints force a finite number of isolated real solutions, so one solution cannot

be obtained by an infinitesimal perturbation of another. For example, Figure 2-1 is

a solvable i1 .-i \-t' it" of three points and three distances between pairs of points. A

. s;'sit, ,tl,/l overconstrained system is one which has at least one real zero.

Note. It is important to distinguish "solvable" from "has a real solution."

Although (inconsistently) overconstrained (or even certain wellconstrained) systems

may have no real solutions at all, by our definition, since their set of real zeroes

is discrete, they would still be considered "solvable." In general, whenever a

-,il -\-t i" of a constraint system is detected that has no real solutions, then the

solution process would have to immediately halt and inform the designer of this

fact. 0

Informally, a geometric constraint solver which solves a large constraint system

E by using a DR-planner -to guide a direct algebraic/numeric solver capable of

solving small -111. -\-t'I li proceeds by repeatedly applying the following three

steps at each iteration i.

(T (ST)( S3

ITTS))

E E2 Tl(E,) E= T2(E2) E = Tn~ T,(E,)

Figure 2-2: Step 1 rectangles, Step 3 ovals

1. Find a small solvable -iil -.\'-f i, Si of the (current) entire system Ei (at the

first iteration, this is simply the given constraint system E, i.e, El = E). This step

is indicated by a rectangle in Figure 2-2. The -iil -\-- itl S could be also chosen by

the designer.

2. Solve Si using the direct algebraic/numeric solver.

3. Using the output of the solver, and perhaps using the designer's help to

eliminate some solutions, replace Si by an abstraction or simplification Ti(S,)

thereby replacing the entire system E by a simplification T,(E,) = E,+1. This

step is indicated by an oval in Figure 2-2. Some informal requirements on the

simplifiers T, are the following: we would like E to be (real algebraically) inferable

from Ei,+; i.e, we would like any real solution of Ei,+ to be a solution of E, as well.

A constraint solver that fits the above structural description, -which we shall refer

to as S in future discussions -is called a decomposition-recombination (DR) solver.

(Formal definition imposes further requirements on simplifier T see the next

section). This solver terminates when the small, solvable -iil-\ -t' in S5 found in

Step 1 is the entire polynomial system E. An optimal DR-plan will minimize the

size of the largest S,. If the whole system is underconstrained, the solver should

still solve its maximal solvable -l1 i- -ti in-

For the purpose of planning a solution sequence apriori, we would like to execute

Steps 1 and 3 alone without access to the algebraic solver, and get a DR-plan,

without actually solving the -ii l-\--t ii- I.e, we would like the constraint solver to

look as in Figure 2-3, with the DR-planner driving the direct algebraic/numeric

solver.

To generate a DR-plan apriori, one would have to locate a solvable -ii1.-\-ti il

S,, and without actually solving it, find a suitable abstraction or simplification of

it that is substituted into the larger system Ei to obtain an overall simpler system

Ei+1 in Step 3. On the other hand, such a DR-plan would possess the advantage

of being robust, or generically independent of particular numerical values attached

to the constraints, and of the solution to the Si, and usually only depends on the

degrees of freedom of the relevant geometric objects and geometric constraints.

2.1.5 Desirable Characteristics of DR-planners for CAD/CAM

We enumerate and informally describe a set C of natural characteristics

desirable for a DR-planner. We begin with four criteria that directly follow

from the overall structural description of a typical DR-planner S in the previous

subsection.

(i) The DR-planner should be general, i.e, it should not be artificially restricted

to a narrow class of decomposition plans; it should output a DR-plan if there

is one; and it should be able to decompose underconstrained systems as

well. Furthermore, if a DR-plan exists, the planner should run to completion

irrespective of how and in what order the solvable -il1 .-\-'ti i,- Si are chosen

for the plan (a Church-Rosser property).

(ii) The planner should potentially output a close-to-optimal DR-plan (i.e,

where the size of the largest solvable -ll.-\ -t' i l S, is close-to-minimal). This

dictates efficiency of solution of the constraint system.

(iii) The DR-planner should be fast and simple; the time complexity should

be low, the planner should be fast in practice, easily implementable, and

compatible with existing algebraic solvers, CAD systems, constraint models

and manipulators.

(iv) The planner should utilize and take advantage of geometric domain knowl-

edge, as well as other special properties of geometric constraints arising from

the relevant design or assembly application or, in some situations, from a

downstream manufacturing application.

Besides critically affecting the speed of constraint solving, a good DR-plan is a

key component of the constraint model which participates in the overall process of

design/assembly. This is especially so, since the constraint system is a component

of the CAD model which the designer directly interacts with, and moreover, a

DR-plan is nothing but a hierarchical, structural decomposition of the geometric

constraint system. Therefore, maintaining a robust DR-plan which reflects design

intent at every level of refinement is invaluable in improving efficiency, flexibility

and transparency in the overall design process. These properties are crucial

for intelligent CAD systems to facilitate designer interaction at a conceptual,

early-design phase; in fact, an effective CAD system based on spatial, variational

adding the following desirable characteristics to the set C.

(v) The DR-plan should be consistent with the design intent: in particular,

the designer often has a multi-layered or hierarchical conceptual, design

decomposition in mind, reflecting features or conglomerates of features

in the case of product design and parts or subassemblies in the case of

assembly. See e.g, Klein (1996); Mantyla et al. (1989); Middleditch and

Reade (1997). The designer would typically wish to further decompose the

components of her design decomposition, as well as treat these components

(recursively) as units that can be independently manipulated. For example,

the geometric objects within a component are manipulated with respect to

a local coordinate system, and a component as a whole unit is manipulated

with respect to a global (next level) coordinate system. The DR-plan should

therefore be a consistent extension and/or refinement of this conceptual

design decomposition.

(vi) While the DR-plan is used to guide the algebraic solver, it should remain

unaffected or adapt easily to the designer's intervention in the solution

process, which is valuable for pruning combinatorial explosion: the designer

can throw out meaningless or undesirable solutions of -it1 -\-- t'i i- at an

early stage. Such designer interference is also crucial for avoiding the use of

point P is to the left of line L, the designer can simply throw out any partial

solutions that put P to the right of line L.

(vii) The DR-plan for solving a geometric constraint system should remain

meaningful in the presence of other than geometric constraints, such as

Finally, the CAD system and the CAD model do not stand alone. In standard

client-server based architectures (see e.g, Bronsvoort and Jansen (1994); Kraker et

al. (1997); Hoffmann and Joan-Arinyo (1998)), the CAD model is just one client's

view of the product master model (Newell and Evans, 1976; Semenkov, 1976),

with which it has to be continually coordinated and made consistent. The master

model in turn coordinates with other downstream production client systems which

maintain other consistent views. These clients include geometric dimensioning

and tolerancing systems (GD& T), and manufacturing process planners (MPP)

for automatically controlled machining or assembly. Each client view contains

potentially proprietary information that must be kept secure from the master

model. Figure 2-3 illustrates this architecture and those parts that we deal with

directly.

Each client view contains its own internal view of the constraint model

and therefore coordination and consistency checks between the various views

Master
model
SMaster
I model
SConstraint
Decomposition Optional design history
consistent with
designed conceptual of constraint models
Master model server decomposition (for nonvanational case)

GD&T Manuf. proc. Other
View/system View/system downstream I iI
View/ Decomp. gebrpcI-
system pln s

GD&T MPP Constraint model manpulato CAD
model model constraint
coD 0 0 ^- A-nodel|

Figure 2-3: CAD/CAM/CAE master model architecture

crucially involve, and viceversa may affect the DR-plan. This leads to the following

additions to the set C of desirable characteristics for DR-planners.

(viii) The DR-plan should be as robust as possible (see e.g. Fang (1992)) to on-line

changes made to the constraint system, and the DR-planner should be able

to quickly alter the DR-plan in response to such changes. In particular,

the DR-plan should ideally not change at all with numerical perturbations

(within large ranges) to the constraints, thereby permitting the DR-plan to

be computed as a pre-processing step. Addition, deletion, and modification

of constraints and geometric objects to the constraint system occurs in

numerous circumstances during the design process. For example:

(a) In the process of solving the system using the DR-plan, a -1ii1 -\- in l Si

deemed solvable by a degree of freedom analysis may be found to have no

real solutions, or may, in a degenerate case, turn out to be underconstrained

or have infinitely many solutions, preventing a continuation of the solution

process S;

(b) In lhi-tm t \--1 ',- I1 CAD systems, the specification of the product takes the

form of a progressive sequence of changes being made to successive partial

specifications and the associated partial constraint systems;

(c) Inferred or understood constraints are often added to an incomplete

constraint specification at the discretion of the CAD system;

(d) In underconstrained situations, as might occur in designing flexible or

moving parts and assemblies, the mapping of the configuration space of

the resulting mechanism would require the repeated addition of constraints

at the discretion of the constraint solver -the solutions to the resulting

wellconstrained systems would represent various configurations of the

mechanism;

(e) For a variational spatial constraint solver to be effective or even usable, it

would have to rely on extensive interactive constraint changes being made by

the designer, sometimes in the course of solving; and

(f) Finally, when one of the various clients in Figure 2-3 makes a change to

its own view of the constraint model, this will result in consistency updates to

the master constraint model which will, in turn, result in updates to the other

views of the constraint model.

(ix) The DR-plan should isolate overconstrained ~li-\-ti iti- which arise in

assembly problems; furthermore, with multiple (possibly proprietary) views

of the constraint model being kept by various clients as in 2-3, constraint

reconciliation often takes place and the DR-plan should facilitate this process,

and viceversa, should be robust against this process. For a precise description

of the constraint reconciliation problem, see Hoffmann and Joan-Arinyo

(1998). The problem requires isolation of overconstrained lli-\-ti iti and is

compounded in the case of non-variational, hi-t 't \ -1 .,-i .1 design.

2.1.6 Formal Definition of DR-solvers using Polynomial Systems

We will formally state the decomposition-recombination (DR) problem for

polynomial systems arising from geometric constraints. This requires formal-

izing the notion of a decomposition-recombination solution sequence, and of

decomposition-recombination solvers that fit the description S given in the previous

section. In addition, we define the performance measures that capture some of the

desirable properties C given in the last section for comparing such sequences and

solvers.

Note. As was mentioned previously, for the sake of gradual exposition, we will

first assume that the constraint system is being solved even as the decomposition is

being generated i.e, the DR-solver exactly fits the structural description S given

in the previous section. Furthermore, we will define the DR-solver in the general

context of polynomial equations that arise from the geometric constraints.

In Section 2.1.7 we will shift our attention to the DR-planning problem. The

DR-planner generates a decomposition plan apriori, which then drives the direct

algebraic/numeric solver -together, they form a DR-solver. The DR-planner,

however, will be defined in the context of constraint graphs that incorporate

geometric degrees of freedom. The DR-planner and its performance measures will

be analogous to the italics terms defined here.

In order to formally define DR-solvers and their performance measures, we need

to specify the model of real-algebraic computation used by the algebraic/numeric

solver in Figure 2-3. However, the issue of the model or how real numbers are

represented is entirely outside the focus of this manuscript for the following

reason: our DR-solvers and performance measures are robust in that they are

conceptually independent of the algebraic/numeric solver being used or how real

numbers are represented. The definition of DR-solvers and performance measures

adapts straightforwardly to other natural models of computation. In other words,

our conceptual definition of DR-solvers and performance measures are such that

if DR-solver A performs better than DR-solver B with respect to one of our

performance measures, then it continues to do irrespective of the (natural) model of

computation being used by the algebraic/numeric solver.

Further, as pointed out in an earlier note, we only emphasize the boldface

components of Figure 2-3, specifically the DR-planner, whose operation defined in

Section 2.1.7 will be seen to be purely combinatorial. In particular, no emphasis

is placed on the non-boldface components, which includes the algebraic/numeric

solver, and the model of computation it uses. <

Having noted the above, the model of computation we assume for the the sake of

completeness and formality of definitions in this manuscript is the Blum-Shub-

Smale model of real algebraic computation Blum et al. (1989). Briefly, in this

model, real numbers are assumed to be representable as such as entities, without

any recourse to rational approximations and finite precision or interval arithmetic.

Real arithmetic operations such as multiplication and addition and division can

be performed in constant time, and finding unambiguous representation of each

real zero of a univariate polynomial p can also be achieved in time polynomial in

the degree of p. Furthermore, these unambiguous representations of real zeroes of

univariate polynomials are treated as entities just like any other real number, and

can for instance, be used as coefficients of other polynomials.

A system of equations E is a pair (P, X) where P is a set of polynomial

equations with real coefficients, and X is a set of formal indeterminates. The union

of two i f. i,,-. E1 = (P1, X) and E2 = (P, X2) is the system (P1 U P2, X1 U X2).

An intersection El n E2 is the system (P1 n P2, X1 n X2).

Within the geometric context, a solved -ill -v-tf' in is a rigid or solvable system

where all the variables have already been "solved for," i.e, they have typically been

expressed explicitly as polynomials of D free variables, where D represents the

number of degrees of freedom of the rigid body (within its local coordinate system)

in the prevailing geometry. In some cases, for example when the rigid body is a line

in 3 dimensions, the number of free variables, including redundant ones, may be

greater than D.

Let E = (P, V) with V = {yi,... D} U X be such that every equation in P

is of type xj = Fj(yi, ..., yD), for all xj E X and Fj is a rational function (of the

form Q1/Q2 where Qi and Q2 are, as usual, polynomials with real coefficients) that

can be evaluated in time polynomial in (|V|, |P|). Such a system of equations, E,

is called a solved system. The variables yi, ..., yD E V of the solved system are free

variables of E, and the variables l,.., .., Xk V are explicitly fixed variables. Note

that all the variables in a solved system are fixed or free, whereas a general solvable

system may have free, explicitly fixed and other, i;,!' li, ,fixed variables.

A typical DR-solver that follows the overall structural description S of Sec-

tion 2.1.4 obtains a sequence E1,... E,, consisting of successively simpler

solvable systems. These are general solvable systems and have successively fewer

implicitly fixed variables; E = El; and Em is a solved system. New variables yi

may be introduced at intermediate stages which represent free variables within

-,Ll--\ t' ii ,s Si that are solved with respect to these variables. These solved sub-

systems represent various rigid bodies located and fixed with respect to their local

coordinate systems. However, these local coordinate systems are still constrained

with respect to each other, and hence in fact only D of the newly introduced vari-

ables are, in effect, free and the remainder are implicitly fixed. Some of these newly

introduced variables may be removed at later stages. Eventually, in the solved

system Em, all the original variables and those newly introduced variables that

might remain become explicitly fixed with respect to D free variables as well. The

set of real solutions to Ei+1 should also be a subset of solutions to Ei, to ensure

that the final solutions to Em actually represent solutions to the original system E.

We formally define real algebraic equivalence and real algebraic inference of

two geometric constraint systems as follows. Given two systems El = (P1, X U Yi)

and E2 = (P2, X U Y2) where the set X represent the original variables that

are currently implicitly or explicitly fixed, and the sets Y, represent the newly

introduced variables (that are "free" within the solved -,1 .-\-t'fI I S1), we say that

the system El = (P1, X U Yi) is real .,i/, 1,,;i, all, inferable (in short, inferable)

from the system E2 = (P2, X U Y2), if for any real solution Y2 = A2; X = B that

satisfies the equations in P2, there is a corresponding assignment A1 of real values

to Y1 such that Y = A1; X = B satisfies P1. Two systems El and E2 are real

.,1./i 1i,/ti ,a11, equivalent if El is real algebraically inferable from E2 and E2 is real

algebraically inferable from El.

Now we are ready to define the notion of a DR-solution sequence. Let E be a

system of equations.

A DR-solution sequence of E is a sequence of systems of equations E, ..., Em

such that E = El, Em is a solved system (so Em has a real solution), every E,

is solvable, each E, is inferable from E,+1. Any solvable system E which has a

real solution in fact has a DR-solution sequence. A trivial DR-solution sequence

E, E* where E* is a solved system equivalent to E will do. (Note that by E,

we denote abstract algebraic systems, rather than their computer representations

that could only have rational coefficients and therefore may only have approximate

DR-solution sequences).

The DR-problem is the problem of finding a DR-solution sequence of a given

constraint system. A DR-solver is a constraint solver or algorithm that solves the

DR-problem. A DR-solver is general if it always outputs a DR-solution sequence

when given a solvable system as input. Aside from being general, we would also

like a DR-solver to have the CI,,,n, l-Rosser property, i.e, the DR-solver should

terminate irrespective of the order in which the solvable -l1 .-\-'fi i,- S, are chosen.

In other words, at each step, a solvable -ill -\-tf itl Si can be chosen greedily

to satisfy the (easily checkable) requirements of the algorithm. This prevents

exhaustive search.

Performance measures. Next, we formally define a set of p if, ,ni, i,'

measures for the DR-solution sequences and DR-solvers. These performance

measures are designed to capture the characteristics C of constraint solvers given in

Section 2.1.5, that are desirable for engineering design and assembly applications.

(Note that many of these measures are Boolean, i.e, either a certain desirable

condition is met or not.) In particular, we would like a DR-solution sequence Q =

E, ... E, of a solvable system E to have several properties. To describe these

properties we formalize a simplifying map from each system Ei to its successor

E,+1. In fact, for generality, we choose these maps T, called subsystem simplifiers

- to map the set of -i1 .-\--t' ir-, of Ei onto the set of -iit -\-f. ti i- of Ei+1.

First, in order to reflect the Church-Rosser property in (i), and points (iv) and (vi)

of C, we would like these -i1l .-\-t int simplifiers T, to be natural and well-behaved,

i.e, to obey the following simple and non-restrictive rules.

(1) If A is a -iil .-\-ft. of B then T,(A) is a -1.-\-t. il" of T,(B)

(2) T,(A) U T,(B) = T,(A U B)

(3) T,(A) n T(B) = T(A n B)

Second, in the description of the typical DR-solver S given in Section 2.1.4, each

system E1+1 in the DR-solution sequence is typically to be obtained from Ei by

replacing a solvable -il .-\-t'i i L S, in Ei (located during Step 1 of S), by a simpler

-,ill -\-ft i (during Steps 2 and 3). For a manipulable DR-solution sequence (again

satisfying the points (i), (iv), (v) and (vi) of C), we would like Ei+1 to look exactly

like E, outside of S,. This leads to another set of properties desirable for the

-,il l'-\'f i" simplifiers Ti.

(4) Each E = Si U Ri U UI 1 < i < m where S, is solvable, Ri is a maximal

-,it.-\'-t itl such that Si and Ri do not share any variables, Si, Ri, U, do not

share any equations, and all variables of Ui are either variables of Si or Ri.

For any AC ?R, T,(A) = A.

Thirdly, in order to address the points (iv)-(vi), and (i) of C simultaneously, i.e,

permitting generality of the -i t '-\-t il Si that is replaced by a simpler system

during the i h step, while at the same time making it convenient for the designer

to geometrically follow and manipulate the decomposition, we would like the

-i1l '-\-t' i" simplifiers to satisfy the following property.

(5) For each i, all the pre-images of S,, T,- ...T 1 (Si), 1 < j < i- 1 are

algebraically inferable from Si, and furthermore, they are solvable or rigid

-,1-l .-'t' iI for the given geometry (recall that the definition of "solvable"

depends on the geometry). It follows from (2) and (3) above that the

inverse T 1(A) = UB where the union is taken over all B C Ei, such that

T(B)C A.

The above property says that while the -1-i\'t-'- it simplifiers enjoy a high degree

of generality and are completely free to map solvable -il -'-t'.i iu-t into solvable

systems, they should never map (convert) -iil -\'-t. ii-v that are originally not

solvable into one of the chosen, solvable systems Si, at any stage i. In other words,

in the act of simplifying, the -1 ti-\-t' itl simplifiers should not create one of the

solvable -l ,-\-t'i i,-t S out of -l ,-\-t'i i,-t that were originally not solvable. A DR-

solution sequence that satisfies the above properties is called valid. Thus a valid

DR-solution sequence for a geometric constraint system E is specified as a sequence

of El,... Em such that El = E, Em is a solved system, every Ei is solvable and

inferable from E,+1, every Ei = Si U Ri U U, as described above.

The motivation given for each of the properties above makes it clear that valid DR-

solution sequences encompass highly general but geometrically meaningful solutions

of the original constraint system E. Next we turn to point (ii) of C, i.e, optimality,

which also competes with generality (point (i)). For optimality, we would like to

minimize the size of the largest solvable -i11 -\-t'i i Si in the DR-solution sequence.

Formally, the size of the DR-sequence Q is equal to maxl
ISl is equal to the total number of it variables less the number of its explicitly

fixed variables. The optimal DR-size of the algebraic system E is a minimum size

of Q, where the minimum is taken over all possible DR-solution sequences Q of

E. An optimal DR-solution sequence Q of E is a solution sequence such that the

size of Q is equal to the optimal DR-size of E. The approximation factor of DR-

solution sequence Q of the system E is defined as the ratio of the optimal DR-size

of E to the size of Q.

As a general rule for optimality, it is clear that the larger the choice for solvable

-,I,-v-tf I- i I S, of Ej available at any stage i, the more likely that one can find

a small solvable S, in Ei. In other words, we would like to make sure that the

-,1i -\-tf iul simplifier does not destroy solvability of too many -i-ii -\t, -i- starting

from the original system E. Note that while the definition of DR-solution sequence

makes sure that the entire system Em is solvable if El is, it does not require the

same for -iil-\ -t'i ii s of Ei. (In fact, even if all the Ei in a DR-solution-sequence

are algebraically equivalent, this would still not imply that solvability is preserved

for the -il -v-ti ii-). In addition, while the definition of a DR-solution sequence

ensures that Ei+1 has a real solution if Ei has one, it does not ensure the same

for -il.- -t l-ii of E,. The next two definitions capture properties of -1i -\-ti ilL

simplifiers that preserve -ii \-'t iii solvability (resp. solutions) to varying degrees,

thereby helping the optimality of the DR-solver, i.e, (ii) of C. The DR-solution

sequence Q = E, ..., Em is ,*,1., ,,l1if,i preserving if and only if for all A c Ei, A is

solvable (resp. has a real solution) and (An Si= 0 or A c S) c=# T (A) is solvable

(resp. has a real solution). A DR-sequence Q = E, ..., Em is strictly .l/,-,,li,;f1,

preserving if and only if for all A C Ei, A is solvable (resp. has a real solution)

-== T(A) is solvable (resp. has a real solution). Requiring such a solvability

preserving simplifier places a weak restriction on the class of valid DR-solution

sequences, but on the other hand, this restriction cannot eliminate all optimal

DR-solution sequences. Furthermore, solvability preservation helps to ensure the

Church-Rosser property in (i) of C.

Note. To be more precise, "solvability preservation" should be termed "solvability

and solution-existence preservation," but we choose the shorter phrase. O

In fact, for DR-solution sequences to be optimal, we would prefer that for all

i > 1, S, does not contain any strictly smaller -iii1-\-'ft it say B that is solvable,

and has not been found (during Step 1 of the description S in Section 2.1.4) and

simplified/replaced at an earlier stage j < i. The DR-sequence Q = E, ..., E, is

complete if and only if for every nontrivial solvable B C Si, B = T,_1T_-2...Tj(Sj)

for some j < i 1. While the completeness requirement restricts the class of valid

DR-solution sequences, it only eliminates sequences that either have size greater

than optimal or the same size as some optimal sequence that does simplify B. In

addition to affecting optimality, i.e, (ii) of C, completeness also strongly reflects

(ix): completeness prevents a DR-solver from overlooking an overconstrained

-'l .-i\ i1 inside a wellconstrained -liil --ti i-" which is also useful for constraint

reconciliation (see Hoffmann and Joan-Arinyo (1998)).

Performance measures. So far, we have discussed performance mea-

sures that measure the desirability of DR-solution sequences. Next we formally

define directly analogous performance measures for DR-solvers which generate

DR-sequences. A DR-solver A is said to be valid, l,.-',l1,ilit';, preserving, strictly

.,'11,,1ili;f, preserving, complete if and only if for every input constraint system E,

every DR-solution sequence produced by A is valid, solvability preserving, strictly

solvability preserving or complete respectively.

Note. Purely as a tool to help analysis and exposition, often we assume that

DR-solvers for producing DR-sequences are randomized or nondeterministic in a

natural way, i.e those steps in the algorithm where arbitrary choices of equations or

variables are made (for example, to be the lowest numbered equation or variable)

are now taken to be randomized or nondeterministic choices. O

The next definition formalizes performance measures related to the characteristic

(ii) of C that differentiates randomized DR-solvers for which some random choice

leads to an optimal DR-solution sequence, from inferior DR-solvers where no choice

would lead to an optimal DR-solution sequence. The worst-choice approximation

factor of a DR-solver A on input system E is the minimum of the approximation

factors of all DR-solution sequences Q of E obtained by the algorithm A over all

possible random choices. The best-choice approximation factor of the algorithm

A on input E is maximum of the approximation factors of all the DR-solution

sequences Q of E obtained by the algorithm A over all possible random choices.

2.1.7 Formal Definition of a DR-planner via Constraint Graphs

The DR-solution sequence defined in the previous section embeds a DR-plan

that is intertwined with the actual solution of the system. Therefore, a DR-solver

that simply outputs a DR-solution sequence (even one that is strictly solvability

preserving, complete, and so on..), may not be modular as shown in Figure 2-3.

How to construct a DR-solver that first generates a DR-plan before using it to

drive the general algebraic/numeric solver? First notice that repeatedly applying

Steps 1 and 3 of the DR-solver in the description S in Section 2.1.4 could poten-

tially generate a DR-plan without actually solving any I1.-\-t. 1i (and without

applying Step 2), provided that the following can be done:

(a) solvability of a l ,-\-fti i' S, of the system E, in Step 1 can be determined

generically, without actually solving it; and

(b) a solvable -il.-\-f' i Si can be replaced in Step 3 without actually solving

S, i.e, by a hypothetical solved -1il \-v-f' ilt to give the new system Ei+1. For

this we need a consistent and adequate abstraction of the systems Ei, and of their

-ll-\-f-t ii- as well as a way to abstract a hypothetical solved -ll -\-t f iil In

other words, we need to adapt the -iil -v-ft itl simplifier maps described in the last

section, so that the iteration can proceed with Step 1 again.

By thus modifying the description S of the DR-solver, we obtain a DR-planner

which can be employed as a preprocessing step to output a DR-plan instead of

a DR-solution sequence. This DR-plan can thereafter be used to direct a series

of applications of Step 2, leading to a complete DR-solution sequence. This

modularity helps, for example, towards maneuverability by the designer (point

(vi)), and towards compatibility of the DR-solver with existing solvers (point (v)).

In order to formally define such a DR-plan, we follow a common practice and view

the constraint system as the constraint hypergraph: this abstraction permits us to

build the DR-planner on the foundation of generalized degree of freedom analysis

which is known to work well in estimating generic solvability of constraint systems

without actually solving them. (This is explained more precisely after the formal

graph-theoretic definitions are in place). Hence using constraint graphs facilitates

both the tasks (a) and (b) described above.

In other words a DR-planner that is based on a generalized degree of free-

dom analysis is robust in the following sense: changing the numerical values of

constraints is not likely to affect the DR-plan. Thus the motivation for using con-

straint graphs includes all of the desirable characteristics (iii) to (vi) of the set C in

Section 2.1.5. In addition, geometry is more visibly displayed via constraint graphs

than via equations, thereby helping interaction with the designer.

Note. DR-plans and planners and their performance measures could also be

defined directly in terms of the algebraic constraint systems, just as we defined

32

1
(2 2
E E
D F
1 /1

2

Figure 2-4: A constraint graph

DR-solvers. In this thesis, however, due to the reasons mentioned above, we define

DR-plans and planners entirely within the context of constraint graphs and degree

of freedom analysis.

Constraint graphs and solvability. First we define the abstraction

that converts geometric constraint system into a weighted graph. Recall that a

geometric constraint problem consists of a set of geometric objects and a set of

constraints between them.

A geometric constraint graph G = (V, E, w) corresponding to a geometric

constraint problem is a weighted graph with n vertices (representing geometric

objects) V and m edges (representing constraints) E; w(v) is the weight of vertex

v and w(e) is the weight of edge e, corresponding to the number of degrees of

freedom available to an object represented by v and number of degrees of freedom

removed by a constraint represented by e respectively. For example, Figure 2-4

is a constraint graph of a constraint problem shown in Figure 2-1. Note that the

constraint graph could be a ,,i,, ,iI,,'1, each hyperedge involving any number of

vertices.

Now we introduce definitions that will help us to relate the notion of solvability of

the geometric constraint system to the corresponding geometric constraint graph.

A subgraph A C G that satisfies

w(e)+D > w(v) (2.1)
eeA veA

is called dense, where D is a dimension-dependent constant, to be described below.

The function d(A) = (CA w(e) EOA w(v) is called /'/;, of a graph A.

The constant D is typically ('d+) where d is the dimension. The constant D

captures the degrees of freedom associated with the cluster of geometric objects

corresponding to the dense graph. In general, we use words "subgraph" and

. IL-t' t" interchangeably. For planar contexts and Euclidean geometry, we expect

D = 3 and for spatial contexts D = 6, in general. If we expect the cluster to be

fixed with respect to a global coordinate system, then D = 0.

A dense graph with density strictly greater than -D is called overconstrained.

A graph that is dense and all of whose subgraphs (including itself) have density at

most -D is called wellconstrained. A graph G is called welloverconstrained if it sat-

isfies the following: G is dense, G has at least one overconstrained subgraph, and

has the property that on replacing all overconstrained subgraphs by wellconstrained

subgraphs, G remains dense. A graph that is wellconstrained or wellovercon-

strained is said to be solvable. A dense graph is minimal if it has no proper dense

subgraph. Note that all minimal dense subgraphs are solvable, but the converse is

not the case. A graph that is not solvable is said to be underconstrained. If a dense

graph is not minimal, it could in fact be an underconstrained graph: the density of

the graph could be the result of embedding a subgraph of density greater than -D.

In order to understand how solvable constraint graphs relate to solvable constraint

systems it is important to remember that a geometric constraint problem has two

aspects combinatorial and geometric. The geometric aspect deals with actual

parameters of the geometric constraint problem, while the combinatorial aspect

deals with only the abstractions of objects and constraints. Unfortunately at the

moment it is not known how to completely separate the two aspects, except for

some special cases. In this thesis we will limit ourselves to the geometric constraint

problems where generally there is a correspondence between solvable constraint

AcD

F

Figure 2-5: Generically unsolvable system

systems and solvable constraint graphs. In order to do that, we need to introduce

and formalize the notion of generic solvability of constraint systems. Informally, a

constraint system is generically (un)solvable if it is (un)solvable for most choices of

coefficients of the system. More formally we use the notion of genericity of e.g, Cox

et al. (1998). A property is said to hold genet iI ,,ll for polynomials fi,... f, if

there is a nonzero polynomial P in the coefficients of the f, such that this property

holds for all fl,... f, for which P does not vanish.

Thus the constraint system E is generically (un)solvable if there is a nonzero

polynomial P in the parameters of the constraint system such that E is

(un)solvable when P does not vanish. Consider for example Figure 2-5. Here

the objects are 6 points in 2d and the constraints are 8 distances between them.

This system is unsolvable since the edge BC can be continuously displaced, unless

the length of AD (induced by the lengths of AE, DE, DF, AF and EF) is equal to

the |ABI + IBC| + |CDI. Since we can create an appropriate nonzero polynomial

P(AB, BC,... EF), such that P() = 0 if and only if IADI = IABI + BCI + ICDI,

this system is generically unsolvable.

While a generically solvable system always gives a solvable constraint graph,

the converse is not always the case. In fact, there are solvable, even minimal dense

graphs whose corresponding systems are not generically solvable, and are in fact

generically not solvable (note that the position of the 'not' changes the meaning,

B G33

Figure 2-6: Generically unsolvable system that has a solvable constraint graph

the latter being stronger than the former). Consider for example Figure 2-6. The

constraint system is shown on the left, it consists of 8 points in 3d and 18 distances

between them. The corresponding constraint graph is shown on the right, weight of

all the edges is 1, weight of the vertices is 3, geometry dependent constant D = 6.

Note that this system is generically unsolvable since rigid bodies ABCDE and

AFGHE can rotate about the axis passing through AE. This example can also be

transformed into a constraint problem in 2d, where the objects are circles and the

constraints are angles of intersections Saliola and Whiteley (1999).

Another example is the graph K7,6 that in 4 dimensions represents distances

between pairs of points. The constraint graph is minimal dense but it does not

represent a generically solvable system.

It should be noted that in 2 dimensions, according to Laman's theorem Laman

(1970) if all geometric objects are points and all constraints are distance constraints

between these points then any minimal dense subgraph represents a generically

solvable system. Also there exists a purely combinatorial characterization of

solvable systems in 1 dimension based on connectivity of the constraint graphs.

However, as examples above indicate, the generalization of Laman's theorem fails in

higher dimensions even for the case of points and distances.

There is a matroid based approach to verifying whether solvability of the

constraint graph implies solvability of the constraint system. It begins by checking

whether a submodular function f(E) = 21VI 3, defined on sets of edges of the

constraint graph creates a matroid Whiteley (1992, 1997), i.e checking whether

the created function is positive on single edges. Thereafter this approach checks

whether the corresponding geometric structure is generically rigid. Some matroid

based approaches to determining generic solvability are fast in practice, for example

those to be discussed later under "generalized maximum matching". However, so

far no match has been established (for all cases) between the created matroid and

generic rigidity of the particular geometric problem.

There are several attempts at characterization of generic solvability in dimen-

sion 3 and higher for the case of points and distances Tay (1999); Graver et al.

(1993). One such characterization, the so called Henneberg construction, checks

whether a given constraint graph can be constructed from the initial basic graph

by applying a sequence of standard replacements. A characterization due to Dress

checks whether the constraint graph satisfies a certain inclusion-exclusion type rule.

A characterization due to Crapo examines whether the constraint graph can be de-

composed into a union of certain edge-disjoint trees. All of these characterizations

though interesting and useful, are so far unproven conjectures.

Note. Due to the above discussion, we restrict ourselves to the class of constraint

systems where solvability of the constraint graph in fact implies the generic

solvability of the constraint system. (As pointed out earlier, the converse is always

true, with no assumptions on the constraint system).

As was indicated above, this class is far from empty, it contains all constraint

problems involving points and distances in 2d, problems resulting from Cauchy

triangulations of the polyhedra in 3d as well as body-and-hinge structures in

3d. Moreover, it should be emphasized that while existing applications stop at

finding subgraphs representing solvable constraint systems, we are interested in

the entire problem of decomposition and recombination, optimizing the size of the

largest -il -v\--tf iln to be solved, i.e we are interested in finding an optimal DR-plan.

Also note that already for the class of generically solvable constraint systems and

I DE

Figure 2-7: Original geometric constraint graph G1 and simplified graph G2

corresponding graphs, the DR-planning problem is, in general, difficult (see later

Note about NP-hardness, after the definition of the optimal DR-plan). O

Formal definition of DR-plans using constraint graphs. Informally,

stated in terms of constraint graphs, the DR-planning problem involves finding

a sequence of graphs Gi a DR-plan such that the original constraint graph

G = G1 and every Gi contains a minimal solvable subgraph Si, which is simplified

or abstracted into a simpler subgraph T,(S,) and substituted into Gi to give an

overall simpler graph GC+I = Ti(Gi). (While Ti(S,) should be simpler than Si,

it should also be somehow equivalent to Si, for example by having same density

value.) If the original graph G1 is wellconstrained, then the process terminates

when G, = S,. (If not, the process terminates with the decomposition of G, into

a maximal set of minimal solvable subgraphs).

Consider for example Figure 2-7 which shows one simplification step. On the left

is the constraint graph G1, the weight of all vertices is 2, weight of all edges is 1,

geometry dependent constant D = 3. Then S1 = {A, B, C} is a solvable subgraph.

On the right is the simplified graph TI(G1) = G2, after subgraph S1 is replaced by

a vertex {V} = Ti(S1). Since density of the subgraph S1 is -3, the weight of the

vertex {V} could be set to 3.

A sequence of simplification steps is shown in Figure 2-8. The top part depicts

a geometric constraint graph G, where the weight of each edge is 1, the weight

of each vertex is 2, and the dimension dependent constant D is equal to 3 (this

S, S3 S5

A B C D E F G HI J K L M N O

Figure 2-8: Geometric constraint graph and a DR-plan

corresponds to points and distances in 2d). One of the plans for decomposing

and recombining G (and the geometric constraint system that G represents) into

small solvable subgraphs (representing solvable -il1 ---\ti i4 -), is to decompose G

into dense subgraphs St = {A, B, C}, S2 = {D, E, F}, S3 = {G, H, I}, S5 =

{J, K, L}, S6 = {M, N, O}, represent their solutions appropriately in a simplified

graph so that they can be recombined, one possibility is to represent them as

vertices P, Q, R, S, T of weight 3 each; recursively decompose the simplified graph

into 4 = {P, Q, R}, S7 = {S, T}; represent and recombine their solutions as

vertices U, W of weight 3; and so on until the entire graph is represented and

recombined as a single vertex.

A corresponding DR-plan is shown at the bottom part of Figure 2-8. Note

that there could be more than one DR-plan for a given constraint graph. For

example, another possible DR-plan for a constraint graph described above is shown

in Figure 2-9.

An optimal DR-plan will minimize the size of the largest solvable subgraph Si

found during the process. I.e, it will minimize the maximum fan-in of the vertices

in the DR-tree shown in Figure 2-8 and Figure 2-9, where by fan-in of a vertex we

mean the number of immediate descendants of the vertex. With this description,

it should be clear, that DR-plans obtained using the weighted, constraint graph

model are generically robust with respect to the changes made to the geometric

constraints; as long as the number of degrees of freedom attached to the objects

and destroyed by the constraints remains the same, the same DR-plan will work

for the changed constraint system as well. Thus, such DR-plans satisfy the initial

robustness requirements of the characteristic (viii) of the set C described in

Section 2.1.5.

Next we formally define a DR-plan for constraint graphs, and the various perfor-

mance measures that capture desirable properties of DR-plans and DR-planners.

40

S1

B3 S, /S7
S2G

Sg

S4

S S S2 S5 \ S6

ABC DE F G H I J K L MN O

Figure 2-9: Another possible DR-plan

The development is parallel to that in Section 2.1.6, where a DR-solution sequence

and various performance measures for these sequences and for DR-solvers were

motivated and defined in terms of polynomial systems.

Note. Due to the strong analogy of DR-planners to DR-solvers defined in the

previous section, the discussion here is more terse. The following are useful generic

correspondences to keep in mind after reading the Note at the end of Section 2.1.7,

and the paragraphs preceding it:

Solvable subgraphs = solvable -ill -\-'t it -.

DR-plans = DR-solution sequences;

DR-planner = DR-solver;

subgraph simplifier =: -ii -\- iti simplifier. <

Let G be a solvable constraint graph. A DR-plan Q for G is a sequence Q of

graphs G1, ..., G, such that G1 = G, Gm is a minimal solvable graph, and every

Gi is solvable. An algorithm is a general DR-planner if it outputs a DR-plan when

given a solvable constraint graph as input.

The union (resp. intersection) of two ub,,1li~,,!.i A and B is the graph induced by

the union (resp. intersection) of sets of vertices of A and B. All subgraphs are

understood to be vertex induced.

The mapping from the graph Gi to GC+1 is called a o,,1li,,,, simplifier and is

denoted by Ti. This mapping should have the following properties.

(1) If A is a subgraph of B then TI(A) is a subgraph of Ti(B).

(2) T,(A) U T,(B) is the same as the graph T,(A U B).

(3) T,(A) n T,(B) is the same as T,(A n B).

As in the case of DR-solution sequences, assume that every constraint graph Gi in

the DR-plan can be written as Si U Ri U Ui, where Si is minimal solvable, Ri is a

maximal subgraph such that S, and Ri do not have common vertices, Si, Ri, U, do

not have common edges and all vertices of Ui are either vertices of Si of vertices of

Ri. Analogous to properties (4) and (5) of the -ii1-\--'f i n simplifier in the previous

section, we would like the subgraph simplifier to have the following additional

properties.

(4) For every A C Ri, T,(A) = A

(5) All the pre-images of S,, i.e -T-1 T ...T, 1(S) for all 1 < < i 1, are

solvable.

A DR-plan that satisfies the above rules is called valid. The size of the DR-plan Q

of G is the maximum of the sizes of Si. The size of an arbitrary subgraph A C Gi

is computed as follows.

Size(A) = 0

For l
B = T_1T,_,...T,(Sj)

If AnB 0

then Size(A) = Size(A) + D, A = A \ B

end if

end for

Size(A) = Size(A) + EZeA w(v)

In other words, the image of any of the Sj contributes D to the size of A, where

D is geometry dependent constant, and the vertices of A that are not in any such

image contribute their original weight. The optimal size of the constraint graph G

is the minimum size of Q, where the minimum is taken over all possible DR-plans

of G. An optimal DR-plan of G is the DR-plan that has size equal to the optimal

size of G. The approximation factor of DR-plan Q of the graph G is defined as the

ratio of the optimal size of G to the size of Q.

Note. The problem of finding the optimal DR-plan for a constraint graph with

unbounded vertex weights is NP-hard. This follows from a result in the authors'

paper Hoffmann et al. (1997) showing that the problem of finding a minimum

dense subgraph is NP-hard, by reducing this problem to the CLIQUE. The

CLIQUE problem is extremely hard to approximate Hastad (1996), i.e, finding

a clique of size within a n'-6 factor of the size of the maximum clique cannot be

achieved in time polynomial in n, for any constant e (unless P=NP). However

our reduction of CLIQUE to the optimal DR-planning problem is not a so-called

gap-preserving reduction (or L-reduction); thus how well this problem could be

approximated is still an open question.

The definition of solvability preservation is largely analogous to the case of DR-

solvers, but is additionally motivated by the following. In the case of DR-solvers,

one condition on a solvability preserving simplifier is that it preserves the ex-

istence of real solutions for certain iil-\-'f in- Here, in the case of DR-plans,

a natural choice is to correspondingly require that the simplifier does not map

wellconstrained subgraphs or overconstrained subgraphs to underconstrained

and viceversa. The DR-plan Q of G is l,11,.l,litf, preserving if and only if for all

A C GC, A is solvable and (A n S = 0 or A C S) > T,(A) is solvable. The

DR-plan Q of G is strictly ',11,.l,ilitf, preserving if and only if for all A c Gi, A is

solvable
nontrivial solvable B C S,, B = T,_IT_2...,Tj(Sj) for some j < i 1.

Next, we formally define DR-planners and their performance measures. An

algorithm is said to be a DR-planner if it outputs a DR-plan when given a solvable

constraint graph as input. As before, we consider DR-planners to be randomized

algorithms. A randomized DR-planner A is said to be valid, Nl.,/-dl,l;-,f preserving,

strictly ,-l,.,i;l;ifi, preserving, complete if and only if for every G every DR-plan

produced by A is valid, solvability preserving, strictly solvability preserving or

complete accordingly. The worst-choice approximation factor of a DR-planner A on

input graph G is the minimum of the approximation factors of all DR-plans Q of

G obtained by the DR-planner A over all possible random choices. The best-choice

approximation factor of the algorithm A on input graph G is the maximum of the

approximation factors of all the DR-plans Q of G obtained by the DR-planner A

over all possible random choices.

In addition to the above performance measures, we define three others that reflect

the Church-Rosser property, the ability to deal with underconstrained -ii1 -\--t' in-

as well as the ability to incorporate an input, design decomposition provided by the

designer.

A DR-planner is said to have the CI,,,,I, -Rosser property, if the DR-planner

terminates with a DR-plan irrespective of the order in which the dense subgraphs

S, are chosen.

A DR-planner A adapts to underconstrained constraint i pi,,/, G if every (partial)

DR-plan produced by A terminates with a set of solvable subgraphs Qi such that

each solvable subgraph Qi has no supergraph that is solvable, and moreover, no

subgraph of G that is disjoint from all of the Qi's is solvable.

A conceptual design decomposition P is a set of solvable subgraphs P,, which are

partially ordered with respect to the subgraph relation. A DR-planner A is said to

incorporate a design decomposition P, if for every DR-plan Q produced by A, the

corresponding sequence of solvable subgraphs Si embeds a topological ordering of P

as a subsequence recall that a topological ordering is one that is consistent with

the natural partial order given by the subgraph relation on P.

When a DR-plan incorporates a design decomposition P, the level of a cluster

Pi in the partial ordering of P can now be viewed as a priority rating which

specifies which component of the design decomposition has most influence over

a given geometric object. In other words, a given geometric object K is first

fixed/manipulated with respect to the local coordinate system of the lowest level

cluster P, E P containing K. Thereafter, the entire cluster P, can be treated as a

unit and can be independently fixed/manipulated in the local coordinate system of

the next level cluster Pj containing (the simplification of) P,, and so on.

Finally, we summarize how the above formal performance measures capture the

informally described characteristics in the set C given in the Section 2.1.5. The

property of being a general DR-planner refers to whether the method success-

fully terminates with a DR-plan in the general case, and reflects characteristic

(i); since we use constraint graphs which yield robust DR-plans that can be ob-

tained efficiently, the property of being a general DR-planner also reflects (iii)

and (viii); dealing with underconstrained graphs also reflects (i); incorporating

input, design decompositions reflects (v); validity influences the Church-Rosser

property and reflects (i) as well as (iv),(v), (vi); solvability preservation, strict

solvability preservation influence the Church-Rosser property and reflect (i), (ii),

(iv), (v); completeness is based on the criteria (ii) and (ix); worst and best choice

approximation factors are based on (ii) and complexity directly reflects (iii).

2.1.8 Two old DR-planners

We concentrate on two primary types of prior algorithms for constructing

DR-plans using constraint graphs and geometric degrees of freedom.

Note. Due to reasons discussed in Section 2.1.7, we leave out those graph rigidity

based methods for distance constraints in dimensions 3 or greater such as Tay

and Whiteley (1985); Hendrickson (1992) as well as methods such as Crippen and

Havel (1988); Havel (1991); Hsu (1996) since they are nondeterministic or rely on

symbolic computation, or they are randomized or generally exponential and based

on exhaustive search. Graph rigidity and matroid based methods for more specific

constraint graphs are discussed in Section 2.1.10 under the so-called Maximum

Matching based algorithms. Also, for reasons discussed in the introduction, we

leave out methods based on decomposing sparse polynomial systems. <

The first type of algorithms, which we call SR for constraint S ,,1i' Recognition

(e.g.Bouma et al. (1995); Owen (1991, 1993); Bouma et al. (1995), Hoffmann

and Vermeer (1994, 1995); Fudos and Hoffmann (1996b, 1997)), concentrates on

recognizing specific solvable subgraphs of known shape, most commonly, patterns

such as triangles. The second type, which we call MM for generalized Maximum

Matching (e.g. Ait-Aoudia et al. (1993); Pabon (1993); Latham and Middleditch

(1996); Kramer (1992)), is based on first isolating certain solvable subgraphs by

transforming the constraint graph into a bipartite graph and finding a maximum

generalized matching, followed by a connectivity analysis to obtain the DR-plan.

In this section, we give a formal performance analysis of SR, and MM based

algorithms -choosing a representative algorithm (typically the best performer) in

each class -using the performance measures defined in the previous section.

Informally, one major drawback of the SR and MM algorithms is their inability

to perform a generalized degree of freedom analysis. For example, SR would

require an infinite repertoire of patterns. In the case of spatial constraints, some

elementary patterns have been identified Hoffmann and Vermeer (1994, 1995). In

the case of extending the scope of planar constraint solvers, adding free-form curves

or conic sections requires additional patterns as well Hoffmann and Peters (1995);

Fudos and Hoffmann (1996a). Similarly, a decomposition of underconstrained

constraint graphs into wellconstrained components is possible for SR algorithms

but only subject to the pattern limitations. In many cases, MM algorithms will

output DR-plans with larger, nonminimal subgraphs Si that may contain smaller

solvable subgraphs. This affects the approximation factors adversely.

This inability to find general minimal dense subgraphs also affects their ability

to deal with overconstrained subgraphs that arise in assemblies, which is in turn

needed to perform constraint reconciliation Hoffmann and Joan-Arinyo (1998).

2.1.9 Constraint Shape Recognition (SR)

Consider the algorithm of Fudos and Hoffmann (1996b, 1997) which relies on a

following strong assumption (SR1):

all geometric objects in 2 dimensions have 2 degrees of freedom and all

constraints between them are binary and destroy exactly one degree of freedom.

Thus, in the corresponding constraint graph, the weight of all the edges is 1 and

of all the vertices is 2. Because of this assumption, the SR algorithm ignores the

degrees of freedom and relies only on the topology of the constraint graph.

Description of the algorithm. We give a terse description of the algorithm

-the reader is referred to Fudos and Hoffmann (1996b, 1997) for details. Our

description is meant only to put the algorithm into an appropriate DR-planner

framework that is suited for the performance analysis.

The algorithm consists of two phases. During Phase One, SR uses the bottom-

up iterative technique of Itai and Rodeh (1978): in the current graph Gi (where

G1 = G), specific solvable graphs (clusters) are found that can be represented as

a union of three previously found clusters that pairwise share a common vertex.

Such configurations of three clusters are called ti i ,ll, ll The vertices of Gi

represent clusters and edges of Gi represent constraints between clusters (initially

due to SRI, every vertex and every edge of G is a cluster. More specifically,

once a triangle formed by three clusters has been found, the three vertices in Gi

corresponding to these three clusters are simplified into one new vertex in the

simplified graph Gi+C. The edges of Gi+I are induced by the edges of the three old

vertices. This is repeated for k steps until there are no more triangles left. If there

is only one cluster left in Gk, then the algorithm terminates. Otherwise Gk serves

as an input to Phase Two of the algorithm.

Before we describe Phase Two, we note that a so-called cluster pui'i, Ci

corresponding to Gi is used as an auxiliary structure for the purpose of finding

triangles. Hence the partial DR-plan produced during Phase One of the algorithm

is of the form:

(G1, C1),... (Gk, Ck). The vertices of the cluster graph Ci correspond to

vertices of the original graph G1;

cluster-vertices in the graph GC; and

edges in G1 which have not been included into one of the cluster-vertices in

the graphs Gi_,... G1.

In particular, the first cluster graph C1 contains one vertex for every vertex

and every edge of G1. The edges of the cluster graph C, connect the vertices of

Gi that represent clusters to the original vertices from G1 that are contained in

these clusters. Due to the structure of cluster graphs, triangles in Gi are found by

looking for specific 6-cycles in the corresponding cluster graph Ci. These 6-cycles

consist of 3 cluster-vertices and 3 original vertices. The new cluster graph Cj+I

is constructed from Ci by adding a new vertex ci representing the newly found

cluster S,, and connecting it by edges to the original vertices from G1 that are in

the cluster S,. We also remove the three old cluster-vertices in C, which together

formed the new cluster S,. We note that this is the only way in which cluster-

vertices are removed from cluster graphs. In particular, situations may arise where

two clusters (that share a single vertex) are represented by the same vertex in the

graph Gi, but they are represented by distinct vertices in the cluster graph Ci.

During Phase Two, SR constructs the remainder of the DR-plan (Gk, Ck),..., (Gm, Cm).

First SR uses a global top-down technique of Hopcroft and Tarjan (1973) to divide

Gk into a collection of triconnected subgraphs. These subgraphs are found by

recursively splitting the original graph using separators of size at most 2. Thus the

triconnected components can be viewed as the leaves of a binary tree T each of

whose internal nodes corresponds to a vertex separator of size at most 2.

For every triconnected subgraph S in Gk, a new cluster vertex is created in

the cluster graph Ck: similar to Phase One, this vertex replaces all vertices that

represent existing clusters contained in S.

The next pair (Gk+i, Ck+1) in the DR-plan is found as in Phase One by

finding a triangle in Gk, or effectively locating a 6-cycle in Ck. However, triangles

in Gk have already been located in the course of constructing the tree T these

triangles were originally split by the vertex separators at the internal vertices.

Thus, if the original constraint graph G is solvable, the remainder of the DR-plan

(Gk+1, Ck+I), ... (Gm, Cm) is formed by repeated simplifications along a bottom up

traversal of the tree T.

If the original graph was underconstrained, then it is still possible to construct

a DR-plan of its maximal wellconstrained subgraphs, by introducing additional

constraints and making the original graph solvable. I.e, in order to complete the

bottom up traversal of the tree T in Phase Two, additional constraints need to be

introduced, to make the whole graph solvable. For details, see Fudos and Hoffmann

(1996b, 1997).

Example. Consider Figure 2-10. During Phase One, the triangles

ADE, ABE, BCE and CEF will be discovered and simplified as P. During

Phase Two, the remainder of the graph will be divided into triconnected subgraphs

P, PGKILN and OKHMJP, then PGKILN and OKHMJP are simplified and

finally the union of P, PGKILN and OKHMJP is simplified.

Defining the Simplifier Map. The same simplifier is used throughout

Phase One and Two: replace a subgraph Si consisting of a triangle of clusters in Gi

(or triconnected subgraphs during Phase Two), by one vertex in Gj+I representing

this triangle. The subgraph S, is found as a 6-cycle in the cluster graph Ci. The

cluster graph C,+i is constructed as described in Phase One.

DG D DG G
AE G FO F
AE G
C1 \Fo p 5 o M 0

0\
OM O O DG 0

Figure 2 10: The original, cluster graph and the simplified graphs

More formally, recalling the definitions in Section 2.1.7: let G be a constraint

graph; the first graph G1 in the DR-plan is the original graph G. Let Gi = (V, E)

be the current graph and let Si be a cluster found. Let A C G, A = (VA, EA).

Then the image of A under the subgraph simplifier T, is T,(A) = A, if the

intersection of A and Si is empty; otherwise T(A) = (VT,(A), ET,(A)) where VT,(A) is

the set of all vertices of A that are not vertices of Si plus a vertex ci that replaces

the cluster S,. The set of edges ET,(A) is a set of all edges of A that are not edges

of S,; the edges of EA that have exactly one endpoint in Si will have this endpoint

replaced by the vertex c1; and the edges of A that have both endpoints in Si are

removed.

Since the cluster Si in Gi is located by finding a 6-cycle in C1, we need to

describe how C+1 is constructed from C1, i.e, the effect that T, has on C, (formally,

T, is a map from (Gi, C,) to (Gi+l, C,+I)). To obtain Ci+i, we start with Ci and

G2 G3
(Cl- C2

Tvv y/ \^^
\0xii )/ ^
,k ~ ~ ~ ~ ~ \ /

C1
AE
DE
EB
AB

Figure 2-11:

C1 A
EB D
AB E
CB

C3
C2 A
CB D
CE E
FC

B B B
C CE FE

S T T2

Action of the simplifier on Gi and Ci during Phase One

first add a new vertex ci representing the cluster Si, which is connected by edges

to all the original vertices that it contains. Finally, vertices in Ci that represent

clusters entirely contained in S, are removed, and edges adjacent to these vertices

are also removed.

Figures 2-11 and 2-12 illustrate the action of the subgraph simplifier T, on

both Gi and Ci.

A Further Concession for SR. Observe that the SR algorithm is not a

general DR-planner when input geometric constraint graphs do not comply with

assumption SRI. For example, for the graph shown in Figure 2-13, during Phase

One, SR would not find any triangles and during Phase Two, it would conclude

that the graph is underconstrained, even though the graph is wellconstrained.

SR implementations may remedy some cases similar to this one. For instance,

weight 2 edges to weight 2 vertices often arise from incidence constraints between

geometric elements of same type (two coincident points or two coincident lines),

and such cases are easily accounted for by working with equivalence classes of

such vertices. Moreover, most planners will evaluate the density of a graph before

G1

I\^

KT8 TTTT G9

L M

C6 C7 C8
P D P A P D
DG I Q D
DI G KH L K
DN K KO K
IK L KJF R
IL N JF O
LN A OF

Figure 2-12: Action of the simplifier during Phase Two

1 22
1 2 21
l2 2

Figure 2-13: This solvable graph would not be recognized as solvable by SR

announcing under or overconstrained situations. Thus, an implementation of the

SR algorithms of Fudos and Hoffmann (1996b, 1997) may or may not construct a

DR-plan for the graph of Figure 2-13 depending on the original problem statement.

It is clear that such heuristics enlarge the class of solvable graphs, but fall short of

decomposing all solvable constraint graphs.

However, even when input graphs do comply with SRI, and SR checks overall

density of a graph, SR could still mistake a graph that is not solvable for solvable.

Figure 2-14 shows an example which the SR algorithm may process incorrectly:

since the graph contains no solvable triangles, SR proceeds immediately to Phase

Figure 2-14: Weight of all vertices is 2, weight of all edges is 1

Two. During Phase Two the triconnectivity of the entire graph could be erro-

neously construed to mean that the graph is solvable. In fact, the graph does

have density -3, which superficially seems to support such a conclusion. However,

the graph is certainly not wellconstrained: the eight vertices on the right form a

subgraph of density -2, that is, an overconstrained subgraph. Moreover, if this is

overconstrained subgraph is replaced a subgraph of density -3, the resulting graph

has density -4 uncovering that it is not welloverconstrained either, and therefore

not solvable.

Figure 2-14 demonstrates that the density calculation heuristics is insufficient

to determine the existence of minimal dense subgraphs of triconnected constraint

graphs. What is needed is a general algorithm for finding minimal dense subgraphs.

In order to give a performance analysis of the SR algorithm for those classes

of constraint graphs where it does produce a satisfactory DR-plan, we make a

strong concession (SR2) that: only "triangular" structures are "acceptable" for

the remainder of this section. I.e, we modify the definitions of validity, solvability

preserving, strictly solvability preserving and completeness by replacing the words

"solvable subgraph" by the following recursive definition: "either a subgraph that

can be simplified into a single vertex by a sequence of consecutive simplifications of

triangles of solvable subgraphs, (using the simplifier described in Section 2.1.9), or

a vertex, edge or triconnected subgraph'.

Performance Analysis. In this section, we analyze the SR algorithm with

respect to the various performance measures defined in Section 2.1.7.

Claim 3 Under the concession SR2, the SR :,li.,' ifll, is valid.

Proof We will show that every pre-image of the "solvable" cluster Si found

at the ith iteration of the SR algorithm is also "solvable," using the stricter

SR2 definition of "solvable". It can be easily checked that the other requirements

necessary for validity from Section 2.1.7 also clearly hold.

Let Si be a cluster that has been found in Gi by the SR algorithm (by locating

a 6-cycle in C1). Then from the description of the algorithm, and by assertion SR2,

there is a sequence of simplifications, say Ti+i, ... Tm whose composition maps S5

to a single vertex. For every pre-image A C Gj A = T .... T-(S), for j < 1,

the composition Tm o... T, o T,_i o ... o Tj will map A to a single vertex, hence A is

"solvable". I

Claim 4 The SR :,li., ill,, is strictly ,11.,lili'fd, preserving under the concession

SR2, and therefore is le.,el ili'fd, preserving as well.

Proof Let A be a "solvable" subgraph, i.e there is a sequence of simplifi-

cations A, ..., Am such that A = A1 and Am consists of only one vertex. If the

intersection of A and the currently found cluster Si is empty then Ti(A) = A and it

remains "solvable". If this intersection is not empty, then a subgraph B = A n S is

simplified into a new cluster vertex c, = T,(S,). We need to show that in this case

Ti(A) remains "solvable" as well. Suppose that some subgraph of B is a vertex of

a 6-cycle in one of the cluster graphs formed during the simplification A, ..., Am.

Then clearly the new cluster vertex c, could perform the same function: i.e, it

could also be a vertex in that 6-cycle of one of the cluster graphs formed during the

simplification A, ..., Am. Also if A was triconnected, then so is T,(A). This proves

that there is a sequence (essentially a sequence A1,... Am modified by replacing

A B

Figure 2-15: Two triconnected subgraphs not composed of triangles

Figure 2-16: Solvable graph consisting of n/3 solvable triangles

B by ci) which terminates in a single vertex, thus demonstrating the "density" of

T,(A). I

Claim 5 Even under concession SR2, SR is not complete.

Proof We will show that there are cases when the SR picks large, nonminimal

"solvable" subgraphs Si to simplify, ignoring smaller "solvable" subgraphs of Si.

Consider Figure 2-15. If subgraphs A and B are triconnected but not com-

posed of triangles, then they are "solvable," but since the entire graph is tricon-

nected neither A nor B will be simplified by SR. However, since the whole graph

A U B is triconnected, it will chosen by SR as S1 during Phase Two. I

Claim 6 The (worst and) best-choice approximation factor of SR under concession

SR2 is at most 0(1).

Proof Consider Figure 2-16. The entire graph consists of n triangles. During

Phase One SR will successfully locate and simplify each one of them. However,

during Phase Two SR will not be able to decompose the entire solvable graph

into smaller solvable ones (since entire graph is triconnected) so the size of the

corresponding DR-plan is O(n). On the other hand, the optimal DR-plan would

simplify neighboring pairs of triangles, one pair at a time, thus the optimal size is a

small constant. I

Next, we consider three other performance measures discussed in Section 2.1.7.

Claim 7 Under the concession SR2, the :ii.,'i Illi, SR adapts to underconstrained

Proof Suppose that the graph G is underconstrained. Let A be a "solvable"

subgraph that is not contained in any other "solvable" graph. Since Phase Two of

the SR algorithm is top-down, either SR will find A and simplify it as one of the

S,'s, or A n Si 0 for one of the S,. In either case, SR adapts to G. I

Observation 1 Under the concession SR2, the SR lw/ ,i;t it,,. has the CI,,,1, 1,-

Rosser property, since the new ,in ,1, Gi+I remains ,-*,IlI,'1 if Gi is ,',*I, a 1,",

i,, -i,,lless of the choice of the 'Il,*Il',1," Si that is simplified at the ith stage.

Claim 8 Under the concession SR2, the SR .l'i't ill,,in can incorporate design

decompositions P if and (,or1,1 if P fulfills the following requirement: ',,, pair of

,u,,hll,.'" .,,,.lit,,al,, Pk and Pt in P satisfy Pk C Pt or Pt C P~ or Pk n Pt contains

no edges.

Proof For the 'if' part we consider the most natural modification of the

original SR algorithm, and find a topological ordering O of the given design

decomposition P which is a set of "solvable" subgraphs of the input graph G,

partially ordered under the subgraph relation such that O is embedded as a

subplan of the final DR-plan generated by this modified SR algorithm; i.e, O

forms a subsequence of the sequence of "solvable" subgraphs Si, whose (sequential)

simplification gives the DR-plan.

We take any topological ordering O of the given design decomposition P and

create a DR-plan for the first "solvable" subgraph P1 in P. I.e, while constructing

the individual DR-plan for P1, we "ignore" the rest of the graph. This individual

DR-plan induces the first part of the DR-plan for the whole graph G. In particular,

the last graph in this partial DR-plan is obtained by simplifying P1 using the

simplifier described in 2.1.9 (and treating P1 exactly as SR would treat a cluster

Sj found at some stage j). Let Gi be the last graph in the DR-plan for G created

thus far. Now, we consider the next subgraph P2 in the ordering 0, and find an

individual DR-plan for it, treating it not as a subgraph of the original graph G, but

as subgraphs of the simplified graph Gi. This individual DR-plan is added on as

the next part of the DR-plan of the whole graph G.

The crucial point is that the simplification of any subgraph, say Pk, will not

affect any of the unrelated subgraphs Pt, t > k, unless Pk C Pt. This is because,

by the requirement on P, Pk and Pt share no edges. Therefore, when the cluster

vertex for Pk is created, none of the clusters inside Pt is removed.

The process of constructing individual DR-plans for subgraphs in the

decomposition P and concatenating them to the current partial DR-plan is

continued until a partial DR-plan for the input graph G has been produced, which

completely includes topological ordering O of the decomposition P as a subplan.

Let G, be the last graph in this partial DR-plan. The rest of the DR-plan of G is

found by running the original SR algorithm on G, and the corresponding cluster

graph C,.

For the 'only if' part, consider Figure 2-17. Let P = {Po, P, P2}, where Po =

ABD, P1 = BCD, P2 = ABCD. Then SR cannot produce any DR-plan that would

incorporate P as subplan. I

2.1.10 Generalized Maximum Matching (MM)

Consider the algorithms of Ait-Aoudia et al. (1993); Pabon (1993); Kramer

(1992); Serrano and Gossard (1986); Serrano (1990) as well as graph rigidity and

matroid based methods for distance constraints in 2 dimensions Hendrickson

(1992), Gabow and Westermann (1988), Imai (1985) as well as more general

constraints Sugihara (1985) all of which more or less use (generalized) maximum

C
Pc
P

B D\

AA A B D B C D A B D C

Figure 2-17: Constraint graph, intended and actual decompositions

matching (or equivalent maximum network flow) for finding solvable subgraphs

in specialized geometric constraint graphs. These methods either assume that

the constraint graph has zero density or they reduce the weight of an arbitrarily

selected set of vertices in order to turn the graph into one of zero density.

In this thesis we will analyze what we consider to be the most general algo-

rithm of this kind Pabon (1993) (it generalizes the algorithm of Ait-Aoudia et al.

(1993), although the latter provides a more complete analysis), supplemented (by

us, as -I-. -tI i1 by a reviewer) with a method from Hendrickson (1992).

Note that while the algorithm of Pabon (1993) both locates solvable subgraphs

and describes how to construct a corresponding DR-plan, Sugihara (1985), Hen-

drickson (1992), Gabow and Westermann (1988) only describe algorithms that

allow to verify whether a given graph is solvable, but do not explicitly state how to

use these algorithms for successively decomposing into small solvable subgraphs, i.e,

for constructing DR-plans.

While neither of the previously known algorithms analyzed in this thesis

perform well according to our previously defined set of criteria, later, we will

describe our network flow based Frontier Algorithm that improves the performance

in several key areas.

Description of the Algorithm. As in the case of SR, we give a terse de-

scription of the MM algorithm -the reader is referred to Ait-Aoudia et al. (1993);

c

A D
A D

B
B

Figure 2-18: Original graph and it decomposition

Pabon (1993) for details. Our description is meant only to put the algorithm into

an appropriate DR-planner framework that is suited for the performance analysis.

By using maximum flow, the input constraint graph is decomposed into a

collection of subgraphs that are strongly connected. The flow information also

provides a partial ordering of the strongly connected subgraphs representing the

order in which these subgraphs should be simplified. This ordering in turn specifies

the DR-plan. It is important to note that these strongly connected components

represent a sequence of solvable subgraphs of the original constraint graph, only if

the input constraint graph is wellconstrained.

Consider Figure 2-18. All the vertices have weight 2, all the edges have weight

1, and the geometry is assumed to be in 2 dimensions (i.e geometry dependent

constant D = 3). The output of the MM algorithm is:

a set of vertices whose total weight is reduced by 3 units, say weight of

vertices A,B and C to be reduced by one unit each, (this corresponds to fixing

3 degrees of freedom the number of degrees of freedom of a rigid body in 2

dimensions);

two strongly connected components ABC and D;

the DR-plan, i.e, the information that the -ii -\--t' il represented by ABC

should be simplified/solved first and then its union with D should be simpli-

fied/solved.

AC A AC A

AB B1S B AB 1 B
BC Ck BC Sink

CD 1 CD (
BD (1 BD (

Figure 2-19: Modified bipartite graph and maximum flow in this graph

In order to produce such an output, the MM algorithm first constructs the

network X = (VX, EX) corresponding to the original weighted constraint graph

G = (V, E) (after the weights of A, B, C were reduced by one unit each). The set

of vertices VX is the union of VX1 and VX2 where the vertices in VX1 correspond

to the vertices in V, vertices in VX2 correspond to the edges in E. An edge

ex E EX would be created between vxl E VX1 and vz2 E VX2 if the vertex in

V corresponding to the vxl is an endpoint of an edge in E corresponding to vz2.

The edge ex has infinite capacity. All the vertices in VXi are connected to the

special vertex called Sink. The capacity of connecting edges is equal to the weight

of the corresponding vertices in V. For example, the left half of Figure 2-19 shows

the bipartite graph corresponding to the constraint graph of Figure 2-18. Next

the maximum flow in the network X is found, with vertices in VX2 being source

vertices of capacity equal to the weights of the corresponding edges in E. See right

half of Figure 2-19 (thick edges have nonzero flow).

The maximum flow found in X induces a partition of the original graph G

into a partially ordered set of strongly connected components -giving a sequence

of solvable subgraphs of G, provided G is wellconstrained -as in Figure 2-18,

according to the following rules: if the flow in X from the vertex z E VX2 that

is connected to the vertices x, y E VX1 is sent toward x, then in the graph G,

the edge corresponding to z (between vertices x and y) becomes an oriented edge

directed from y to x. If the flow from z is sent toward both x and y, then the edge

C

AB

/ BD S2ink

2
BC

A DC

Figure 2-20: Constraint graph and network flow with 3 extra flow units at AB

is bidirected toward both x and y. (Recall that a strongly connected component S

in this directed graph is a subgraph such that for any two vertices a, b E S, there

is an oriented path from a to b.) The partial ordering of components is induced as

follows. Let K and L be two strongly connected components in the oriented version

of G. If all the edges between vertices of K and L are pointing toward L, then L

should be simplified after K.

Defining the Simplifier Map. We capture the transformations performed

by the MM DR-planner described above, by describing its simplifier maps (recall

the definitions in Section 2.1.7).

Let G = (V, E) be the geometric constraint graph. Denote by G1 = (Vi, 'E =

E) the directed graph after weights of some vertices have been reduced as described

above and a partial ordering of strongly connected components has been found.

First S1 C G1 is located such that S1 is strongly connected. Then S1 is simplified

into the vertex vi of weight zero. The other vertices of G1 remain unchanged.

Edges of G1 that had both endpoints outside of S1 are unchanged, edges that had

both endpoints in S1 are removed, edges that had exactly one endpoint in S1 have

this endpoint replaced by vl. In the next step, C2 the next strongly connected

component in a topological ordering of the components in G2 = T1(G1) is located.

The subgraph S2 is set to be {vl} U C2. Then S2 is simplified into the vertex v2

A E
12 2

C

B D
2 2)

Figure 2-21: DR-plan depends on the initial choice

of weight zero. This process is continued until Gk consists of one vertex. The

simplifier maps are formally defined as follows.

-T(S,) = vi where vi is the vertex in G1+I of weight 0.

If B C G, B n Si = 0, then Ti(B) = B.

If B C G,, B n S, # 0, then the image of B, T,(B) = {vi} U (B \ S,).

Redefining Solvability.

Claim 9 The MM ,-i/., t ifllr is a not a ./, u ,l DR-planner.

Proof While the MM algorithm can correctly classify as solvable and de-

compose wellconstrained and welloverconstrained graphs and correctly classify

underconstrained graphs that have no overconstrained subgraphs as being un-

solvable, it is unable to correctly classify an underconstrained graph that has an

overconstrained subgraph. Consider Figure 2-21, the weight of all the edges is

1, of the vertices as indicated. Graph ABCDE has density -3, it contains over-

constrained subgraphs AC, BC, CD, CE, and removal of say edge AC will result

in ABCDE becoming non-dense, hence ABCDE is not welloverconstrained and

not solvable. The MM algorithm may or may not detect this, depending on the

initial choice of vertices whose weights are to be reduced. Suppose that the weight

of the vertex E was reduced by 1 and the weight of the vertex D by 2. Then the

corresponding maximum possible flow f and the corresponding decomposition

into strongly components are shown in Figure 2-22. Note that it is impossible to

simplify the strongly connected component A, and thus there can be no DR-plan

AC A A /.
AB B1
BC Sink

CED

Figure 2-22: Maximum flow and decomposition given the bad initial choice

Figure 2-23: Decomposition given the good initial choice

for this initial choice of vertices for reducing weights. On the other hand, if the

weight of A had been reduced by 2 and weight of D by 1 then the maximum flow

is larger than f (in fact, no choice of vertices for weight reduction can give a larger

flow) and it can be checked that MM would yield a DR-plan, see Figure 2-23

(S1 = A, S2 = AC, S3 = ABC, S4 = ABCD, S5 = ABCDE). I

Thus MM runs into problems in the presence of overconstrained subgraphs, unless

G happens to be welloverconstrained, which cannot be apriori checked without

relying on an algorithm for detecting overconstrained subgraphs and replacing

them by wellconstrained ones. Therefore it is necessary and sufficient to locate

overconstrained subgraphs and replace them by wellconstrained ones, in order to

guarantee that MM will work.

The author of Pabon (1993) does not specify how to do this. The following

modification similar to that of Hendrickson (1992) could be used, as was -I-.- -t Id

by a reviewer, and completed here.

First a maximum flow in the unmodified network X (i.e where vertex weights

are not reduced) is found. After that, for every source vertex v E VX2, except one

vertex vm the following two steps are repeated.

Three extra units of flow are sent from v, possibly rearranging existing flows

in X.

These 3 units of flow are removed, without restoring original flows.

For vertex v,, 3 units of flow are sent but not removed.

For example, consider Figure 2-20. The constraint graph is shown on the

left, weights of all vertices is 2, the weights of all edges is 1. The resulting network

flow is shown on the right, assuming that AB was the last vertex v,. The extra 3

units of flow and their destination vertices A and B are marked by asterisks. This

particular flow induces a weight reduction of A by 1 unit and of B by 2 units.

This modification identifies overconstrained subgraphs since it is impossible

to send all 3 extra units from at least one edge of an overconstrained graph.

These overconstrained graphs have to be modified by the designer to become

wellconstrained, and the flow algorithm is run again.

This is important because otherwise the DR-algorithm cannot proceed, since

in underconstrained graphs with overconstrained subgraphs, strongly connected

components do not necessarily correspond to a sequence of solvable subgraphs.

To reflect the modification above, we make the following concession for MM,

(MM1): the input constraint graph G has no overconstrained subgraphs in G. A

subgraph A C G is "solvable" if it has zero density, after the vertices for weight

reduction by D are chosen (these can be chosen arbitrarily, provided there are no

overconstrained subgraphs).

Performance Analysis. In this section, we analyze the MM algorithm with

respect to the various performance measures defined in Section 2.1.7.

Note that despite the concession MM1, the MM algorithm has the following

drawbacks: the DR-plan is uniquely determined, once vertices whose weights are

reduced are chosen. For example, in Figure 2-20 after the weight of A is reduced

by 1 and the weight of B by 2 units, solvable subgraphs to be simplified are

S1 = {A, B}, S2 = {TI(S), D}, S3 = T2(S2), C}. While the subgraph BCD is also

solvable (if say initially weights of vertices B and C were reduced instead of A and

B), it cannot be chosen as one of the Si after weights of A and B are reduced.

This causes MM to have bad worst-choice and best-choice approximation

factors and to be unable to incorporate general designer decomposition.

The following is straightforward from the description of the simplifiers.

Claim 10 Under the concession MM1, the MM :ldi./ ilim is a valid DR-planner.

Claim 11 Under the concession MM1, the MM :li/.,, ill/,, is strictly 11,.,ldilif,

preserving (and therefore ,' 1lift e, preserving).

Proof Suppose that a subgraph A of the input graph G is "solvable," i.e

d(A) = 0. Let Si be the "solvable" subgraph to be simplified at the current

stage. Let B = A n S, C = A \ B. Since we assume that G does not contain

any over-constrained subgraphs, d(B) < 0 and d(A U S) < 0 => d(A U S) =

d(A) + d(Si) d(B) < 0 = 0 + 0 d(B) < 0 = d(B) = 0. Thus d(TJ(A))

d(T,(C)) + d(T,(B)) = d(C) + 0 = d(A) d(B) = 0, therefore T,(A) is also solvable.

I

Claim 12 Under the concession MM1, the MM ,l1./,, illcp is complete.

Proof Let A be a proper solvable subgraph of the Si, i > 2. Since A is

solvable, there cannot be any edges outside of A pointing toward A (this is because

there is no room for flows of "outside" edges toward the vertices of A). Recall that

Si is the union of Ci U {v,_i}, where Ci is the first strongly connected component

-3 3
hn2 3 R 2
Ln/2l1 R n/2-1

L2 R2
2 2

Figure 2 24: Bad best-choice approximation

in the topological ordering of the components at the stage i, and vi-1 is the

simplification of S,_1. However, unless A = v_l1, the first strongly connected

component at this stage would have been A \ {vi-l}, not Ci, which contradicts the

choice of C, at stage i. I

Claim 13 Under the concession MM1, the best-choice (and worst-choice) approxi-

mation factor of MM is at most 0(l).

Proof To prove the bound on the best-choice approximation factor consider

Figure 2 28. The left and right columns contain n/2 vertices each. The weights of

all the vertical edges are 2, the weights of all other edges are 1, the weights of the

vertices are as indicated, and the geometry dependent constant D = 3.

Note that all solvable subgraphs in Figure 2 28 could be divided into 3 classes.

The first class consists of the subgraphs CLL2; CL1L2L3; ... ; CL1L2 ... L /2-1L /2

The second class consists of the subgraphs CRR2; CR1R2R3; ... ; CR1R2... R /2-1R,/2

The third class contains the solvable subgraphs that contain both L and R vertices.

There is only one element in this class the entire graph CL1L2... L /2R1R2... R,~/2

There is an optimal DR-plan of constant size that takes S1 = CL1L2, S2 =

S1 U L3,... ,Sn/2-1 = Sn/2-2 U Ln/2. After that it takes Sn/2 = CR1R2, S/2+1 =

Sn/2 UR3, ... Sn = S-_1 U R,/2. Finally it takes S,+I = Sn/2-1 U S,.

However all DR-plans found by MM will have size O(n). The reason for this

is that MM is unable to simplify solvable subgraphs on the left of the Figure 2-28

independently from the solvable subgraphs on the right. More formally let S1

be the first subgraph simplified by MM under some DR-plan Q. If S1 belongs to

the third class of solvable subgraphs then size of Q is O(n). Otherwise, without

loss of generality we can assume that S1 belongs to the first class. According to

the definition of MM, the simplification of S1 is a vertex vi of weight 0. After

this simplification any strongly connected component that contains some Ri

should also contain all of the R1... R/2. Hence there is an S, in Q such that

RiR2 ... R n/2 C Si. Hence the size of Q is O(n). I

Next, we consider three other performance measures discussed in Section 2.1.7.

Claim 14 Under the concession MM1, a simple modification of the MM :d,,wi I;fli

is able to adapt to underconstrained vi,b,,,l' This modification, however, increases

the, ',w,1, pb if,, by a factor of n.

Proof Suppose that the graph G is underconstrained. Consider the max-

imum flow found by MM in the network X corresponding to G, as described in

subsection 2.1.10. There are two cases.

The first case is when the last vertex vm in X corresponds to an edge of some

solvable subgraph A1. Then all vertices vi in X corresponding to vertices of A1

have their capacities completely filled, since A1 is solvable. On the other hand,

at least one vertex v of G will not have its capacity filled completely, since G is

underconstrained. Let W be the set of such vertices v. The new modification of

MM would proceed by removing all vertices of X corresponding to vertices of W

and edges adjacent to W, as well as flows originating at such edges. Once this is

done, a new set W is recomputed and removed, until all remaining vertices of X

that represent vertices of G have their capacities filled completely. These vertices

comprise a solvable subgraph A1 such that no supergraph of A1 is solvable. Once

the subgraph A1 (and its DR-plan Qi) is found, it could be removed from G and

the process applied recursively to G\A1 to find a solvable subgraph A2, its DR-plan

Q2 and so on.

The second case is when the last vertex vm in X is not contained in any solv-

able subgraph. Then constructing and removing sets W as described above will

completely exhaust G, without finding any solvable graphs. The new modification

of MM would proceed by finding another maximum flow in network X correspond-

ing to G such that the last vertex v^ is different from vn. Again two cases are

considered for vertex vo. If all voE G are not contained in any solvable subgraph,

then G does not contain any solvable subgraph and the process terminates.

Note that this modification offhand increases the complexity of MM by a

factor of n, and there does not seem to be any obvious way to prevent this factor.

This modification of MM outputs a DR-plan Q = Q1,... Qk for set of

solvable subgraphs A1,... Ak such for any i no supergraph of Ai is solvable and

there are no solvable subgraphs B C G such that B n Aj = 0, Vj. Thus this

modification of MM is able to adapt to underconstrained graphs. I

Observation 2 (i) Under the concession MM1, MM has the Cl,i1,, I-R .-. r

property since i;,!,lifi,;i.'i any Si that is solvable at the current stage preserves the

.1, ,.si;i of the whole intol, Gi+l (i.e Gi+l is solvable if and (orl,, if Gi is).

(ii) Under the concession MM1, MM is able to incorporate a design decomposition

P specified by the designer if and ol.1,1 if for every Pk, Pt E P such that Pk n Pt # 0

either Pk C Pt or Pt C Pk .

Proof is similar to the corresponding proof for SR algorithm in Claim 8.

2.1.11 Comparison of Performance of SR and MM

Next, we give a table comparing the SR and MM DR-planners with respect

to the performance measures of Section 3. "Uii. u-tu refers to the ability to

deal with underconstrained graphs, "Design decomp, -it i ii refers to the ability to

incorporate design decompositions specified by the designer.

The superscript '*' refers to a narrow class of DR-plans: those that require

the solvable -il1 i-\--' iri S, to be based on triangles or a fixed repertoire of pat-

terns. The superscript 't' refers to results that were not true for the original MM

algorithm developed by (Ait-Aoudia et al. (1993); Pabon (1993)) and proved in

this thesis through a modification of MM described in Section 2.1.10. The super-

script < refers to a restricted class of graphs in which there are no overconstrained

subgraphs. It also refers to a further modification of MM described in Claim 14,

which however, increases the complexity by a factor of n. The superscript 'o' refers

to strong restrictions on the design decompositions that can be incorporated into

DR-plans by SR and MM.

2.1.12 Analysis of Two New DR-planners

We present two new Decomposition-Recombination (DR) planning algorithms

or DR-planners. The new planners follow the overall structural description of a

typical DR-planner, based on the DR-solver S given in previous sections. Fur-

methods such as SR (shape recognition) and MM (generalized maximum matching)

that were analyzed and compared previously. In particular, those methods as well

Performance measure SR MM
Generality No Yest
Underconstr. No(Yes*) Yest,
Design decomposition No(Yes*o') No(Yest0)
Validity No(Yes*) Yest
Solvability No(Yes*) Yest
Strict solvability No(Yes*) Yest
Complete No(No*) YesT
Worst approximation factor 0 (0(Q)*) O(i)t
Best approximation factor 0 (0(6)*) O(i)
Church-Rosser No(Yes*) Yest
Complexity 0((m + n)2) O(n(m + n))t

as the new planners are based on degree of freedom analysis of geometric constraint

It should be noted that the SR and MM based algorithms Bouma et al. (1995);

Owen (1991, 1993); Bouma et al. (1995); Hoffmann and Vermeer (1994), Hoffmann

and Vermeer (1995); Latham and Middleditch (1996); Fudos and Hoffmann

(1996b, 1997), Ait-Aoudia et al. (1993); Pabon (1993); Kramer (1992); Serrano

and Gossard (1986); Serrano (1990), were being developed even as the issue

CAD/CAM -was still in the process of crystallization.

In contrast, our development of the new DR-planners is systematically guided

by the new performance measures, to closely reflect several desirable characteristics

C of DR-planners for CAD/CAM applications.

An important building block of both the new DR-planners is the routine used to

isolate the solvable -i l.-\'.-t' i-, S, at each step i. In both new DR-planners, the

solvable i1 -\-ti ili/subgraph Si is isolated using an algorithm developed by the

authors based on a subtle modification of incremental network flow: this algorithm,

called "Algorithm Dense," first isolates a dense subgraph, and then finds a minimal

dense subgraph inside it, which ensures its solvability. The interested reader is

referred to earlier papers by the authors: Hoffmann et al. (1997) for a description

as well as implementation results, and Hoffmann et al. (1998) for a comparison

with prior algorithms for isolating solvable/dense subgraphs. Here, we shall only

note several desirable features of Algorithm Dense.

(a) A useful property of the dense subgraph G' found by the Algorithm

Dense (in time O(n(m + n))) is that the densities of all proper subgraphs are

strictly smaller than the density of G'. Therefore, when G' corresponds to a

wellconstrained -i l.--t\i i" then G' is in fact minimal, and hence it is unnecessary

to run the additional steps to obtain minimality. Ensuring minimality crucially

affects completeness of the new DR-planners.

(b) A second advantage of Algorithm Dense is that it is much simpler to

implement and faster than standard max-flow algorithms, This was borne out

by our C++ implementation of Algorithm Dense both for finding dense and

minimal dense subgraphs. By making constantD a parameter of the algorithm, our

method can be applied uniformly to planar or spatial geometry constraint graphs.

Furthermore, the new algorithm handles not only binary but also ternary and

higher-order constraints which can be represented as hyperedges.

(c) A third specific advantage of our flow-based algorithm for isolating dense

subgraphs (solvable -ll -\ --t' in-t) is that we can run the algorithm on-line. That is,

the constraint graph and its edges can be input continuously to the algorithm, and

for each new vertex or edge the flow can be updated accordingly. This promises a

good fit with interactive, geometric constraint solving applications, i.e, the criterion

(viii) of C.

As will be seen below the main difference between the two new planners, however,

lies in their simplifier maps T,, i.e, the way in which they abstract or simplify a

solvable subgraph S, once it has been found.

Comparison. As a prelude to the ,it.l\ -li of the new DR-planners Condense

and Frontier we give a table below which extends the comparison between the SR

and MM to the new DR-planners Condense and Frontier.

The "complexity" entries for the 2 new DR-planners are directly based on

the complexity of a building block Algorithm Dense (briefly discussed above) for

isolating minimal dense subgraphs Si. "Under-o cni- refers to the ability to deal

with underconstrained systems, "Design-dec" refers to the ability to incorporate

design decompositions specified by the designer, "Solv." and "Strict solv." refer

to (strict) solvability preservation,

best choice approximation factor.

\\WV -t and Best ;I1I1'lc.:

refer to the worst and

Note. The variable s in the

n, plus the number of edges,

complexity expressions denotes the number of vertices,

m, of the constraint graph. Recall that D refers to the

number of degrees of freedom of a rigid object in the input geometry (in practice,

this could be treated as a constant). <

The superscripts '*' and '+' refer to narrow classes of DR-plans: those that

require the solvable -1il -\--' -t'i Si to be based on triangles or a fixed repertoire

of patterns, or to represent rigid objects that fixed or grounded with respect to

a single coordinate system. The superscript 'f' refers to results that were left

unproven by the developers of the MM based algorithms (Ait-Aoudia et al. (1993),

Pabon (1993)) and proved in this paper through a crucial modification of MM

described previously. The modification also results in the improvement of the

complexity of the best MM algorithm to O(n(s = n + m)).

Perf. meas. SR MM Condense(new) Frontier

Generality No Yest Yes Yes

Under-const No(Yes*) No Yes Yes

Design-dec No(Yes*') No No(Yes) Yes

Validity No(Yes*) Yes+ Yes Yes

Solv. No(Yes*) Yes+ Yes Yes

Strict solv. No(Yes*) Yes+ No Yes

Complete No(No*) No Yes Yes

Worst approx. 0 (O()) 0(1) O(1) O(1)

Best approx. 0 (O(Q)*) O()t O(1) 0(1)

Church-Rosser No(Yes*) Yest Yes Yes

Complexity O(s2) O(nD+1s)t O(n3s) O(n3s)

(d

(b

Figure 2-25: Sequential extension

The superscript 'o' refers to a strong restriction on the design decompositions

that can be incorporated into DR-plans by SR and the new DR-planner Condense.

In fact, the other new DR-planner Frontier also places a (however, much weaker)

restriction on the design decompositions that it can incorporate, as will be shown

later.

2.1.13 The DR-planner Condense and its Performance

This DR-planner was sketched by the authors already in Hoffmann et al.

(1998). The authors' flow-based algorithm discussed above Hoffmann et al. (1997)

is applied repeatedly to constraint graphs to find minimal dense subgraphs or

clusters (which we know to be generically solvable containing more than one vertex.

DR-planner Condense consists of two conceptual steps. A minimal dense cluster

can be ", ,'ifidllh1 extended under certain circumstances by adding more geometric

objects one at a time, which are rigidly fixed with respect to the cluster. After

a cluster has been thus extended, it is then simplified into a single geometric

object, and the rest of the constraint graph is searched for another minimal dense

subgraph. The following example illustrates sequential extension. Consider the

constraint graph G of Figure 2-25. We assume that all vertices have weight 2 and

all edges have weight 1. The geometry-dependent constant D = 3. The vertex

set {a, b} induces a minimal dense subgraph of G which could be chosen by DR-

planner Condense as the initial minimal dense cluster, which could be extended

sequentially by the vertices c, d, e, f, one vertex at a time, until it cannot be

3c' 332Z 2

22 2

Figure 2-26: Sequence of simplifications from left to right

extended any further. The resulting subgraph is called an extended dense ,,1I',I,,,I

or cluster.

The simplification of an extended cluster is taken to be a single geometric object

with D degrees of freedom. This is done as follows: an extended cluster A is

replaced by a vertex u of weight D; all edges from vertices in A to a vertex w

outside A are combined into one edge (u, w), and the weight of this induced edge

is the sum of the weights of the combined edges. After the simplification, another

solvable subgraph is found, and the process is continued until the entire graph is

simplified into a single vertex.

This is illustrated by the sequence of simplifications of Figure 2-26. Initially

all vertices have weight 2, all edges have weight 1. The vertices connected by

the heavy edges constitute minimal or sequentially extended clusters. After four

simplifications the original graph is replaced by one vertex.

Defining Subsystem Simplifiers. We capture the transformations per-

formed by the DR-planner Condense by describing simplifier maps. Let G be the

input constraint graph; the first graph G1 in the DR-plan is the original graph G.

Let Gi = (V, E) be the current graph and let Si be a cluster found at the current

stage. Let A be any subgraph of Gi. Then T,(A) is defined as follows.

If A n S, = 0 then T,(A) = A.

If A n S 0 then T,(A) = (VTA, ETA) where VTA is the set of all vertices of

A that are not vertices of Si plus one vertex c, of weight D which represents

the simplification of the cluster Si. The set of edges ETA is formed by

removing edges with all endpoints in Si, and combining edges with at least

one endpoint outside S,, (as well as their weights) as described earlier in this

section.

Performance Analysis. In this section, we analyze the DR-planner

Condense with respect to the various performance measures that were defined in

proceeding sections.

Claim 15 Condense is a valid DR-planner with the CI,,,,, l-Rosser property. In

addition, DR-planner Condense finds DR-plans for the maximal solvable ol,,1iI,'l,

of underconstrained i, ,,,l,

Proof If the graph G is not underconstrained, then it will remain so after

the replacement of any solvable subgraph by a vertex of weight D, i.e, after a

simplification step by DR-planner Condense. Thus, if G = G1 is wellconstrained,

it follows that all of the Gi are wellconstrained. Moreover, we know that if the

original graph is solvable, then at each step, DR-planner Condense will in fact find

a minimal dense cluster S, that consists of more than one vertex, and therefore

Gi+1 is strictly smaller than Gi for all i. Thus the process will terminate at stage

k when Gk is a single vertex. This is independent of which solvable subgraph Si is

chosen to be simplified at the i'h stage, showing that DR-planner Condense has the

Church-Rosser property.

On the other hand, if G is underconstrained, since the subgraphs Si chosen to

be simplified are guaranteed to be dense/solvable, the process will not terminate

with one vertex, but rather with a set of vertices representing the simplification of

a set of maximal solvable subgraphs (such that no combination of them is solvable).

This completes the proof that DR-planner Condense is a DR-planner that can

E E
B
D B D F Fs
A U

M M

L L
Figure 2-27: Original and simplified graphs

The proof of validity follows straightforwardly from the properties of the

simplifier map. I

Claim 16 DR-planner Condense is ',l,.,li;l;if, preserving.

Proof The simplifier maps T, do not affect subgraphs outside of Si. I

Claim 17 DR-planner Condense is not strictly .',/,1, il;if, preserving.

Proof Consider the constraint graph of Figure 2-27. The vertex weights are

2, the edge weights are 1, and the geometry-dependent constant D = 3. The graphs

ABCDNO, EFHGID and NMKLIJ are all dense/solvable. Suppose that first

the cluster S1 = ABCDNO was found and simplified into one vertex S of weight 3.

Now the graph SEFHGI = T1(EFHGID) is not dense/solvable anymore. I

Intuitively, the reason why DR-planner Condense is not strictly solvability

preserving is that the removal of the vertices D and N loses valuable information

about the structure of the solvable graph.

Claim 18 DR-planner Condense is complete.

Proof This is because DR-planner Condense finds minimal dense subgraphs

at each stage. I

Claim 19 Best-choice (and worst-choice) approximation factor of DR-planner

Condense is at most O(1/n).

3 3

2 2
L2n/2

L I \\ /

C

Figure 2 28: Bad best-choice approximation

Note. This proof mimics the MM approximation factor proof, except that now the

subgraph S is not a strongly connected component.

Proof To prove the bound on the best-choice approximation factor consider

Figure 2 28. The left and right columns contain n/2 vertices each. The weights of

all the vertical edges are 2, the weights of all other edges are 1, the weights of the

vertices are as indicated, and the geometry dependent constant D = 3.

Note that all solvable subgraphs in Figure 2 28 could be divided into 3 classes.

The first class consists of the subgraphs CL1L2; CL1L2L3; ... ;CL1L2 ... L /2-1 Ln /2

The second class consists of the subgraphs CRiR2; CRIR2R3; ... ; CR1 R2... Rn/2-1 Rn/2

The third class contains the solvable subgraphs that contain both L and R vertices.

There is only one element in this class the entire graph CL1L2... L 2R1R2... R2n/2

There is an optimal DR-plan of constant size that takes S1 = CL1L2, S2 =

S1 UL3,..., Sn~/2-1 = Sn/2-2 U Ln/2. After that it takes Sn/2 = CR1R2,Sn/2+1

Sn/2 U R3,... Sn = Sn1 U Rn/2. Finally it takes S>+ = Sn/2-1 U Sn.

However all DR-plans found by DR-planner Condense will have size O(n).

The reason for this is that DR-planner Condense is unable to simplify solvable

subgraphs on the left of the Figure 2 28 independently from the solvable subgraphs
subgraphs on the left of the Figure 2-28 independently from the solvable subgraphs

on the right. More formally let S1 be the first subgraph simplified by DR-planner

Condense under some DR-plan Q. If S1 belongs to the third class of solvable

subgraphs then the size of Q is O(n). Otherwise, without loss of generality we

can assume that S1 belongs to the first class. According to the definition of DR-

planner Condense, Ti(S1) is a single vertex of weight 3 that replaces several vertices

including vertex C. Now for any graph A that belongs to the second class, Ti (A)

is not solvable anymore (it has density -4). Hence there is an Si in Q such that

R1R2 ... R /2 C S,. Hence the size of Q is O(n). I

Next, we consider the last performance measure incorporating input decomposi-

tion.

Observation 3 In general, DR-planner Condense cannot incorporate a design

decomposition of the input viul, for reasons similar to those given for the SR

,l.i/.. ;ll,, It can incorporate a design decomposition, ow(lj, if every pair A and B of

,,11,i.liii,iN in the decomposition are either ,.,,!,1 t,. /l, disjoint or A C B or B C A.

2.1.14 The DR-planner Frontier and its Performance

Intuitively, the reason why DR-planner Condense is not strictly solvability

preserving is that the simplification of a minimal or extended dense cluster into

a single vertex loses valuable information about the structure of the cluster. The

algorithms described in this section preserve this information at least partially by

designing a simplifier that keeps the structure of the frontier vertices of the cluster,

i.e, those vertices that are connected to vertices outside of the cluster. However,

the way in which the minimal dense clusters and their sequential extensions are

found is identical to that of DR-planner Condense i.e, by using the authors'

Algorithm Dense from Hoffmann et al. (1997).

Informally, under DR-planner Frontier, all internal (i.e not frontier) vertices of

the solvable subgraph S, found at the ith stage are simplified into one vertex called

the core c1. The core vertex is connected to each frontier vertex v by an edge whose

79
D

N

Figure 2-29: The simplified graph after three clusters has been replaced by edges

weight is equal to that of v. All other edges, except ones between frontier vertices,

of S, are removed. This is repeated until the solvable subgraph S, found is the

entire remaining graph Gm.

If the solvable subgraph Si has only two frontier vertices, then all internal

vertices of Si should be removed and no new core vertex created. Instead the

two frontier vertices of S, should be connected by an edge whose weight w is

chosen so that the sum of the weights of the two frontier vertices less w is equal

to the constant D. This ensures that the graphs Gi become steadily smaller as i

increases. For instance, Figure 2-27 is such a special case, where every cluster has

only 2 frontier vertices. Hence after three iterations it would be simplified by the

DR-planner Frontier into Figure 2-29.

Defining the Subsystem Simplifier. We capture the transformations

performed by the Frontier DR-planner by describing simplifier maps.

Let Si be the solvable subgraph of Gi found at stage i, SI be the set of inner

vertices of Si, FI be the set of frontier vertices of Si, and A be any subgraph of Gi.

Then the simplifier map T,(A) is defined as follows

-T(Si) = ci, where the weight of c, is equal to the geometry-dependent constant D.

If A n S, = 0, then T,(A) = A.

If An S, ~ 0 and A n SI = 0, then the image of A under the map T, is A

minus those edges that A shares with S,.

If A n SI # 0, then T,(A) = (VTA, ETA), where VTA is the set of all vertices

of A that are not vertices of SI, plus the vertex c, representing the simplification of

S,. The set of edges ETA is formed as described earlier in this section.

Performance Analysis. In this section, we analyze the DR-planner Frontier

with respect to the various performance measures defined in previous sections.

Claim 20 DR-planner Frontier is a valid DR-planner with the CI,,l l-Rosser

property. In addition, DR-planner Frontier finds DR-plans for the maximal solvable

s.,,'1ilipl of underconstrained ,ii,,/'l,

Proof The proof is identical to the one used for DR-planner Condense. I

Before we discuss the solvability preserving property of DR-planner Frontier,

we would like to consider the following example shown in Figure 2-30. All edges

have weight 1, vertices as indicated. Initially, the graph BCDEIJK is solvable.

The graph ABCDEFGH is also solvable, vertices A and B are its inner vertices,

vertices C, D, E, F, G and H are its frontier vertices. After ABCDEFGH has been

simplified into the graph MCDEFGH, the new graph MCDEIJK is no longer

dense (edges MC, MD, ME, MH and MF have weight 2 now). However, note that

according to the definition of the DR-planner Frontier simplifier map, the image of

BCDEIJK is not the graph MCDEIJK, but the graph MCDEFGHIJK which

is dense. This graph MCDEFGHIJK is welloverconstrained, since it has density

-1 and it could be made wellconstrained by say removing constraints FG and FH.

Thus the image of ABCDEFGH is also solvable.

In general, the following claim holds.

Claim 21 Let A and B be solvable No,,I,11,I, such that A n B # 0 and A n B v

where v is a single vertex of weight less than geometry dependent constant D, then

A U B is solvable.

Figure 2-30: Original graph BCDEIJK is dense, new graph MCDEIJK is not

Proof Since A is solvable, the density of A n B, d(A n B) < -D (a

solvable graph cannot contain an overconstrained subgraph, unless it is a wellover-

constrained graph in which case it can be replaced by an equivalent wellcon-

strained graph). Hence the density of A U B, d(A U B) = d(B) + d(A \ B)

-D + d(A) d(A n B) > -D. Equality occurs when d(A n B) = -D, otherwise

A U B is overconstrained. If A U B is overconstrained it is welloverconstrained,

since it could be converted into wellconstrained by reducing weights of some edges

of B\Aor A \B. I

This property can be used to show that

Claim 22 DR-planner Frontier is .,' *,1/l;ifd, preserving as well as strictly solvabil-

ity preserving.

Proof Let B be a solvable graph, and suppose that the solvable graph Si

was simplified by DR-planner Frontier. Then B would only be affected by this

simplification if B contains at least one internal vertex of Si (recall that frontier

vertices of Si remain unchanged). But then, by the definition of the DR-planner

Frontier simplifier, T,(B) = T(B U S). Since T1(B U S) is obtained by replacing

Si by solvable T((Si), and according to the previous claim, the union of two solvable

graphs is solvable, thus T((B) = T,(B U S) is also solvable. I

Claim 23 DR-planner Frontier is complete

Proof This is because DR-planner Frontier (just as Condense) finds minimal

dense subgraphs at each stage. I

Claim 24 DR-planner Frontier has worst-choice approximation factor O(1/n).

Proof Consider Figure 2-31 the solvable constraint graph G. Initially

DR-planner Frontier will locate the minimal dense subgraph ABC (since this

is the only minimal dense subgraph of G). It will not be able to locate any

dense subgraphs disjoint from ABC, or including only frontier vertices of ABC.

If it attempts to locate a dense subgraph that includes the entire (simplified)

cluster ABC, and does so by inspecting the other vertices in the sequence

A, B, C, H, I... F, E, (i.e going counterclockwise), then DR-planner Frontier

would not encounter any dense subgraphs after ABC, until the last vertex of the

sequence E is reached. The minimal dense subgraph found at this stage is the en-

tire graph G. Thus the size of the DR-plan corresponding to this choice of vertices

is proportional to n. On other hand, there is a DR-plan of constant size. This

DR-plan would first locate the minimal dense subgraph St = ABC and simplify it.

After that it would simplify S2 = ABCE = St U {E}, after that S3 = S2 U {F}

etc going clockwise until the vertex H is reached. At every stage i, the size of Si is

constant, hence the size of this DR-plan is constant. I

E
C 2 F
5 3 52

B A

H 1 2

I

Figure 2-31: 1/n worst-choice approximation factor of DR-planner Frontier

Claim 25 The best-choice approximation factor of DR-planner Frontier is at least
1
2'

Proof Let G be the weighted constraint graph. Let P be an optimal DR-plan

of G, let p be the size of P (i.e the size of every cluster Si simplified under the

optimal DR-plan is less than p + 1). We will show that there is a DR-plan P' that

is "close to" P. Complete resemblance (P = P') may not be possible, since the

internal vertices of the cluster S found by DR-planner Frontier at the i' stage,

are simplified into one core vertex, thereby losing some information about the

structure of the graph. However we will show that there is a way of keeping the

size of P' within the constant D of the size of P.

Suppose that DR-planner Frontier is able to follow the optimal DR-plan up to

the stage i, i.e Si = Sf. Suppose that there is a cluster Sj in the DR-plan P such

that i < j and Sj contains some internal vertices of Si. Therefore the simplification

of Sj by DR-planner Frontier may be different from simplification of Sj by P.

However, since the union of S, and Sj is solvable, DR-planner Frontier could use

S' = T1 (Si) U Sj instead of Sj. The size of S. differs from the size of Sj by at

most D units, where D is the constant depending on the geometry of the problem.

Hence the size of P' is at most p + D, and since p is at least D, the result follows.

I

Next, we consider the last performance measure ability to incorporate design

decomposition.

Observation 4 DR-planner Frontier can incorporate a design decomposition of

the input inpl,, if and (,oil, if all pairs of u',,1,i,,,.il A and B in the given design

decomposition satisfy: the vertices in A n B are not 'i,", ."i the internal vertices of

either A or B.

Note. This condition on the design decomposition puts no restriction on the sizes

of the intersections of the subgraphs in the decomposition, and is far less restrictive

than the corresponding conditions for SR and Condense. <

Proof The proof is similar to the case of the DR-planner SR. For the 'if'

part, we find a topological ordering O of the given design decomposition P which

is a set of solvable subgraphs of the input graph G, partially ordered under the

subgraph relation such that O is embedded as a subplan of the final DR-plan

generated by DR-planner Frontier; i.e, O forms a subsequence of the sequence of

solvable subgraphs Si, whose (sequential) simplification gives the DR-plan.

We take any topological ordering of the given design decomposition P and

create a DR-plan for the first solvable subgraph A in P. I.e, while constructing

the individual DR-plan for A, we "i- ilnl the rest of the graph. This individual

DR-plan induces the first part of the DR-plan for the whole graph G. In particular,

the last graph in this partial DR-plan is obtained by simplifying A using the

simplifier described in Section 2.1.14 (and treating each A exactly as DR-planner

Frontier would treat a cluster Sj found at some stage j). Let Gi be the last graph

in the DR-plan for G created thus far. Next, we consider the next subgraph in the

ordering 0, and find an individual DR-plan for it, treating it not as a subgraph of

the original graph G, but as a subgraph of the simplified graph Gi. This individual

DR-plan is added on as the next part of the DR-plan of the whole graph G.

The crucial point is that the simplification of any subgraph, say A, will not

affect any of the unrelated subgraphs B in P. This is because by the requirement

on the decomposition P, A and B share at most frontier vertices. As a result, by

the functioning of the DR-planner Frontier, when the core vertex for A is created,

none of the solvable subgraphs inside B are affected.

The process of constructing individual DR-plans for subgraphs in the

decomposition P and concatenating them to the current partial DR-plan is

continued until a partial DR-plan for the input graph G has been produced, which

completely includes some topological ordering of the decomposition P as a subplan.

Again, let Gk be the last graph in this partial DR-plan. The rest of the DR-plan of

G is found by running the original DR-planner Frontier on Gk.

For the 'only if' part, we consider a DR-plan Q produced by DR-planner Frontier.

We first observe that the sequence of (original) solvable subgraphs whose

sequential simplification gives Q can never contain two subgraphs A and B such

that A n B contains both internal vertices of A and internal vertices of B. This

is because if, for instance, A is simplified before B, then B (on its own) cannot be

simplified at a later stage (although A U B could), since an internal vertex of B

that is also an internal vertex of A will disappear from the graph (will be replaced

by a core vertex representing everything internal to A), the moment A has been

simplified.

Next, we consider the remaining case where A n B contains some internal

vertices of A but only frontier vertices of B. In this case, potentially B could be

simplified before A, and A will not be affected, since the frontier vertices of B are

unchanged by the simplification. However, since a given design decomposition

P could contain an arbitrary number of overlapping subgraphs, we can choose

decompositions P such that all topological orderings of P are infeasible i.e, no DR-

plan can incorporate them as a subsequence. For instance, if there are 3 subgraphs

A, B and C in P such that A n B contains only frontier vertices of B but some

internal vertices of A; B n C contains only frontier vertices of C, but some internal

vertices of B; and C n A contains only frontier vertices of A, but some internal

vertices of C. This forces the DR-plan to simplify B before A, C before B and A

before C, which is impossible. I

2.2 Relating Problems of Chapter 1 to some Measures of Chapter 2

DR-planning problem of geometric constraint solving community directly

corresponds to our RD-dag problem. Underlying weighted graph G is the same for

both problems, number of geometric degrees of freedom D = -K, density function

d() is the same etc. Stably dense graphs correspond to solvable graphs/systems.

Solving optimal RD-dag problem is equivalent to finding optimal DR solution

sequence.

Maximal stably dense subgraphs correspond to the roots of tree in DR-plans.

Clearly size of the optimal DR solution sequence is greater or equal to the size

of the minimum stablyy) dense subgraph of G, and to the size of the minimum

stablyy) dense subgraph of T1(G) and and to the size of the minimum stablyy)

dense subgraph of T2(G) etc. Hence an efficient solution of minimum dense

subgraph problem would be helpful for finding optimal DR solution sequence. Since

it desirable that every S, were cluster minimal (see Section 1.1) a minimal dense

subgraph problem is also important for obtaining optimal DR solution sequence.

CHAPTER 3
MAXIMAL, MAXIMUM AND MINIMAL STABLY DENSE PROBLEMS

3.1 Finding Maximum Dense Subgraph

Unbounded case.

Claim 26 Problem of finding maximum dense i,,oi1,ili,,/ when weights are allowed

to be unbounded is NP-Complete

Proof By straightforward reduction from CLIQUE. I

Bounded case. Suppose that the weight of all vertices is 3, of all edges 1 (it

important that weights be bounded, because unbounded maximum dense subgraph

problem is NP-complete). Constant K = 0.

Consider a following LP:

max Exa

s.t

EI 3Exi > 0

xi > yij, Vi, j

0 < x, < l, Vi, j

Lemma 4 Let S(G) be a solution of the LP above, for a given ,,Ii'l, G. Then

vertices vi s.t xi = 1 form the 1,,,i, 1/ dense ub,,inl,,,ii A of G (dimension dependent

constant K = 0). (If there are more than 1 dense ,,/,ii,,l, of size JAI then the one

that has highest /, 1. ,I;, is formed).

Proof Let A be the (densest) maximum dense subgraph. Let B be a sub-

graph constructed as described above. Suppose that A # B. Consider an xp that

has the smallest positive value in B. If xp = 1 then B = A, contradiction. Suppose

that 0 < x2 < 1. Consider edges ypj such that 0 < ypj = xp < 1. If there are

3 or more such edges than both xp and ypj could be increased by some positive

6, resulting in a better LP solution than B, contradiction. Therefore there are at

most 2 such edges. Remove xp and its adjacent edges from B and set some vertex

vi E A \ B equal to xp. This creates another optimal LP solution B'. Consider an

x' that has the smallest positive value in B' and repeat. Eventually the value of x'

will reach 1 and by construction |A \ BI = B \ Al and density of A is equal to the

density of B. I

Note that since maximum dense subgraph is also maximal, technique above

locates maximal dense subgraph as well.

3.2 Finding a Stably Dense Subgraph

3.2.1 Distributing an Edge

In Hoffmann et al. (1997) we have described an algorithm Distribute(ei, A, K)

that "distributes" weight of an edge ei to its endpoints (v v') (or into graph A),

assuming that all other edges in A has already been "distributed". Formally edges

of A, {ej} = A are distributed if there is a mapping (distribution) of edge-weights

w(e ) into two parts fj, f (corresponding to the endpoints of the edge ej) such

that (**)

Ve, E A both fp and fj are non-negative integers and