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Mixed-Integer Nonlinear Algorithms and Analysis for Spatial Network Interdiction Problems

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Title:
Mixed-Integer Nonlinear Algorithms and Analysis for Spatial Network Interdiction Problems
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1 online resource (104 p.)
Language:
english
Creator:
Romich, Andrew N
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Smith, Jonathan Cole
Committee Co-Chair:
Lan, Guanghui
Committee Members:
Geunes, Joseph Patrick
Xia, Ye

Subjects

Subjects / Keywords:
algorithms -- complexity -- interdiction -- networks -- nonlinear -- optimization
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
We address problems related to locating network components and routing flow. First, we examine the problem of placing stationary sensors in a continuous space, with the goal of minimizing an adversary's maximum probability of traversing an origin-destination route without being detected. In particular, we consider the deployment of sensors whose probability of detecting an intruder is a function of the distance between the sensor and the intruder. Under the assumption that the detection probabilities are mutually independent, we construct a two-stage mixed-integer nonlinear programming formulation for the problem. We provide an algorithm that optimally locates sensors in a continuous space. We examine this problem for the case in which sensor locations are restricted to two different discretized subsets of continuous space. Our analysis provides optimization algorithms for each case, and derives bounds on the worst-case optimality gap between the restrictions and the initial (continuous-space) problem. We then consider a situation where sensors must be placed so that they cover a set of targets in the region, and should be deployed in a manner that allows sensors to communicate with one another. Complicating the sensor location problem are uncertainties related to sensor placement, e.g., as caused by drifting due to air or water currents to which the sensors may be subjected. Our problem seeks to maximize a metric regarding intra-sensor communication effectiveness, subject to the condition that all targets must be covered by some sensor, where sensor drift occurs according to a robust (worst-case) mechanism. We also consider several sparsity-inducing restrictions of the standard maximum flow problem.  Each restriction is a unique combination of the number of nodes having an out-degree greater than one and the inclusion of either a restriction on the total amount of flow on all arcs or the total number of arcs used in the network. Assuming an option of initially allowing continuous or integer flow, we analyze a total of twelve problems.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Andrew N Romich.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Smith, Jonathan Cole.
Local:
Co-adviser: Lan, Guanghui.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

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UFRGP
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Applicable rights reserved.
Classification:
lcc - LD1780 2013
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UFE0045765:00001

MISSING IMAGE

Material Information

Title:
Mixed-Integer Nonlinear Algorithms and Analysis for Spatial Network Interdiction Problems
Physical Description:
1 online resource (104 p.)
Language:
english
Creator:
Romich, Andrew N
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Smith, Jonathan Cole
Committee Co-Chair:
Lan, Guanghui
Committee Members:
Geunes, Joseph Patrick
Xia, Ye

Subjects

Subjects / Keywords:
algorithms -- complexity -- interdiction -- networks -- nonlinear -- optimization
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
We address problems related to locating network components and routing flow. First, we examine the problem of placing stationary sensors in a continuous space, with the goal of minimizing an adversary's maximum probability of traversing an origin-destination route without being detected. In particular, we consider the deployment of sensors whose probability of detecting an intruder is a function of the distance between the sensor and the intruder. Under the assumption that the detection probabilities are mutually independent, we construct a two-stage mixed-integer nonlinear programming formulation for the problem. We provide an algorithm that optimally locates sensors in a continuous space. We examine this problem for the case in which sensor locations are restricted to two different discretized subsets of continuous space. Our analysis provides optimization algorithms for each case, and derives bounds on the worst-case optimality gap between the restrictions and the initial (continuous-space) problem. We then consider a situation where sensors must be placed so that they cover a set of targets in the region, and should be deployed in a manner that allows sensors to communicate with one another. Complicating the sensor location problem are uncertainties related to sensor placement, e.g., as caused by drifting due to air or water currents to which the sensors may be subjected. Our problem seeks to maximize a metric regarding intra-sensor communication effectiveness, subject to the condition that all targets must be covered by some sensor, where sensor drift occurs according to a robust (worst-case) mechanism. We also consider several sparsity-inducing restrictions of the standard maximum flow problem.  Each restriction is a unique combination of the number of nodes having an out-degree greater than one and the inclusion of either a restriction on the total amount of flow on all arcs or the total number of arcs used in the network. Assuming an option of initially allowing continuous or integer flow, we analyze a total of twelve problems.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Andrew N Romich.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Smith, Jonathan Cole.
Local:
Co-adviser: Lan, Guanghui.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045765:00001


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MIXED-INTEGERNONLINEARALGORITHMSANDANALYSISFORSPATIALNETWORKINTERDICTIONPROBLEMSByANDREWN.ROMICHADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013AndrewN.Romich 2

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Idedicatethisworktomyfamily 3

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ACKNOWLEDGMENTS Iwouldliketorstthankmyadvisors,Dr.J.ColeSmithandDr.GuanghuiLan.Theknowledgeandguidanceyouprovidedinhelpingmetacklechallengingproblemsarethereasonthisdissertationwaspossible.Ourweeklymeetingsnotonlyencouragedme,butalsofocusedandguidedmyefforts,providingperspectivefromtwoexpertsintheeld.Thetools,lessons,andadvice,bothacademicallyandpersonally,thatyouhaveprovidedmethroughoutmyyearsinthePh.D.programhavebuiltaninvaluablefoundationthatwillhelpmeimmenselyfortherestofmylife.IwouldalsoliketothankDr.JosephGeunesandDr.YeXiaforservingonmysupervisorycommitteeandprovidingperceptivecommentsandsuggestions.IwouldalsoliketothankallofmyfriendsinthegraduateprogramoftheDepartmentofIndustrialandSystemsEngineering.ThewonderfulmemoriesyouhaveprovidedIwillneverforget;Ionlyhopethatwehavemanymoretocome.IwouldliketosendaspecialthankstomyPh.D.GeneralExamstudygroup:SoheilHemmati,CinthiaPerez,JohannaAmaya,ClayKoschnick,andMikePrince.Withoutthoseweekendstudysessions,Iwouldnotbewritingthisacknowledgment.AveryspecialthankstoBrittanyNethers,whoworkedaroundmyridiculousstudyscheduleforveyears;youwereinstrumentalinthecompletionofthisdissertation.Mostimportantly,Iwouldliketothankmyparents.Withoutyourconstantsupport,guidance,advice,andlove,noneofthiswouldbepossible.Iamforevergratefulforallthatyouhavesacricedtoprovidemewithanopportunitytoobtainthehighestdegreeinmyeld.Iloveyoubothsomuch. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2OPTIMIZINGPLACEMENTOFSTATIONARYMONITORS ........... 13 2.1MotivationandLiteratureReview ....................... 13 2.2Continuous-SpaceFormulation ........................ 16 2.2.1Preliminaries .............................. 16 2.2.2MINLPFormulation ........................... 18 2.2.3Two-StageLower-BoundFormulation ................. 20 2.3OptimalContinuous-SpaceAlgorithm .................... 26 2.3.1OverallApproach ............................ 27 2.3.2ImprovementStep ........................... 29 2.4Midcolumn-RestrictedFormulation ...................... 36 2.5Full-DiscretizationFormulation ........................ 38 2.6ComputationalResults ............................. 39 3AROBUSTSENSORCOVERINGANDCOMMUNICATIONPROBLEM .... 46 3.1MotivationandLiteratureReview ....................... 46 3.2FormalProblemDescriptionsandFormulation ............... 48 3.3SolutionApproaches .............................. 54 3.3.1EquivalentFormulationExpressions ................. 54 3.3.2ComputationofSampleSizeof-Vectors .............. 55 3.3.3Cutting-PlaneSolutionMethods .................... 56 3.3.3.1GeneralizedBendersDecomposition ........... 58 3.3.3.2Kelley'sMethod ....................... 60 3.4ComputationalResults ............................. 60 4ONTHECOMPLEXITYOFSPARSEMAXIMUMFLOWPROBLEMS ..... 66 4.1MotivationandLiteratureReview ....................... 66 4.2ProblemsofInterest .............................. 68 4.3ComplexityAnalysisandPolynomial-TimeAlgorithms ........... 69 4.3.1UnrestrictedFlowSplitting ....................... 69 4.3.2Fully-RestrictedFlowSplitting ..................... 73 5

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4.3.3Partially-RestrictedFlowSplitting ................... 79 5CONCLUSIONSANDFUTURERESEARCH ................... 83 APPENDIX APROOFS ....................................... 86 BEXTENDEDCOMPUTATIONALRESULTS .................... 94 REFERENCES ....................................... 99 BIOGRAPHICALSKETCH ................................ 104 6

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LISTOFTABLES Table page 2-1MLIMEffectivenessinExactAlg;nc=f4,7,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)g,andp=f0.01,0.25g. ................... 41 2-2ComparisonofSolutionMethodswithnc=f10,15,20g,nr=10,jSj=f2,4g,R=100,p=0.75,andN=10. .......................... 42 2-3EffectofponExactAlgwithnc=f4,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf -435.35 -14.44 Td[(1)g,andp=f0.01,0.25,0.75g. ........................... 43 2-4EffectofNonFDFSolutionwithnc=f20,40g,nr=f10,15g,jSj=f2,4g,R=200,andp=0.75. ............................... 43 2-5FDFSolutionQualityonLargeInstanceswithnc=f80,100g,nr=15,jSj=f2,4g,R=f100,200g,p=f0.75,0.95g,andN=6. ............... 44 3-1EffectofDTonSolutionQuality,D1=25 ..................... 62 3-2EffectofD1onSolutionQuality,DT=200 ..................... 63 3-3EffectofCPTypeonSolutionQuality ........................ 64 5-1ComplexityResults .................................. 85 B-1MLIMEffectivenessinExactAlg;nc=f4,7g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)g,andp=f0.01,0.25g .................... 94 B-2MLIMEffectivenessinExactAlg;nc=f7,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)g,andp=f0.01,0.25g .................... 95 B-3EffectofponExactAlgwithnc=f4,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf -435.35 -14.44 Td[(1)g,andp=f0.01,0.25,0.75g(Full) ........................ 96 B-4EffectofNonFDFSolutionwithnc=f20,40g,nr=f10,15g,jSj=f2,4g,R=200,andp=0.75(Full) ............................ 97 B-5FDFSolutionQualityonLargeInstanceswithnc=f80,100g,nr=15,jSj=f2,4g,R=f100,200g,p=f0.75,0.95g,andN=6(Full) ............ 98 7

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LISTOFFIGURES Figure page 2-1Afeasibleadversarypathandmonitorlocationswithnc=5,nr=4,andjSj=3. ............................................ 17 2-2Curvecorrespondstoformulation( 2 ),whilepiecewise-linearconcavefunctioncorrespondstoformulation( 2 ). ......................... 21 2-3Illustrationofmidcolumnpoints. ........................... 36 3-1Afeasiblesetofeight(initial)sensorlocationsguaranteeingcoverageofninetargets. ........................................ 47 3-2ArcsetforFigure 3-1 sensorsandtheresultingpartial-capacity(dashed)andfull-capacity(solid)arcs. ............................... 51 3-3CPUTimeComparisonofTable 3-1 Instances .................. 63 3-4CPUTimeComparisonofTable 3-2 Instances .................. 64 3-5CPUTimeComparisonofTable 3-3 Instances .................. 65 4-1Reductiongraphwitharccapacities(capacityoneonunlabeledarcs)forthe3-SATformula123^234^134^234^134(4variablesand5clauses). 71 4-2ExampleGraphwithn=6. ............................. 73 4-3ExampleExpandedNetworkforGraphinFigure 4-2 (s0=4andL=minf6,4+1g=5).EachsolidarcisduplicatedfromFigure 4-2 andeachdashedarcisadditionalarcwithinnitecapacity. ......................... 74 4-4ExampleReductionGraphfor3-dimensionalmatchingtoD-MFfS(x)l,jjxjj1s1g(withcircledelementsofamatching). ................. 81 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMIXED-INTEGERNONLINEARALGORITHMSANDANALYSISFORSPATIALNETWORKINTERDICTIONPROBLEMSByAndrewN.RomichAugust2013Chair:J.ColeSmithCochair:GuanghuiLanMajor:IndustrialandSystemsEngineeringWeaddressproblemsrelatedtolocatingnetworkcomponentsandroutingow.First,weexaminetheproblemofplacingstationarysensorsinacontinuousspace,withthegoalofminimizinganadversary'smaximumprobabilityoftraversinganorigin-destinationroutewithoutbeingdetected.Inparticular,weconsiderthedeploymentofsensorswhoseprobabilityofdetectinganintruderisafunctionofthedistancebetweenthesensorandtheintruder.Undertheassumptionthatthedetectionprobabilitiesaremutuallyindependent,weconstructatwo-stagemixed-integernonlinearprogrammingformulationfortheproblem.Weprovideanalgorithmthatoptimallylocatessensorsinacontinuousspace.Weexaminethisproblemforthecaseinwhichsensorlocationsarerestrictedtotwodifferentdiscretizedsubsetsofcontinuousspace.Ouranalysisprovidesoptimizationalgorithmsforeachcase,andderivesboundsontheworst-caseoptimalitygapbetweentherestrictionsandtheinitial(continuous-space)problem.Wethenconsiderasituationwheresensorsmustbeplacedsothattheycoverasetoftargetsintheregion,andshouldbedeployedinamannerthatallowssensorstocommunicatewithoneanother.Complicatingthesensorlocationproblemareuncertaintiesrelatedtosensorplacement,e.g.,ascausedbydriftingduetoairorwatercurrentstowhichthesensorsmaybesubjected.Ourproblemseekstomaximizeametricregardingintra-sensorcommunicationeffectiveness,subjecttothecondition 9

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thatalltargetsmustbecoveredbysomesensor,wheresensordriftoccursaccordingtoarobust(worst-case)mechanism.Wealsoconsiderseveralsparsity-inducingrestrictionsofthestandardmaximumowproblem.Eachrestrictionisauniquecombinationofthenumberofnodeshavinganout-degreegreaterthanoneandtheinclusionofeitherarestrictiononthetotalamountofowonallarcsorthetotalnumberofarcsusedinthenetwork.Assuminganoptionofinitiallyallowingcontinuousorintegerow,weanalyzeatotaloftwelveproblems. 10

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CHAPTER1INTRODUCTIONNetworks,whichconsistofasetofnodesandedgesconnectingsomesubsetofnodes,areusedtomodelavarietyofscenarios([ 52 ]illustratesseveralexamples).Specically,wefocusonnetworkmodelsastheyrelatetoplacementofsecuritygridsensorsandmaximizingow.Appropriatenetworkoperationsdecisionsarecrucialtoeffectivefunctioningofthesenetworks.Tooptimizethesedecisions,mathematicalprogrammingisawidelyutilizedtool[ 5 ].Whentailoredtoreal-worldapplications,thenaturalformulationsoftheseproblemstendtobenonlinearandhighlynonconvex.Thus,methodsthatobtainoptimal(ornear-optimal)solutionswithinpracticalcomputationallimitsaretransformativetonetworkoperationsproblems.Recently,reportsfromDepartmentofHomelandSecurityofcialsstatethatal-Qaedaisunderenormouspressuretoexecutealarge-scalenuclearattack,whichcouldinvolvesmugglingofnuclearweaponsintotheU.S.throughoneofthenation's360seaports[ 42 ].TheU.S.StateDepartmentassertsthatthepervasiveuseofradioactivesubstancesformedicinalpurposescreatesanopeningforterroriststogainaccesstomaterialforaradiologicaldirtybomb[ 3 4 ].Couplethisthreatwiththedocumentedlackofsufcientsecurityatports[ 2 ],andtheU.S.economycouldexperiencelossesofaround$58billionifsuchanattackoccurs[ 1 ].Acountermeasuretopreventsuchadisasteristostrategicallyplacedetectionsensorsatports.Optimizingsensorplacementhasbeentackledinanumberofdifferentwaysintheliterature.Accountingforaworst-casescenarioinwhichtheadversarycorrectlyidentiesthesensorlocations,theproblemcanbesimilarlythoughtofasinterdictingarcsinanetworktominimizeanadversary'smaximumow([ 57 ]providesasurveyofnetworkinterdiction).IsraeliandWood[ 38 ]provideafoundationalone-stagemixed-integerprogrammingformulationoftheproblemthatmodelsthebasictwo-stagenetworkinterdictionmodel.However,researchinthisareafocusesmainlyona 11

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discretizationandapproximationoftheregionunderconsiderationanddoesnotcapturecontinuous-spaceproblemsthataresovitaltoportsecurityoperations.Whensensorsareassignedwithcooperativelydisseminatinginformation,inadditiontomonitoringduties,itmaybeadvantageoustoplacethesesensorsinawaythatmaximizesthenumberofqualityofpairwisecommunicationexistingamongthem[ 35 ].Intheliterature,numerousstudies[ 34 ]addresstheissueofsensorplacementasitrelatestomaximizingcoverageofasetoftargetsandconnectivityamongthesensors.However,thesestudiesdonottakeintoaccountuncontrolledsensordisplacementwherethereisnooptionofsensorre-positioningforsomeperiodoftime,whichoccurs,forexample,iftraveltosensorlocationsisdelayed.Aplethoraofevents,fromnaturaldisasterstoman-madeattacks,arepossiblethatcoulddisplacethesesensorsfromtheiroriginallocationsanddisruptconnections[ 36 ].Asaresult,thosetaskedwithplacingsensorsmustdosoinawaythatallowsacertaindegreeofcoverageandconnectivity.Thisdissertationproceedsasfollows.InChapter 2 ,wedevelopandanalyzethreealgorithmstosolveatwo-stagenetworkinterdictionmodeltoplacesensorsinanareaofinterest.(Notethatthetermsmonitorsandsensorscanbeusedinterchangeably,althoughweemploytheformerterminourtechnicaldescriptioninChapter 2 .)InChapter 3 ,weanalyzeformulationsandsolutiontechniquesforaproblemwhosegoalistoplacesensorsinlocationsthatguaranteecoverageofasetoftargetsandmaximizeintra-sensorcommunication.InChapter 4 ,wedocumentthecomplexityofseveralsparsemaximumowproblemsarisinginavarietyofnetworksituations. 12

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CHAPTER2OPTIMIZINGPLACEMENTOFSTATIONARYMONITORS 2.1MotivationandLiteratureReviewDetectionofanapproachingadversaryiscrucialtothwartinganattack.Asanexampleofanattack-vulnerableenvironment,weconsiderUnitedStates(U.S.)seaports.Almost11millioncontainersarriveinU.S.portseachyear,transporting95percentofU.S.non-NorthAmericantradebyweightand75percentbyvalue[ 1 30 ].PortsalsocontributetotheU.S.economythroughrecreationalactivitiessuchasshing,boating,andcruises.Becauseportsplaysuchakeyrole,theyalsoserveasaprimetargetforaterroristattack.SimulationsshowthataterroristattackforcingeveryU.S.porttoshutdowncouldresultinmanufacturingandproductionlossesof$58billion[ 1 2 ].AccordingtoarecentreportbytheU.S.StateDepartment,thepervasiveuseofradioactivesubstancesformedicinalpurposescreatesanopeningforterroriststogainaccesstomaterialforaradiologicaldirtybomb[ 3 4 ].Additionally,recentreportsfromDepartmentofHomelandSecurityofcialsstatethatal-Qaedaisunderenormouspressure[ 42 ]toexecutealarge-scalenuclearattack,whichcouldinvolvesmugglingofnuclearweaponsintotheU.S.throughitsports.Inresponsetothisthreat,weexamineinthischaptertheplacementofstationarymonitorsinanareaofinterestsoastominimizetheadversary'smaximumprobabilityofevasion.Inanefforttooptimizeplacementofthesemonitors,wedevelopatwo-stagemixed-integernonlinearprogramming(MINLP)formulationoftheproblem.Therststagedeploysthemonitors,whilethesecondstagemodelstheadversary'schoiceofanoptimalpaththroughtheareaofinterest.ThemodelfallsunderthecategoryofStackelbergleader-followergames,inwhichthefollower(adversary)isassumedtocorrectlyidentifythemonitorlocationsinthenetwork[ 58 ].Optimizingsensorplacementwiththegoalofminimizinganadversary'smaximumevasionprobabilityisrelatedtotheproblemofinterdictingarcsinanetworktomaximize 13

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anadversary'sshortestpath(see[ 57 ]forasurveyofnetworkinterdiction).Inthiscontext,theworkofIsraeliandWood[ 38 ]ispivotal,asitprovidesaone-stagemixed-integerprogrammingformulationoftheproblemthatmodelsthebasictwo-stageshortestpathnetworkinterdictionmodel.Brownetal.[ 18 ]createaframeworkforidentifyinganear-optimaldefensestrategybasedonagivenbudgetviaStackelberginterdictionmodels.Cormicanetal.[ 25 ]studyastochasticnetworkinterdictionmodel,whichseekstominimizetheexpectedmaximumowinwhichthecapacityofanarcafterinterdictionisuncertain.BienstockandVerma[ 16 ]andPinaretal.[ 53 ]explorethevulnerabilityofelectricalgridnetworkstothesimultaneousfailureofakeysubsetoflinks.Additionally,studiesin[ 8 61 62 ]analyzetheeffectivenessofemployingdefensivevehiclestointerdictanadversarywhoattemptstotransportaradiological/nuclearweaponintoacity.Variantsoftheproblemweconsiderhavebeenexaminedunderassumptionsthatcertaincharacteristicsofthenetworkareunknown[ 37 ],probabilitiesofevasionareparameters[ 50 ],andinformationbetweentheadversaryandinterdictorareasymmetric[ 12 ].WilhelmandGokce[ 63 ]focusonsolvingthisproblemasitrelatestotheportenvironment,wheretheprobabilityofdetectinganadversaryisknowngivenadiscretesetofsensorcombinations,sensorlocations,surveillancepoints,andenvironmentalconditions.Thevariationofthetraditionalshortestpathnetworkinterdictionmodelthatweconsiderexaminesacontinuousfeasibleregioninwhichthemonitorscanbeplaced.However,theadversary'spathislimitedtomovementsonadiscretizedspace.Evasionprobabilityisbasedonthequantityandlocationofmonitorsandthenumberofdiscreteadversarymoves,withanindependentprobabilityofdetectingeachmove.Additionally,weoperateundertheassumptionthattheadversaryknowsthemonitorlocationssoastoaccountforaworst-casescenario.Thisisincontrasttotheworkin[ 60 ]whereboththeleader,whochoosesmonitorlocations,andfollower,whochoosesapaththrough 14

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thenetwork,makedecisionssimultaneously,resultinginamixed-strategysolution.Anotheropposingexaminationoftheproblemisfromarobustoptimizationperspective,wheretheadversary,ratherthanknowingexactlythemonitorlocations,mustaccountforanypossiblesetofmonitorlocationswhendeterminingapath[ 14 15 ].Thistypeofmodel,however,doesnotallowtheextentofrecourseontheadversary'sdecisionthatisallowedinourmodel.Thecontributionsofthischapterareasfollows.Wepresenttherstalgorithmthatcanconvergetoanoptimalsolutionfortheforegoinginterdictionmodel.Throughconvergenceanalysisofthisalgorithm,weestablishaboundonthemaximumnumberofmixed-integerlinearprogramssolvedtoobtainaparticularsolutionaccuracy.Furthermore,wederiveboundsonthemaximumdifferencebetweentheoptimalobjectivefunctionvaluesofamodelwithmonitorsplacedinparticulardiscretespacesandthatofacontinuous-spacemodel.Wealsodiscusshowouranalysisallowsonetodeterminewhetheritismostappropriatetousetheexactalgorithmoroneoftherestrictiontechniquesmentionedabove.Theremainderofthischapterisorganizedasfollows.InSection 2.2 wepresentatwo-stageMINLPformulationoftheprobleminwhichmonitorscanbedeployedanywhereinthecontinuousspace.WepresentinSection 2.3 analgorithmtosolvethecontinuousspaceproblem,andprovethatthisalgorithmcanconvergetoanoptimalsolution,andprovideanestimateonitsrateofconvergence.Section 2.4 providesarestrictedformulationthatemploysapartialdiscretization,resultinginprovablynear-optimalsolutions.Section 2.5 discussesafulldiscretizationofthemonitors'feasibleregion,alongwithananalysisoftheimpactoffulldiscretizationonthe(worst-case)optimalitygap.Section 2.6 presentscomputationalresultsoftheproposedmethodsonrandomlygeneratedtestinstances. 15

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2.2Continuous-SpaceFormulationWerstprovidesomepreliminaryassumptionsandmodelnotationinSection 2.2.1 ,andnextformulatethemonitorplacementproblemasaMINLPinSection 2.2.2 .WethendeveloparelaxationofthisMINLPthatcanbesolvedmoreefcientlyinSection 2.2.3 2.2.1PreliminariesWebeginbyconsideringarectangularareaonwhichourmonitorsaredeployed.Thisareaisdividedintonccolumns,indexedbyI=f1,...,ncg,andnrrows,indexedbyJ=f1,...,nrg,withnc2andnr2.Therstcolumnislocatedwherethex-axiscoordinateequalszerooverthearea,thelastcolumnislocatedwherethex-axiscoordinateequalsL,andallcolumnsareevenlyspacedapart.Thenrrowsarelikewiseevenlyspacedfrom0toH.Wecreatencnrnodes,oneateachcoordinate(i,j)ofthegrid,8i=1,...,ncandj=1,...,nr.Itisalsousefultodeneafunction:R2!R2thatmapsavector(k,l)overthe(continuous)gridspace,toitscorrespondingvectorintheoriginalspace[(0,0),(L,H)],i.e.,(k,l)=(k)]TJ /F7 11.955 Tf 11.96 0 Td[(1)L nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1,(l)]TJ /F7 11.955 Tf 11.96 0 Td[(1)H nr)]TJ /F7 11.955 Tf 11.95 0 Td[(1.Anadversarycanentertheareaatanynode(1,j)intherstcolumn.Theadversarywillthenmovefromcolumnitocolumni+1,foreachi=1,...,nc)]TJ /F7 11.955 Tf 12.85 0 Td[(1.Avalidmoveacrossanarc,indexedbyijk,isthusfromlocation(i,j),i2Infncg,j2J,tolocation(i+1,k),wherek2J.ThesetofmonitorsisindexedbyS=f1,...,jSjg.Figure 2-1 illustratestheforegoingmodelenvironment.Theadversaryisassumedtohavefullknowledgeofthemonitorplacementsandwillprogressfromcolumnitoi+1,i2Infncg,initspathinordertotakeadirectroutetothetarget.(Thisextensioninwhichanadversarycanmovealongapaththatvisitsmultiplenodesineachcolumncanalsobehandledinouranalysis.)Furthermore,we 16

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Figure2-1. Afeasibleadversarypathandmonitorlocationswithnc=5,nr=4,andjSj=3. assumethattheprobabilityofevadings2Salongamovefromcolumni2Infncgtoi+1isindependentofallpreviousmoves.Wenowdiscussourmodelingassumptionsthatdictatetheprobabilitythattheadversaryisdetectedoneacharc.First,foreveryarcijk,letvijk2(0,1]betheprobabilitythattheadversaryescapesdetectiononarcijkinthenetworkiftherearenomonitorsthatobservethisarc.Werefertothesev-parametersasarcfactors,whichaccountforexistingdetectionresourcesnativetotheport.Next,denethedis-tancefromamonitortoarcijkastherectilineardistance(`1-norm)fromthemonitortothemidpointlocation,cijk,ofthearc.Wemodeltheprobabilityofevadingamonitorduringamoveacrossanarcasafunctionofthedistancefromthemonitortothearc.Inparticular,ifthisdistanceexceedssomemaximumdetectionradius,R,thentheprobabilityofevadingthatmonitoris1.Furthermore,wedenep2(0,1)tobeamonitor-dampeningparameter,whichdenotestheminimumprobabilityofevadingamonitor(evenifthatmonitorweredirectlyonanarcmidpointtraversedbytheadversary).TheadversarythereforeevadesdetectionintraversingarcijkwithprobabilityvijkYs2Smink(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)k R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p)+p,1. 17

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2.2.2MINLPFormulationDenevariablesfijk=1,8i2Infncg,j,k2J,iftheadversarymovesonarcijk,andfijk=0otherwise.Thefollowingisatwo-stageMINLPformulationoftheproblem,whereX=fx1,...,xjSj:(1,1)xs(nc,nr),8s2Sg. minx2XmaxYi2Infncg Xj2JXk2J vijkfijkYs2Smink(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)k R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1!! (2a)s.t.Xj2JXk2Jf1jk=1 (2b)Xj2J()]TJ /F5 11.955 Tf 9.3 0 Td[(fijk+fi+1,k,j)=08i2Infncg,k2J (2c)fijk2f0,1g8i2Infncg,j,k2J. (2d)Constraints( 2b )and( 2c )aretheadversary'sowbalanceconstraintsforeverygridpointexceptforthoseincolumnnc,whichareimpliedby( 2b )and( 2c ).Constraints( 2d )restrictowonallarcstobebinary.Intheremainderofthispaper,weexpressconstraints( 2b )and( 2c )asAf=b,anddene=ff2f0,1gn2r(nc)]TJ /F8 7.97 Tf 6.59 0 Td[(1):Af=bgasthesetofallfeasibleows.Thesecond-stage(adversary's)problemisthefollowing,givenxedvaluesx1,...,xjSj:maxf2Yi2Infncg Xj2JXk2J vijkfijkYs2Smink(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)k R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1!!. (2)TheobjectivefunctionisnonconvexbecauseitsHessianisindenite.However,recallingthatthemonitor-dampeningparameterpandallarcfactorsvijkarepositive,andthatPj2JPk2Jfijk=1,8i2Infncg,eachtermXj2JXk2J vijkfijkYs2Smink(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)k R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1! 18

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ispositive.Becausethenaturallogarithmfunctionisnondecreasing,wecaninsteadtakethenaturallogarithmofthefunctionin( 2 ),yieldingthefollowingformulation:maxf2Xi2Infncgln Xj2JXk2J vijkfijkYs2Smink(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)k R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1!!. (2)Becauseallf-variablesarerestrictedtobebinary-valued,( 2 )isequivalenttomaxf2Xi2InfncgXj2JXk2J Xs2Slnminjj(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p)+p,1+ln(vijk)!fijk. (2)UsingthefactthatAistotallyunimodularandbisabinaryvector,wecanreplacebinarinessrestrictionsonf2withnonnegativityrestrictions[ 13 ].Thus,theoveralltwo-stageformulationisminx2Xmaxf2(G(x,f)=Xi2InfncgXj2JXk2J Xs2Slnminjj(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p)+p,1+ln(vijk)!fijk). (2)Thesecond-stageproblemcanberewrittenforsimplicityas maxq(x)Tf (2a)s.t.f2, (2b)wherex=)]TJ /F9 11.955 Tf 5.48 -9.69 Td[(x1,...,xjSj2R2xjSj,thecoefcientoffijkisgivenbyq(x)ijk=Xs2S (k(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)k)+ln(vijk),8i2Infncg,j,k2J, (2)where (d)=lnmind R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1. (2) 19

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Associatingdualmultipliersywith( 2b ),thedualto( 2 )minimizesy1,subjecttoATyq(x),wherey2Rnr(nc)]TJ /F8 7.97 Tf 6.59 0 Td[(2)+1,notingthatbTy=y1.Since( 2 )isafeasibleandboundedlinearprogram,strongdualitymusthold.Combiningthisdualwiththerst-stageproblem,weobtain miny1 (2a)s.t.ATyq(x) (2b)x2X, (2c)whichisequivalentto( 2 ). 2.2.3Two-StageLower-BoundFormulationBecauseq(x)isnotaconvexfunctionofx,formulation( 2 )isnotaconvexprogramandisthereforeverydifculttosolve.Inanefforttoobtainamoretractableformulation,weconsidereverycombinationofmonitorssandarcsijk,andcreateapiecewise-linearconcavefunctionthatunderestimatesthenaturallogarithmoftheprobabilityofevadingmonitorswhilemovingalongarcijk.Werstpartitiontherangeofallpossiblevaluesofthedistancebetweenxsandcijkintoasetofdistanceintervals.Inparticular,wedeneasetBsijksothattheb-thdistanceintervalisgivenby[dsijk,b,dsijk,b+1],forb2Bsijk=f1,...,jBsijkjg.Morespecically,wechoosethevaluesofdsijk,b,8b=1,...,jBsijkj+1,sothattherstjBsijkj)]TJ /F7 11.955 Tf 19.1 0 Td[(1distanceintervalspertainingtodistanceslessthanorequaltoRareofequallength,i.e.,dsijk,b=b)]TJ /F8 7.97 Tf 6.59 0 Td[(1 Bsijk)]TJ /F8 7.97 Tf 6.59 0 Td[(1R,8b2Bsijk,withonemoredistanceintervalforalldistancesgreaterthanR,i.e.,dsijk,jBsijkj+1=L+H.Tocreatethispiecewise-linearconcavefunction,weintroduceparameterspsijk,b,8b=1,...,jBsijkj+1,s2S,i2Infncg,j,k2J,sothatpsijk,b= (dsijk,b),8b=1,...,jBsijkj+1,wheretheb-thprobabilityintervalisgivenby[epsijk,b,epsijk,b+1].Figure 2-2 depictsthisfunctionforaninstancewherejBsijkj=5,H=1000,L=1000,R=100,andp=0.01. 20

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Figure2-2. Curvecorrespondstoformulation( 2 ),whilepiecewise-linearconcavefunctioncorrespondstoformulation( 2 ). Subsequently,weintroduceavectorofbinaryvariableswsijk,b,8s2S,i2Infncg,j,k2J,b2Bsijk.Variablewsijk,^b=1ifmonitor-arccombinationsijkoccupiesdistanceinterval^b2Bsijkandwsijk,b=0,8b6=^b.Accordingly,deneW=(w:Xb2Bsijkwsijk,b=1,wsijk,b2f0,1g,8s2S,i2Infncg,j,k2J,b2Bsijk) (2)torequirethatamonitor-arccombinationsijkoccupiesexactlyoneofitspossiblejBsijkjdistanceintervals.Usingtheforegoingelements,weproducethefollowingtwo-stagelower-boundformulation: min(x,w)2X( 2 )maxXi2InfncgXj2JXk2Jt(x,w)ijkfijk (2a)s.t.f2, (2b)whereX( 2 )=fx2X,w2W: 21

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k(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)kXb2Bsijkdsijk,b+1wsijk,b8s2S,i2Infncg,j,k2J (2c)xs,1xs+1,18s2SnfjSjgg (2d)andt(x,w)ijkincorporatesthelower-boundingpiecewise-linearfunctionandarcfactors,givenbyt(x,w)ijk= Xs2SXb2Bsijk psijk,b+1)]TJ /F5 11.955 Tf 11.95 0 Td[(psijk,b dsijk,b+1)]TJ /F5 11.955 Tf 11.95 0 Td[(dsijk,b(jj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj)]TJ /F5 11.955 Tf 21.26 0 Td[(dsijk,b+1)+psijk,b+1!+ln(vijk)!wsijk,b. (2)Constraints( 2c )ensurethatthemonitorlocationiswithinanupper-bounddistance,dsijk,b+1,ofdistanceintervalb,ifthisparticularmonitor-arccombinationsijkhaswsijk,b=1.Notethatconstraintsensuringthatthemonitorlocationisgreaterthanorequaltoitscorrespondinglower-bounddistancedsijk,barenotneededbecausetheoptimalobjectivefunctionvaluetotheinnermaximizationproblemof( 2 )isanondecreasingfunctionofjj(xs)]TJ /F9 11.955 Tf 12.38 0 Td[(cijk)jj.Therefore,therealwaysexistsanoptimalsolutioninwhichtheouterminimizationproblemsetswsijk,b=1forthesmallestindexsuchthatk(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)kdsijk,b+1.Constraints( 2d )eliminatesomealternativeoptimalsolutionsbyensuringthat,foranymonitor^s,everyothermonitors^sisnotlocatedatasmallercolumn-axiscoordinatethanmonitor^s. Theorem2.1. Theoptimalobjectivefunctionvalueof( 2 )isalowerboundonG(x,f),where(x,f)isoptimalto( 2 ). Proof. DeneF( 2 )andF( 2 )asthefeasibleregionof( 2 )and( 2 ),respectively.Weshowthateveryfeasiblesolution(x,f)toF( 2 )correspondstoafeasiblesolution(x,w)toF( 2 ),wheretheobjectivefunctionvaluefor(x,w)in( 2 )doesnotexceedG(x,f)in( 2 ).Consideranyfeasiblesolution(x,f)2F( 2 ),andnotethatf2.Foreachs2Sandarcijk,let^b2Bsijkbesuchthatdsijk,^bjj(xs)]TJ /F9 11.955 Tf 12.77 0 Td[(cijk)jjdsijk,^b+1(assumedtoexistbyconstruction),andsetwsijk,^b=1.Thus,wehaveestablisheda 22

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w:(x,w)2F( 2 ).WenowshowthatG(x,f)isatleastaslargeas( 2a )evaluatedat(x,w,f).Foranysijksuchthats2S,i2Infncg,j,k2J,sothatjj(xs)]TJ /F9 11.955 Tf 11.91 0 Td[(cijk)jjliesintheb-thdistanceintervalforsandijk,wehavelnminjj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1+1 jSjln(vijk)=lnmindsijk,^b+1 R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p)+p+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()dsijk,^b R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1+1 jSjln(vijk), (2)for=jj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj)]TJ /F5 11.955 Tf 21.25 0 Td[(dsijk,^b dsijk,^b+1)]TJ /F5 11.955 Tf 11.96 0 Td[(dsijk,^b. (2)Continuingthisanalysis,theexpressionin( 2 )isatleastlnmindsijk,^b+1 R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+p,1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()lnmindsijk,^b R(1)]TJ /F3 11.955 Tf 11.96 0 Td[(p)+p,1+1 jSjln(vijk) (2)=psijk,^b+1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()psijk,^b+1 jSjln(vijk) (2)=jj(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj)]TJ /F5 11.955 Tf 21.26 0 Td[(dsijk,^b dsijk,^b+1)]TJ /F5 11.955 Tf 11.95 0 Td[(dsijk,^bpsijk,^b+1+dsijk,^b+1)-222(jj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj dsijk,^b+1)]TJ /F5 11.955 Tf 11.96 0 Td[(dsijk,^bpsijk,^b+1 jSjln(vijk) (2)= psijk,^b+1)]TJ /F5 11.955 Tf 11.96 0 Td[(psijk,^b dsijk,^b+1)]TJ /F5 11.955 Tf 11.96 0 Td[(dsijk,^b)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj)]TJ /F5 11.955 Tf 21.25 0 Td[(dsijk,^b+1+psijk,^b+1+1 jSjln(vijk)!wsijk,^b (2)=t(x,w)sijk, (2)where( 2 )followsbyconcavityofthenaturallogarithmfunction,( 2 )followsfrom( 2 ),and( 2 )followsfromdeningt(x,w)sijkasthes-thtermintheoutersummationof( 2 )plus(1=jSj)ln(vijk)andthefactthatwsijk,b=0forallb6=^b.Since 23

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( 2 )( 2 )holdstrueforanysetofmonitorlocationsxcorrespondingtoanyfeasiblesolution(x,f)2F( 2 )(anditscorrespondingsolution(x,w)2F( 2 )),G(x,f)isgreaterthanorequaltotheobjectivefunctionvalueof( 2 )at(x,w). Toobtainasingle-stagelower-boundformulation,weperformasimilartransformationon( 2 )asperformedon( 2 ).Thedualtothesecond-stageproblemof( 2 )minimizesu1,subjecttoATut(x,w);combiningthisdualwiththerststageof( 2 ),weobtaintheone-stageproblem minu1 (2a)s.t.ATut(x,w) (2b)(x,,w)2X( 2 ), (2c)whereX( 2 )=fx2X,0,w2W: (2d)2Xo=1(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk+sijk)oXb2Bsijkdsijk,b+1wsijk,b8s2S,i2Infncg,j,k2J (2e)xs,o)]TJ /F5 11.955 Tf 11.96 0 Td[(cijk,o+sijk,ocijk,o)]TJ /F5 11.955 Tf 11.96 0 Td[(xs,o8o2f1,2g,s2S,i2Infncg,j,k2J (2f)xs,2xs+1,28s2SnfjSjgg. (2g)Constraints( 2f )ensurethatthefollowingrelationshipholds:2Xo=1(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk+sijk)o=k(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)k, (2)whereelementsattaintheirminimumpossiblevaluesasdictatedby( 2b )and( 2e ),whichguaranteesthatxs)]TJ /F9 11.955 Tf 12.59 0 Td[(cijk+sijk=jjxs)]TJ /F9 11.955 Tf 12.59 0 Td[(cijkjj.Observethatt(x,w)ijkisnonlinearin( 2 )duetothemultiplicationofxandw.Tolinearizetheseterms,we 24

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canaddadditionalvariablesandconstraintsasprescribedin[ 49 ],butattheexpenseofincurringafarmorecomplexmodel.Instead,dene(x,)sijk,btobethelinearsegmentoft(x,w)ijkcorrespondingtomonitorsandsegmentb,i.e.,(x,)sijk,b=psijk,b+1)]TJ /F5 11.955 Tf 11.95 0 Td[(psijk,b dsijk,b+1)]TJ /F5 11.955 Tf 11.95 0 Td[(dsijk,b 2Xo=1(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk+sijk)o)]TJ /F5 11.955 Tf 11.96 0 Td[(dsijk,b+1!+psijk,b+1. (2)Wethenemploythefollowingformulation,inwhich( 2b )and( 2c )belowreplaceconstraints( 2b ). minu1 (2a)s.t.ATug (2b)sijk(x,)sijk,b)]TJ /F5 11.955 Tf 11.95 0 Td[(Msijk,b(1)]TJ /F5 11.955 Tf 11.96 0 Td[(wsijk,b)8s2S,i2Infncg,j,k2J,b2Bsijk (2c)(x,,w)2X( 2 ), (2d)wheregijk=Xs2Ssijk+ln(vijk),8i2Infncg,j,k2J, (2)andMsijk,b=max()]TJ /F5 11.955 Tf 13.15 8.08 Td[(psijk,b+1)]TJ /F5 11.955 Tf 11.95 0 Td[(psijk,b dsijk,b+1)]TJ /F5 11.955 Tf 11.95 0 Td[(dsijk,bdsijk,b+1+psijk,b+1)]TJ /F7 11.955 Tf 11.95 0 Td[(ln(p),psijk,b+1)]TJ /F5 11.955 Tf 11.96 0 Td[(psijk,b dsijk,b+1)]TJ /F5 11.955 Tf 11.96 0 Td[(dsijk,b(L+H)]TJ /F5 11.955 Tf 11.95 0 Td[(dsijk,b+1)+psijk,b+1), (2)8s2S,i2Infncg,j,k2J,b2Bsijk.Constraints( 2c )denesijktobetheadversary'sevasionprobabilitywithrespecttomonitorsandarcijk,asmeasuredbythepiecewise-linearunderestimationfunction.Observethat( 2c )isbindingforthesegmentb2Bsijkforwhichwsijk,b=1,providedthattheM-constantsin( 2 )aresufcientlylargetoguaranteethat( 2c )issatisedwhenwsijk,b=0.Toachievethis,wecomputethemaximumamountbywhichagivenlinesegmentoverestimates 25

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thepiecewise-linearfunction,whichoccursateitherboundaryoftherangeofpossibledistancesfromstoijk.Accordingly,therst(second)termofthemaximizingexpressionof( 2 )isthedifferencebetweenthesegmentfunctionandthepiecewise-linearfunctioncomputedatadistanceofzero(L+H).(NotethatthetermL+Husedin( 2 )forthemaximumdistancebetweenmonitorsandarcijkisconservative;inourcomputations,weinsteadusethetighterboundmaxfnc)]TJ /F5 11.955 Tf 11.96 0 Td[(cijk,1,cijk,1)]TJ /F7 11.955 Tf 11.95 0 Td[(1gL nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1+maxfnr)]TJ /F5 11.955 Tf 11.95 0 Td[(cijk,2,cijk,2)]TJ /F7 11.955 Tf 11.96 0 Td[(1gH nr)]TJ /F7 11.955 Tf 11.95 0 Td[(1.Preliminaryresultsrevealthatthissubstitution,whichresultsinatightenedvaluefortheM-parameters,leadstoimprovedrunningtimes.) 2.3OptimalContinuous-SpaceAlgorithmObservethatasidefromthebinaryrestrictionsonw,( 2 )isaconvexproblem,andisfareasiertosolvethanthe(exact)nonconvexproblem( 2 ).Wethusprescribeanalgorithmthatsolvesaseriesofinstances( 2 ),whichyieldlowerandupperboundsontheoptimalobjectivefunctionvalueof( 2 )inSection 2.3.1 .Section 2.3.2 presentsanupper-boundingimprovementschemeforacceleratingtheconvergenceofourproposedalgorithm.Beforepresentingourapproach,werstpresentthefollowingresult. Lemma1. Letn2Z+,0LiUi1,8i=1,...,n,and0Ui)]TJ /F3 11.955 Tf 12.73 0 Td[(=n,8i=1,...,n.Usingtheforegoingassumptions,wehavenYi=1Ui)]TJ /F6 7.97 Tf 17.31 14.95 Td[(nYi=1Li
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where( 2 )followsbyL1>U1)]TJ /F3 11.955 Tf 11.1 0 Td[(=n,and( 2 )isduetotheassumptionthat=n>0and0Li1,8i=1,...,n.Repeatingtheanalysisin( 2 )( 2 )forj=2yieldsnYi=1Ui)]TJ /F6 7.97 Tf 17.31 14.95 Td[(nYi=1Li0,andinitializeUB=0,LB=,andxandftobeempty. 1. Solve( 2 ),obtaining(^u,^,^x,^,^w).SetLB=^u1. 2. Fixingx=^x,solve( 2 ),andobtainanoptimalevasionpath_f. 3. ComputeG(^x,_f).IfG(^x,_f)
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5. Foreachs2S,i2Infncg,j,k2J,b2Bsijk,suchthat_fijk=1and^wsijk,b=1,computeGAPsijk,basthedifferencebetweentheactualprobabilitythattheevaderavoidsdetectionbymonitorsonarcijkandtheprobabilityestimatedbysegmentb.Thatis,GAPsijk,b=e (jj(^xs)]TJ /F8 7.97 Tf 6.59 0 Td[(cijk)jj))]TJ /F5 11.955 Tf 11.96 0 Td[(e((^x,^)sijk,b). (2)IfGAPsijk,bgap jSj(nc)]TJ /F8 7.97 Tf 6.59 -.01 Td[(1),thendothefollowing: (a) IncrementjBsijkjby1. (b) Forallq=jBsijkj+1,...,b+1(indescendingorder),setdsijk,q+1=dsijk,qandpsijk,q+1=psijk,q. (c) Setdsijk,b+1=jj(^xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jjandln(psijk,b+1)= (jj(^xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj).ReturntoStep 1 InStep0,UBandLBaresettobetheinitialupperandlowerboundsontheoptimalobjectivefunctionvalueto( 2 ),respectively,whilexandfaretheincumbentmonitorlocationsandadversarypath,respectively.InStep 1 ,weupdatethecurrentlowerboundtotheobjectivefunctionvalueobtainedfromsolvingthelower-boundproblem( 2 ).InStep 2 ,wendtheadversary'soptimalpath_fgivenmonitorlocationsat^x.Step 3 computesG(^x,_f),decideswhetherthisvalueisbetterthanthebestupperboundfoundsofar,andupdatesUB,x,andfaccordingly.InStep 4 ,weterminatewithanear-optimalsolutionifthecurrentoptimalitygapissufcientlysmall.Otherwise,wecomputeGAP-valuesasin( 2 )foreachmonitor-arccombinationsijk,anddistanceintervalb2Bsijksuchthat_fijk=^wsijk,b=1.Thereexistsatleastonesuchtermthatisatleastgap(jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)))]TJ /F8 7.97 Tf 6.58 0 Td[(1;weprovethisclaimasfollows.WeobservethatatmostjSj(nc)]TJ /F7 11.955 Tf 11.37 0 Td[(1)nonzeroGAP-valuesexistbecause_fijk=1fornc)]TJ /F7 11.955 Tf 11.37 0 Td[(1arcs,therearejSjmonitors,andforeachs2S,exactlyonevalue^wsijk,b=1.FromthesejSj(nc)]TJ /F7 11.955 Tf 12.28 0 Td[(1)terms,createasetofvaluesLl=e((^x,^)sijk,b)andUl=e (jj(^xs)]TJ /F8 7.97 Tf 6.58 0 Td[(cijk)jj),8l=1,...,jSj(nc)]TJ /F7 11.955 Tf 12.05 0 Td[(1),forsomesijkandbsatisfying_fijk=1and^wsijk,b=1.ByLemma 1 with=gap,n=jSj(nc)]TJ /F7 11.955 Tf 12.38 0 Td[(1),andUlandLldenedasabove,8l=1,...,n,weknowthatGAPsijk,bgap(jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)))]TJ /F8 7.97 Tf 6.58 0 Td[(1forsomes2S,i2Infncg,j,k2J,b2Bsijk.InStep 5 ,foreachsuchtermGAPsijk,bgap(jSj(nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)))]TJ /F8 7.97 Tf 6.59 0 Td[(1,letd02[dsijk,b,dsijk,b+1]be 28

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thedistancefrom^xstocijk.Denethetermbreakpointastheendpointofadistanceorprobabilityintervalb2Bsijk.Wecreateanewbreakpointinthepiecewise-linearfunctiont(x,w)sijkatd0,andreplacethisintervalwith[dsijk,b,d0]and[d0,dsijk,b+1].Asaresult,ifmonitorsisplacedatadistancejj(^xs)]TJ /F9 11.955 Tf 12.24 0 Td[(cijk)jjfromarcijk,thentherenolongerexistsagapatthispointbetweentheupdatedpiecewise-linearconcavelower-boundfunctionandthefunctionitapproximates.Wenowanalyzetheconvergenceofouralgorithminthefollowingtheorem. Theorem2.2. InExactAlg,GAP
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Case1:wsijk,jBsijkj=1:Inthiscasewehavejj(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jjR,implyingthat (jj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj)=0;thus,thefunctionin( 2a )andf,w(x)areequivalent.Case2:wsijk,jBsijkj=0:Inthiscase,wehave (jj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj)=lnjj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cjik)jj R(1)]TJ /F3 11.955 Tf 11.95 0 Td[(p)+pandthefunctionin( 2a )andf,w(x)areequivalent. Ifweperturbeachjj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jjin( 2 )bysomesmall >0,f,w(x)iscontinuouslydifferentiablewithrespecttoxs.Usingthisperturbation,observethatgivenanoptimaladversarypath_fbasedonasetofmonitorlocations^x,wecomputethenegativegradientdirection,rxq_f,^w(^x),withrespecttosomemonitorq2S.Wethenseekanoptimalplacementofmonitorqalongthisdirectionsuchthat_fremainsoptimal,whichisequivalenttosolving miny,Xi2InfncgXj2JXk2J'()ijk_fijk (2a)s.t.(0,0)^xq)]TJ 19.79 8.45 Td[(rxq_f,^w(^x) jjrxq_f,^w(^x)jj(L,H) (2b)ATy'() (2c)y1Xi2InfncgXj2JXk2J'()ijk_fijk (2d)0, (2e)where'()ijk=Xs2Snfqg (jj(^xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj)+ ^xq)]TJ 19.79 8.45 Td[(rxq_f,^w(^x) jjrxq_f,^w(^x)jj)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk+ln(vijk).Constraints( 2b )statethattheupdatedlocationofmonitorqmustbefeasibletotheoriginalproblem( 2 ).Constraint( 2c )ensuresfeasibilityofthedualto( 2 ).Constraint( 2d )requiresthatstrongdualityholdsbetween( 2 )anditsdual,while( 2e )guaranteesthatthestepsizeisnon-negative.(Notethatanti-symmetry 30

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constraints,( 2d ),havebeenomittedfrom( 2 )toallowforfeasbilityofthemost-improvedlocationformonitorq.)Aslongasconstraints( 2b )( 2e )aresatisedfortheupdatedlocationofmonitorq,thenwehaveasolutioninwhichtheadversary'soptimalpath_fremainsthesame,buttheevasionprobabilityfor_fislessthanorequaltoitspreviousvalue.Unfortunately,( 2 )isnotaconvexproblem,notingthat'()isnotaconvexfunctionof.Ratherthanattemptingtosolve( 2 )directly,weinsteadseeka2Proj(F( 2 ))=f:9ysuchthat(,y)2F( 2 )gsuchthatweobtainanoveralldecreaseintheobjectivefunctionvalueof( 2 )basedontheadjustedmonitorlocations,whereF( 2 )isthefeasibleregionof( 2 ).Tocomputeastepsize,weexecuteabisectionsearchprocedure,whichwecallSTEPSIZE.Forthisprocedure,wedenotetheinputmonitorplacementvectorby_xandtheadjustedmonitorsolutionbyx0.Notethatgiven,x0s=_xs,8s2Snfqg,andx0q=_x)]TJ /F3 11.955 Tf 11.96 0 Td[(rxq_f,^w(_x)=jjrxq_f,^w(_x)jj.WerstpresentasynopsisofthisalgorithminSTEPSIZE(COMPACT)andthenprovideamoredetaileddescriptioninSTEPSIZE(EXPANDED). STEPSIZE(COMPACT) 0. Selectalgorithmicparameters>0,step>0assmallterminationparameters,andMasamaximumstepsize.Initialize=1,stepL=0andstepU=M,respectively.ContinuetoStep 1 1. IfstepU)]TJ /F16 11.955 Tf 12.18 0 Td[(stepLstep,thenterminatewith=stepL.Otherwise,gotoStep 2 ifisvalid,andStep 4 ifisnotvalid. 2. Ifr_f,^w()=0andr_f,^w()()]TJ /F3 11.955 Tf 12.46 0 Td[()<0,thenthereexistsalocalminimumof( 2a )at.Terminatewith=stepLinthiscase.Otherwise,gotoStep 3 3. SetstepL=.IfstepU=M,thenset=min2,M,andotherwiseset=(stepU+stepL) 2.GotoStep 1 4. SetstepU=and=(stepU+stepL) 2,andgotoStep 1 Wesaythatisvalidifandonlyifr_f,^w()0,2Proj(F( 2 )),and_f,^w(x0)_f,^w(_x),i.e.,theobjectivefunctionisnotascendinginoursearchdirectionatastepsize 31

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STEPSIZE(EXPANDED) 1: Choose>0,step>0assmallterminationparameters,andMasamaximumstepsize 2: Initialize=1,stepL=0,stepU=M,Ubool=False 3: whilestepU)]TJ /F16 11.955 Tf 11.96 0 Td[(stepL>stepdo 4: whileisvalidandstepU)]TJ /F16 11.955 Tf 11.96 0 Td[(stepL>stepdo 5: ifr_f,^w()=0then 6: ifr_f,^w()()]TJ /F3 11.955 Tf 11.96 0 Td[()<0then 7: return=stepL 8: endif 9: endif 10: stepL= 11: ifUbool=Falsethen 12: =min2,M 13: else 14: =(stepU+stepL) 2 15: endif 16: endwhile 17: whileisnotvalidandstepU)]TJ /F16 11.955 Tf 11.96 0 Td[(stepL>stepdo 18: stepU= 19: Ubool=True 20: =(stepU+stepL) 2 21: endwhile 22: endwhile 23: =stepL 24: return 32

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of,andispartofafeasiblesolutionto( 2 )thathasasmallerobjectivefunctionvaluethantheonecorrespondingto_x.Inlines 1 and 2 ofSTEPSIZE(EXPANDED),weinitializesomealgorithmicparametersandsettheinitialstepsizelowerboundasstepL=0andupperboundasstepU=MforsomeparameterM.Thestepsizeisinitializedat=1.ABooleanvariableUbool,whichequalsFalse(True)ifthecurrentupperboundis(isnot)stillatitsinitialvalue,andishenceinitiallysetasFalse.Thewhileloopinlines 3 22 iteratesuntilthedifferencebetweenstepLandstepUisnotmorethansometerminationparameterstep>0.Therstinnerwhileloop,line 4 16 ,iterateswhileisstillvalidandthedifferencestepU)]TJ /F16 11.955 Tf 12.22 0 Td[(stepLisgreaterthanstep.Insidethiswhileloop,wecheckifthereexistsalocalminimumof( 2a )atinlines 5 8 .Ifso,weterminatewith=stepL;otherwise,inline 10 ,weupdatestepLtothevalueof,becausealocalminimumexistsin[,stepU].Lines 11 15 update.Iftheupperboundstillequalsitsinitialvalue,weupdatetotheminimumof2andtheinitialupperbound;otherwise,weupdatetothemidpointvalueofstepLandstepU.Eventually,thealgorithmterminateswithalocalminimum,orexitstherstinnerwhileloopandentersthesecondinnerwhileloop(lines 17 21 )giventhatstepUandstepLstilldifferbymorethanstep.Insidethissecondinnerwhileloop,weupdatestepUtothevalueofinline 18 ,becauseentranceintothiswhileloopimpliesthatthereexistsalocalminimumin[stepL,].SincestepUisnolongeratitsinitialvalue,weupdatestepUsetaccordinglyinline 19 .ThenewvalueofissettothemidpointvalueofstepLandstepUinline 20 .Finally,ifthealgorithmreachesline 23 (insteadofterminatingatline 7 ),thenthedifferencebetweenstepUandstepLissufcientlysmall.WethensetthenalvalueofequaltostepLandterminate.WearenowreadytodevelopthefollowingMONITORLOCATIONIMPROVEMENTMETHOD(MLIM),given^x,^w,and_f,toiterativelyadjustmonitorlocations.WerstpresentthealgorithmsuccinctlyinMLIM(COMPACT)andthenfollowwithamorespecicdescriptioninMLIM(EXPANDED). 33

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MLIM(COMPACT) 0. Weinitializebychoosingiminandilimitaspositiveintegerparameters,whereimin0asaterminationparameter.Thealgorithmwillattempttoimproveeachmonitor'slocationuntileitheramaximiumlimitofilimitmonitorimprovementiterationsisperformed,oruntiltheobjectiveimprovementoveriminiterationsbecomeslessthanimin.Initializetheupdatedlocations_x=^x.Setq=1,whichwillrepresenttheindexofthemonitorwhoselocationisbeingupdated. 1. Initializetheiterationcounttoi=0,andsetthecurrentobjectivefunctionvaluez(0)to_f,^w(_x).GotoStep 2 2. Ifi=ilimit,orifbothiiminandz(i)]TJ /F6 7.97 Tf 6.59 0 Td[(imin))]TJ /F5 11.955 Tf 11.84 0 Td[(z(0)0(lines 7 11 ).Lines 12 14 updatethecurrentmonitorlocation_xqtox0qbymovingastepsizeinthenegativegradientdirection,rxq_f,^w(_x),whereisdeterminedbySTEPSIZE.Basedontheseupdatedmonitorlocations_x,theobjectivefunctionvaluez(i)ofthecurrentiterationiiscomputedinline 15 .Whenthealgorithmterminatesinline 18 ,theupdatedmonitorlocations_xarereturned.WenowdiscusstheconvergenceofMLIM.NotethatStep 3 ofMLIM,whichisthebottleneckofthealgorithm,iteratesjSjilimittimes.EachiterationofthisstepexecutesSTEPSIZE,whichisitselfnitelyconvergent,provenasfollows.SupposethatwedonotterminateinStep 2 ofSTEPSIZE(orelseniteterminationiscertainlyachieved). 34

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MLIM(EXPANDED) 1: Chooseimin0asaterminationparameter 2: Initialize_x=^x 3: forq=1tojSjdo 4: i=0 5: z(0)=_f,^w(_x) 6: whilei
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2.4Midcolumn-RestrictedFormulationDuetothefactthatevasionprobabilitiesarecomputedasanondecreasingfunctionofdistancefromamonitortoanarcmidpoint,ourinitialexperimentsindicatethatnumerousinstanceshaveoptimalornear-optimalsolutionsinwhichthemonitorsarelocatedatcolumncoordinatescorrespondingtothesearcmidpoints.Inanefforttoobtainaformulationthatexploitsthistendency,weinvestigatearestrictionthatonlyallowsmonitorstobeplacedsuchthatallmonitors'column-axiscoordinatesbelongtothesetf1.5,2.5,...,nc)]TJ /F7 11.955 Tf 12.48 0 Td[(0.5g.Visually,themonitorswillbelongtothemidcolumnofthespace,depictedbydottedlinesinFigure 2-3 .Accordingly,wewillrefertotheregionmidwaybetweencolumnslandl+1asmidcolumnl,8l=1,...,nc)]TJ /F7 11.955 Tf 12.52 0 Td[(1.Toaccount Figure2-3. Illustrationofmidcolumnpoints. forthisrestriction,weamendthepreviousmonitorplacementvariablestoincludebinaryvariablesxc,wherexcs,l=1ifmonitorsislocatedatmidcolumnlandxcs,l=0,otherwise.Asaresult,wewillrestrictvariablesinXtoadheretothefollowingadditionalconstraints: nc)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xl=1(l+0.5)xcs,l=xs,18s2S (2a)nc)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xl=1xcs,l=18s2S (2b)xcs,l2f0,1g8s2S,l=1,...,nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1. (2c) 36

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Thealgorithmforthemidcolumn-restrictedversionisthesameonedescribedinSection 2.3 ,exceptfortheincorporationofxcwithintheupdatedsetXusedinallformulationssolvedinthealgorithm.Also,thegradientsusedintheimprovementstepfromSection 2.3.2 areonlycomputedwithrespecttoxs,2-variables,withallxs,1-variablesremainingxed.Thefollowingtheoremestablishesaboundontheabsoluteoptimalitygapbetweentheoptimalobjectivefunctionvaluetothemidcolumn-restrictedversionof( 2 )andthatforproblem( 2 )itself. Theorem2.4. Let(x,f)and(x,m,f,m)beoptimaltoproblems( 2 )andthemidcolumn-restrictedversionof( 2 ),respectively.Then,deningQ=b(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)R=L+0.5cwehaveG(x,f)+mG(x,m,f,m)G(x,f), (2)wherem=Xs2S" (H))]TJ /F3 11.955 Tf 11.95 0 Td[( H+L 2(nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)+QXq=1 qL nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[( (q)]TJ /F7 11.955 Tf 11.95 0 Td[(0.5)L nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1#. (2) Proof. SeeAppendix A ByTheorem 2.4 ,themaximumdifferenceinprobabilityofevasionbetweentheoptimallocationfoundbysolving( 2 )andthemidcolumn-restrictedversionof( 2 )iseGm)]TJ /F5 11.955 Tf 11.96 0 Td[(eGm)]TJ /F8 7.97 Tf 6.58 0 Td[(m, (2)whereGm=G(x,m,f,m). 37

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2.5Full-DiscretizationFormulationWhiletheexactalgorithmpresentedinSection 2.3 forsolving( 2 )andthemidcolumn-restrictedversionofthisalgorithmbothconvergetotheirrespectiveoptimalsolutionsinanitenumberofiterations,thecomputationaltimeneededtosolvelargerprobleminstancesisoftenprohibitivelylarge.Thus,inanefforttoobtainnear-optimalsolutionsusingalesscomputationally-intensiveformulation,werestrictthefeasiblemonitorlocationstoasubsetofthemidcolumn-restrictedsetoflocationsXfromSection 2.4 .Deneabinaryvariablevectorsuchthats,lr=1ifmonitors2Sislocatedatdiscretelocationr2f1,...,Ngofmidcolumnl2f1,...,nc)]TJ /F7 11.955 Tf 12.4 0 Td[(1g;otherwise,s,lr=0.TheactualcoordinatelocationsxFDFwillbesuchthatxFDFs=nc)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xl=1NXr=1s,lr!lr8s2S, (2)where!lr=)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(l+0.5,(r)]TJ /F7 11.955 Tf 11.96 0 Td[(1)(nr)]TJ /F7 11.955 Tf 11.95 0 Td[(1)(N)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F8 7.97 Tf 6.58 0 Td[(1+1.Weformulatethefully-discretizedformulation,denotedasproblemFDF,asfollows. miny1 (2a)s.t.ATijkyXs2Ssijk8i2Infncg,j,k2J (2b)sijknc)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xl=1NXr=1s,lr,ijks,lr+ln(vijk)8s2S,i2Infncg,j,k2J (2c)nc)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xl=1NXr=1s,lr=18s2S (2d)s,lr2f0,1g8s2S,r=1,...,N, (2e)whereATijkistherowvectorofthetransposeofconstraintmatrixAcorrespondingtoarcijkands,lr,ijk= (jj(!lr)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj), (2) 38

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8s2S,i2Infncg,j,k2J,r=1,...,N.Constraints( 2b )and( 2c )areequivalentto( 2b )by( 2 ),( 2 ),( 2 ),and( 2 ).Constraints( 2d )and( 2e )restricteachmonitortobelocatedatexactlyonemidcolumnlocation.WeestablishinTheorem 2.5 aboundonthemaximumdifferencebetweenanoptimalobjectivefunctionvalueofthemidcolumn-restrictedversionof( 2 )andthatofFDF. Theorem2.5. Let(x,m,x,c,f,m)andy,FDF1beoptimaltothemidcolumn-restrictedversionof( 2 )andFDF,respectively.Then,deningQ=f)]TJ /F5 11.955 Tf 15.27 0 Td[(Qmd,...,0,...,QmdgandQmd=j(nc)]TJ /F8 7.97 Tf 6.59 0 Td[(1)R Lk,wehaveG(x,m,f,m)+FDFy,FDF1G(x,m,f,m), (2)whereFDF= H 2(N)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[( (0)+Xq2Qnf0g H 2(N)]TJ /F7 11.955 Tf 11.95 0 Td[(1)+qL nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[( qL nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1. (2). Proof. SeeAppendix A ByTheorem 2.5 ,themaximumdifferenceinevasionprobabilitybetweentheoptimalmonitorlocationsfoundbysolvingthemidcolumn-restrictedversionof( 2 )andmonitorlocationsfoundbysolvingFDFiseFDF)]TJ /F5 11.955 Tf 11.96 0 Td[(eFDF)]TJ /F8 7.97 Tf 6.59 0 Td[(FDF, (2)whereFDFistheoptimalobjectivefunctionvalueof( 2 ). 2.6ComputationalResultsInthissection,wecomparetheefcacyofExactAlg,MidcolAlg,andFDF,alongwiththequalityofsolutionsobtainedbythesealgorithms.Alloptimizationproblemsare 39

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solvedwithCPLEX12.3usingConcert2.9TechnologyonanIBMSystemx3650withtwoIntelE5640Xeonprocessorsand24GBmemory,andareimplementedusingC++.EachExactAlgandMidcolAlginstanceisinitializedwithjBsijkj=2,andthealgorithmsareallowedtorununtileitherreachingaterminationgapof0.01oracomputationaltimeof3600seconds,whichevercomesrst.Toevaluatetheeffectivenessofourproposedsolutionmethods,wegenerateasetoftestinstancesbyvaryingparametersnc,nr,jSj,andR.WesetL=1000andH=1000forallinstances.Werandomlygenerateasetofarcfactor(AF)vectors,denotedbyva,vb,andvc,andlabelthemasa,b,andc,respectively.Eachv-vectorisalistof22,275elements,representingthelargestnumberofvijk-parametersneededoverallinstancestested.Therstncn2relementsfromeachv-vectorservesasthearcfactorsforthatinstance,whereonlythreearcfactorvectorswereusedtobetterisolatetheimpactofthevariedparametersonthesolutionmethods.TablesthatomitacolumnforAFincludetheaverageoutputstatisticsovera,b,andcforeachinstance(onlyincludingintheaveragesthoseinstancesterminatinginunder3600seconds),whereexponentsonoutputstatisticsdenotethenumberoftheseAFinstancesunsolvedafter3600seconds,andaf3gdenotestheinstanceisunsolvedforallthreeAFvectors.InTable 2-1 ,weshowtheeffectivenessofincorporatingtheimprovementstep,MLIM,withinExactAlg.ThecolumnlabeledTimereferstotheCPUtime(inseconds)requiredtosolvetheinstances,andIterationsreferstothenumberofExactAlgiterationsreachedbeforethealgorithmterminated.Foreach(nc,nr,p)-instanceofthistable,wetesteddetectionradiiof100unitsand1.25L=(nc)]TJ /F7 11.955 Tf 12.98 0 Td[(1)units(i.e.,alengthresultinginadetectionradiusspanningatleasttwocolumnmidpoints),andxedjSj=2.IncorporatingMLIMintheseinstancesneversubstantiallyincreasescomputationaltime,andreducescomputationaltimebyatleast10%in12ofthe24instances,andbyatleast50%intwoinstances.(WefoundsimilarresultsforMidcolAlg.)Thus,weutilizeMLIMwithinExactAlgandMidcolAlgintheremainingcomputational 40

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Table2-1. MLIMEffectivenessinExactAlg;nc=f4,7,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)g,andp=f0.01,0.25g. NoMLIMMLIM ncnrRpIterationsTimeIterationsTime 451000.2566612451000.012713927116454170.251211012129454170.0143f1g679f1g43f1g556f1g471000.25319320471000.018104448474170.251521214229474170.0136f1g2252f1g35f1g1712f1g751000.25333330751000.011128111269752080.25334334752080.012078118699771000.2583463129771000.016f1g1199f1g4916772080.2584166365772080.01f3gf3gf3gf3g1051000.2539621111051000.013f1g204f1g2f1g138f1g1051390.25321132121051390.0181176710451071000.25461335071071000.01f3gf3gf3gf3g1071390.2561310611001071390.01f3gf3gf3gf3g experiments.(NotethatacompletelistingofresultsforTable 2-1 canbefoundinTables B-1 and B-2 inAppendix B .Allinstancesdenotedwithaunderacolumndonotreachaterminatingcriterioninunder3600seconds.)Table 2-2 comparesthesolutionqualityobtainedbyourthreeapproaches,whereColumnBndsGapgivesthedifferencebetweenthenalupper-boundvalueofMidcolAlg(orFDF)andthenallower-boundvalueofExactAlg.ColumnTheoGapreportsthemaximumgapthatcanexistbetweenthenalupperboundforMidcolAlg(oroptimalobjectivefunctionvalueofFDF)andthatof( 2 ),whichisgivenbytheworst-casegapvaluesgivenin( 2 )(orthecombinationof( 2 )and( 2 )forFDF).MidcolAlgrequiresanaverageof286secondslesscomputationaltimethanExactAlg.Importantly,fortheinstanceswherebothExactAlgandMidcolAlgterminateinunder3600seconds,theBndsGapisnomorethan0.002overtheseinstances,which 41

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Table2-2. ComparisonofSolutionMethodswithnc=f10,15,20g,nr=10,jSj=f2,4g,R=100,p=0.75,andN=10. ExactAlgMidcolAlgFDF ncjSjAFTimeTimeBndsGapTheoGapTimeBndsGapTheoGap 102a801090.0040.06610.0270.126102b93980.0010.07110.0010.102102c9911700.06820.0050.107104a17408340.0020.03530.0120.064104b10833880.0010.04230.0150.074104c33378290.0020.03720.0070.063152a2052040.0030.03060.0050.085152b4322810.0020.02940.0100.086152c25328400.03160.0060.089154a360036000.0270.01090.0290.024154b360036000.0230.00760.0250.020154c360036000.0220.00870.0250.021202a135411290.0010.022120.0050.066202b18998630.0040.02070.0060.060202c4056960.0030.019100.0120.059204a360036000.0090.006190.0090.008204b360036000.0080.003130.0080.007204c360036000.0060.004190.0070.006 showsthatallofourinstanceshaveoptimalornear-optimalsolutionsatmidcolumnlocations.Amonginstancesconvergingwithintheallottedtime,MidcolAlgrequiresanaverageof486secondstoconverge,whileFDFtakesonly5seconds.ThemaximumprobabilitydifferencebetweentheoptimalobjectivefunctionvalueforFDFandthatof( 2 )isnomorethan0.029overall18instancesofthistable,andlessthan0.01theterminationgapforExactAlgin11instances.Also,notethatthecomputationaltimeofExactAlgtendstodecreaseasthemonitordampeningparameter(p)increasesfrom0.01to0.25(withsimilarresultsforMidcolAlg).Thisresultislogicalbecauseforanymonitor-arccombinationsijk,thepiecewise-linearconcavefunctionthatunderestimatesthenaturallogarithmcurveontheprobabilityinterval[p,1]becomestighteraspincreases.ThecomputationaltimesofthesmallerinstancesinTable 2-2 showthatExactAlgisefcientinsolvingtheseinstances.Hence,ExactAlgisthemostappropriatealgorithmofthethreefortheseinstances.However,astheinstancesizesgrowlarger,FDFbecomesthemostappropriatemethod:Itscomputationaltimesareverysmallcomparedtothat 42

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ofExactAlg,anditsTheoGapvaluesdecreaseasthesizeofthegridandnumberofmonitorsplacedincreases. Table2-3. EffectofponExactAlgwithnc=f4,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)g,andp=f0.01,0.25,0.75g. ncnrRpTimencnrRpTime 451000.7511051000.7511451000.25121051000.25111451000.011161051000.01138f1g454170.7521051390.7510454170.251291051390.25212454170.01556f1g1051390.011045471000.7531071000.7530471000.25201071000.25507471000.01481071000.01f3g474170.7541071390.7531474170.252291071390.251100474170.011712f1g1071390.01f3g ComputationaltimeofExactAlgtendstodecreaseasthemonitordampeningparameter(p)increases,asshowninTable 2-3 .(MidcolAlgexhibitssimilarresults.)Thisresultislogicalbecauseforanymonitor-arccombinationsijk,thepiecewise-linearconcavefunctionthatunderestimatesthenaturallogarithmcurveontheprobabilityinterval[p,1]becomestighteraspincreases.(NotethatthecompleteresultsforTable 2-3 canbefoundinTable B-3 inAppendix B .) Table2-4. EffectofNonFDFSolutionwithnc=f20,40g,nr=f10,15g,jSj=f2,4g,R=200,andp=0.75. ncnrjSjNTimeTheoGapncnrjSjNTimeTheoGap 20102650.06325401026240.00504201021070.050444010210450.00435201046110.004924010461060.000032010410190.0038540104101430.00002201526150.15519401526970.032422015210250.1287740152102100.02949201546500.027414015463360.000832015410930.02378401541010020.00070 Tables 2-4 and 2-5 reportthesolutionqualityobtainedbyFDFonlargerinstances,whicharetoodifculttosolvewithin3600secondsusingExactAlgorMidcolAlg.Table 2-4 illustratesthatthesolutiontimeincreasesnotonlywithanincreaseingridsizeandnumberofmonitorsplaced,butalsowithanincreaseinthenumberoffeasible 43

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Table2-5. FDFSolutionQualityonLargeInstanceswithnc=f80,100g,nr=15,jSj=f2,4g,R=f100,200g,p=f0.75,0.95g,andN=6. ncjSjRpTimeTheoGapncjSjRpTimeTheoGap 8021000.954990.002526410021000.958700.00070598021000.758510.002947210021000.751331f1g0.0006585f1g8022000.955390.002365610022000.958630.00061308022000.7511890.001226910022000.751703f1g0.0002057f1g8041000.9512080.000016310041000.9525190.00000118041000.751756f1g0.0000068f1g10041000.75f3gf3g8042000.9513150.000010410042000.9527060.00000068042000.752362f2g0.0000003f2g10042000.75f3gf3g locationspercolumn(N).ForTable 2-4 instances,theaveragesolutiontimegrowsfrom81to193whenNjumpsfrom6to10.Furthermore,thistablealsoshowsthattheaverageTheoGapdecreasesfrom0.036to0.030whenNincreasesfrom6to10.TheseresultsareintuitivebecauseanincreaseinNleadstoanincreaseinthenumberofbinaryvariablesinFDF.Thisincreaseinproblemsizetendstoincreasethetimerequiredtosolveeachlower-boundproblem,andalsoleadstoasmallermaximumpossiblegapbetweentheoptimalobjectivefunctionvalueof( 2 )andthatofFDF.Table 2-5 includesthelargestinstancessolvedwithin3600secondsusingthegivenmemorylimits.Amongtheinstancessolvingwithin3600seconds,theaverageTheoGapis0.001,andneverexceeds0.003.NotethatinTables 2-4 and 2-5 ,ouralgorithmsdonotutilizeMLIMbecausetheverysmallTheoGapvaluesshowthatMLIMisnotnecessarytoimprovethemonitorpositions.(AcompletelistingofresultsforTables 2-4 and 2-5 canbefoundinTables B-4 and B-5 ,respectively,inAppendix B .)Wealsoobservethatasthedetectionradiusincreases,TheoGapdecreasesforFDF,butatamuchgreaterratethanthesolutiontimeincreases.FortheresultsinTable 2-5 ,asRincreasesfrom100to200,weobservea64%decreaseinTheoGapwithonlyan18%increaseincomputationaltime.TofurtherexpresstheeffectivenessofFDFonachievinghigh-qualitysolutions,notethatFDFTheoGapvaluescomputedforallTable 2-2 instancesisaslargeas0.126,eventhoughFDFBndsGapisnomorethan0.029.Thisresultimpliesthatalthoughourtheoreticalboundonthegapbetween 44

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theoptimalobjectivefunctionvalueofFDFandthatof( 2 )issmallforinstancesofthelargestgridsize,itisconservativeandmaystillbesignicantlygreaterthanthetruegap. 45

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CHAPTER3AROBUSTSENSORCOVERINGANDCOMMUNICATIONPROBLEM 3.1MotivationandLiteratureReviewManycommunicationapplicationsrequiretheplacementofsensorsinsomeareaofinteresttogatherinformationfromtargets.Additionally,itmaybeadvantageoustoplacethesesensorsinawaythatmaximizesweightedcommunicationowamongthem,e.g.,torelayinformationamongsensors,ortoconstitutesomefully-orsemi-connectedsecuritygrid[ 35 ].Thus,twoconceptsparamounttosensordeploymentincludecover-age,whichdescribestheabilityofsensorstomonitorparticulartargets,andconnec-tivity,whichdescribestheextentofcommunicationamongsensors[ 24 ].Tomeasuretheabilityofsensorstocommunicatewithoneanother,weanalyzethequantityofinformationthatcanbetransferredfromonesensortoanotherbothdirectlyandthroughother(intermediate)sensors.Thestrengthofcommunication(whichrestrictsow)amongapairofsensorsistypicallyafunctionofthedistancebetweenthem.Avastarrayofevents,includingairandwindcurrents,naturaldisasters,andadversarialactions,couldcausethesesensorstodriftfromtheiroriginallocationsanddisruptconnections[ 36 ].Thus,thereissomeuncertaintyastothenallocationsofthesesensors.Givenconstraintsonthetypesofsensordisplacementsthatcouldoccur,thereexistsatarget-coveragezoneforeachsensor,suchthatthesensorcoversanytargetwithinthisareaevenafterthesensorisdisplaced.Ourproblem,whichwecalltherobustsensorcoverandcommunicationproblem(RSCCP),seekstoplaceasetofsensorsinacontinuousspacesuchthateachtargetlieswithinthetarget-coveragezoneofatleastonesensor,whilethesumof(weighted)communicationowamongallsensorpairsismaximized.Figure 3-1 illustratestheRSCCPwithafeasiblesensorlocationsolutionthatcoversalltargets.Optimizingsensorplacementwithagoalofmaintainingcoverageandconnectivityrequirementsiswidelystudiedintheliterature(see[ 34 ]forasurvey).TheRSCCP 46

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Figure3-1. Afeasiblesetofeight(initial)sensorlocationsguaranteeingcoverageofninetargets. arises,forinstance,insonarapplications,wheresensorsmustshareinformationaboutobservedobjects,withadegradationincommunicationoccurringasthedistancebetweensensorsincreases.Anotherexampleisawirelesssensornetworkinchargeofrelayingspatialinformation,suchasthetemperatureofvariousareaswithinafacility.Asthesensorsmovefartherapart,thetransmissionqualitydegrades.Thismayinturnrequireretransmissionofdata,whichcanbeverycostlyandmaydecreasetheoverallnetworklifetime[ 45 ].Wangetal.[ 59 ]useageometricapproachtosystematicallyplacesensors,testingtheirmethodsthroughsimulation.Baietal.[ 9 ]analyzeadiscretesetofsensordeploymentpatternsandtheirabilitytomaintaincoverageandconnectivity.Guptaetal.[ 35 ]examineavariantoftheproblemwhereaminimumsubsetofsensors(whoselocationsaregivenasparameters)arechosentocompletelycoveranarea.Baumgartneretal.[ 11 ]takeintoconsiderationthefactthatsensorlocationsmaychangeovertimeduetovariousevents(e.g.,weather)andusesequentialquadraticprogrammingtodeterminenear-optimalsensorlocations.However,theyassumethat 47

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futuredisplacementscanbeestimatedfromsensor-specicforecastsandmodels,whichwedonotassumeinourmodel.AlHasanetal.[ 7 ]formulateandsolveanintegerprogrambyabranch-and-cutscheme,whichseekstore-positionrandomlydeployedsensorstocoverasetoftargets.Bycontrast,ourproblemconsidersthecaseinwhichsensorsarestrategicallyplacedinitiallyandthenundergosome(constrained)displacement.Inourmodel,disruptionsdonotcompletelydisconnecttwosensors,butrathercausesomedegreeofdisconnectionbetweenthesensorsintermsofcapacitydegradation[ 23 ].Theideathatsensorscanperfectlyobservealltargetswithintheirdetectionradiusiswidelyutilizedintheliterature[ 24 35 41 ].However,variabilitiesinlinkqualitywithinasensingregionmayexistduetovariousphysicalobstructions[ 45 ],orotherstochasticphenomena.Chakrabartyetal.[ 24 ]solveavariantoftheproblemrequiringthatasubsetofsensorsnotonlycoversasetoftargets,butalsocorrectlyidentiesthetargets'locations.Whereastheirmodelrestrictssensorlocationstobechosenfromadiscretesubsetofgridpoints,ourmodelassumesthatthesensorscanbeplacedanywhereinthecontinuousspace.Therestofthechapterisorganizedasfollows.Section 3.2 providesaformaldescriptionandmathematicalprogrammingmodelsoftheproblem.Section 3.3 proposesapairofcutting-planemethodsforsolvingarelaxationoftheproblem,andboundsthemaximumoptimalitygapbetweentheoptimalsolutionsoftheoriginalproblemandtherelaxation.Section 3.4 discussesresultsfromcomputationalexperimentsthatutilizeeachoftheproposedsolutionmethods. 3.2FormalProblemDescriptionsandFormulationTheRSCCPseekstoplaceasetNofsensorsinacontinuoustwo-dimensionalrectangularspacecontainingasetTofxedtargetsatlocationst1,...,tjTj(tj2[(0,0),(L1,L2)],8j)thatmustbecoveredbyasensor.Thissectiondescribesa 48

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mathematicalmodelforthefeasibilityproblemofcoveringthetargets,formallystatesthecommunicationmeasureweseektooptimize,anddiscussesathree-stageformulationfortherobustmaximizationofthecommunicationmeasure.AsensorcoversatargetifthedistancebetweentheirlocationsisnogreaterthanDT.However,afterdeterminingtheinitialsensorlocations,thesensorscanthenbedisplacedbyadistanceofuptoR(DT)units.Thetarget-coveragezonearoundasensoristhusacircularregioncenteredatthesensorlocationhavingaradiusofDT)]TJ /F5 11.955 Tf 12.49 0 Td[(R.Thus,arobustfeasiblesolution(i.e.,onethatremainsfeasibleregardlessofsensorsdisplacements)requiressensorstobeplacedatlocationsx1,...,xjNj,suchthatforeveryj2T,jjxi)]TJ /F9 11.955 Tf 12.73 0 Td[(tjjjDT)]TJ /F5 11.955 Tf 12.73 0 Td[(Rholdstrueforatleastonei2N.(Inorderforsensoritocovertargetj,adisplacementofuptoRunitsmustnotforcethepairtobeatadistancegreaterthanDTfromeachother.)Tocapturethisdisjunctivecondition,wedenebinaryvariablesij,whichequal1ifsensori2Nisresponsibleforcoveringtargetj2T,and0otherwise.Also,forcomputationalease,weusethe`1-normdistancehereandreplacejjxi)]TJ /F9 11.955 Tf 12.69 0 Td[(tjjjwithP2o=1(xi,o)]TJ /F5 11.955 Tf 12.69 0 Td[(tj,o+wij,o),wherewij,o,8i2N,j2T,o=1,2,arenonnegativecontinuousvariablessatisfyingtj,o)]TJ /F5 11.955 Tf 11.96 0 Td[(xi,oxi,o)]TJ /F5 11.955 Tf 11.95 0 Td[(tj,o+wij,o. (3)Letxrepresentthecollectionofallxi,8i2N,letbethevectorofallij-variables,8i2N,j2T,andletwbethevectorofallwij,o-variables,8i2N,j2T,o=1,2.DeningMasalargeconstant(whosevaluewelaterspecify),thefollowingconstraintsrepresentfeasibilitycriteriaforthesensorlocations. Xi2Nij=18j2T (3a)2Xo=1(xi,o)]TJ /F5 11.955 Tf 11.95 0 Td[(tj,o+wij,o))]TJ /F5 11.955 Tf 11.95 0 Td[(M(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ij)DT)]TJ /F5 11.955 Tf 11.96 0 Td[(R8i2N,j2T (3b)tj,o)]TJ /F5 11.955 Tf 11.96 0 Td[(xi,oxi,o)]TJ /F5 11.955 Tf 11.95 0 Td[(tj,o+wij,o8i2N,j2T,o=1,2 (3c) 49

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xi,1xi+1,18i2NnfjNjg (3d)x1,1t1,1+DT)]TJ /F5 11.955 Tf 11.96 0 Td[(R (3e)xjNj,1tjTj,1)]TJ /F5 11.955 Tf 11.95 0 Td[(DT+R (3f)wij,o08i2N,j2T,o=1,2 (3g)ijbinary8i2N,j2T (3h)Constraints( 3a )guaranteethateverytargetiscoveredbyasensor.Constraints( 3b )statethatthe`1-normdistancebetweenxiandtjisnomorethanDT)]TJ /F5 11.955 Tf 13.1 0 Td[(Rwheneverij=1.WesetM=L1+L2)]TJ /F5 11.955 Tf 12.68 0 Td[(DT+Rtoensurethat( 3b )issatisedwhenij=0.Constraints( 3c )establishthe`1-normdistance.Toseethis,rstnoteby( 3b )thatafeasiblesolutionexistsinwhichallw-valuestaketheirsmallestpossiblevalues,i.e.,wij,o=maxf2(tj,o)]TJ /F5 11.955 Tf 12.9 0 Td[(xi,o),0gasrequiredby( 3c ).If(tj,o)]TJ /F5 11.955 Tf 12.89 0 Td[(xi,o)0,thenwij,o=2(tj,o)]TJ /F5 11.955 Tf 12.63 0 Td[(xi,o),andxi,o)]TJ /F5 11.955 Tf 12.64 0 Td[(tj,o+wij,o=tj,o)]TJ /F5 11.955 Tf 12.64 0 Td[(xi,oasdesired.Otherwise,if(xi,o)]TJ /F5 11.955 Tf 12.19 0 Td[(tj,o)>0,thenwij,o=0,andxi,o)]TJ /F5 11.955 Tf 12.19 0 Td[(tj,o+wij,o=xi,o)]TJ /F5 11.955 Tf 12.19 0 Td[(tj,oasdesired.Constraints( 3d )eliminatesomealternativeoptimalsolutionsbyensuringthathorizontal-axiscoordinatesofsensors1,...,jNjarenon-decreasing.Thus,sensors1'shorizontal-axiscoordinatecannotexceedt1,1+DT)]TJ /F5 11.955 Tf 12.54 0 Td[(Rorelsethehorizontal-axiscoordinatesofallsensorswouldbetoolargetocovertarget1.Thisconditionisenforcedby( 3e )toaidintighteningthemodel;ananalogousconditionisgivenin( 3f )forsensorjNj.(Notethattheseconditionsdonotrequiresensor1tocovertarget1,orsensorjNjtocovertargetjTj,neitherofwhichisnecessarilytrueatoptimality.Adiscussionofsymmetrybreakinginmixed-integerprogrammingcanbefoundin[ 33 48 56 ].)Constraints( 3g )and( 3h )staterequiredboundsonthe-andw-variables.Forconvenience,wedeneSasthesetofallpoints(x,,w)thatsatisfy( 3a )( 3h ),andX=ProjxS=fx:(x,,w)2Sg.IdentifyingafeasiblepointinSisequivalenttosolvingtheDISC-COVERproblem,knowntobeNP-complete[ 31 ];thus,RSCCPisNP-hard.DeneAasthesetofallsensorpairs,suchthat(i,j)2Aifandonlyifsensors 50

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Figure3-2. ArcsetforFigure 3-1 sensorsandtheresultingpartial-capacity(dashed)andfull-capacity(solid)arcs. i,j2Ncancommunicate.Dependingontheapplication,Amaybeastrictsubsetofalldistinctnodepairs.Denethearccapacitybetweentwosensorsiandj,(i,j)2A,asthelargestfractionofaunitofinformationthatcanbetransferredalongarc(i,j).Furthermore,weassumethat(i,j)2Aifandonlyif(j,i)2A,andthatarccapacitiesfor(i,j)and(j,i)areidentical.Arccapacitiesareassumedtobeconcave,nonincreasingfunctionsofdistancebetweensensors.Thearccapacitybetweensensorsiandj(for(i,j)2A)is1ifandonlyifthedistancefromitojisnomorethansomegivendistancelimitD10.Anarccapacityintheinterval[0,1)existsbetweentwosensorsthatarelocatedmorethanadistanceofD1fromeachother.Forexample,considerarcsetAoverthesensorsinFigure 3-1 thatcontains(only)arcs(i,j)forwhichj=i+1orj=i+2,8i2N,whereadditionismodulojNj.Figure 3-2 depictsthesearcs,wheresolidarcshavefullcapacity(equalto1),anddashedarcshavefractionalcapacity(duetothedistanceseparatingthesensors).Oncethesensorshavebeenplaced,thesecondstageinvolvesadisplacementofthesensorsbysomeentity(referredtoasanadversary)thatseekstominimizethe 51

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pairwisecommunicationowamongsensors.Itisusefultoenvisionanadversarythatdisplacesthesesensors,althoughinpractice,wesimplyseektoguaranteeaworst-caselevelofperformanceinoursensordeploymentsolution,evenifdisplacementisentirelyaccidental.Deneasthesetofallfeasibledisplacements.Wenowturnourattentiontothecommunicationmetricthatweseektomaximize.Thevalueofcommunicationowbetweentwo(distinct)sensorsq,r2Nisthefractionofoneunitofinformationthatcanbetransferredfromsensorqtosensorr(andvice-versa)usingthecapacitatedarcsinA.Thismeasureiscomputedastheminimumof1andthemaximumowoverarcsinAfromqtor.Theoverallobjectivethenseekstomaximizethesumofcommunicationowsoveralldistinctsensorpairs.Fornotationalease,letN=f(q,r):q2N,r2N,andq
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whereFS(k)=fi2N:(k,i)2Agistheforwardstarofsensork2NandRS(k)=fi2N:(i,k)2Agisthereversestarofsensork2N.Constraints( 3b )areow-balanceconstraintsforallsensorsexceptthesourceq2Nandsinkr2N.Constraints( 3c )ensurethattheowbetweensensorsdoesnotexceedtheirarccapacity.Constraints( 3d )ensurethatthetotalowbetweensensorq2Nandsensorr2Nisatmost1.Inordertotransformthethree-stageformulationin( 3 )toatwo-stageproblem,werstassociatedualvariablesqriwithconstraints( 3b ),uqrijwithconstraints( 3c ),andqrwithconstraints( 3d ).Treatingxandasxedvalues,thedualoftheinnerproblemof( 3 )isthefollowing: (x,)=minX(q,r)2N0@qr+X(i,j)2AuqrijCij(x,)1A (3a)s.t.qri)]TJ /F3 11.955 Tf 11.95 0 Td[(qrj+uqrij08(q,r)2N,(i,j)2A:i2Nnfq,rg,j2Nnfqg (3b)qr)]TJ /F3 11.955 Tf 11.95 0 Td[(qrj+uqrqj18(q,r)2N,j2FS(q) (3c)uqrij,qr08(q,r)2N,(i,j)2A (3d)qr08(q,r)2N (3e)qriunrestricted8(q,r)2N,i2Nnfq,rg, (3f)whereqrr=0,(q,r)2N.Combining( 3 )withtherst-stageproblem,weobtainthefollowingtwo-stageproblem: maxx2XminX(q,r)2N0@qr+X(i,j)2AuqrijCij(x,)1A (3a)s.t.Constraints( 3b ){( 3f ) (3b)2. (3c) 53

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Givenasetofsensorlocationsx,theadversary'sproblemisdenedbythesecond-stageminimizationproblemof( 3 ). 3.3SolutionApproachesNotethattheadversary'sobjectivefunctionin( 3a )isnonconvexwithrespecttouand,althoughtheadversary'sproblemissolvableasalinearprogramwhenisxed.However,solvingtheadversary'sproblembyenumeratingallpossible-vectors(givenx)isimpossible,becausethesetofpossible-vectorsisinniteinourmodel.Weinsteadobtainanitesampleof-vectors,suchthatjjissufcientlylargetoprovideaguaranteeonthemaximumdifferencebetweenascenario-approximationobjectivefunctionvalue(using)andtheoptimalobjectivefunctionvalueof( 3 )(using).InSection 3.3.1 ,wedeneaconvexformulationthatisequivalentto( 3 ).InSection 3.3.2 ,weboundthemaximumoptimalitygapasafunctionofjj.InSection 3.3.3 ,weproposetwocutting-planesolutionapproachesforsolvingthescenario-basedapproximation. 3.3.1EquivalentFormulationExpressionsFirst,deneConv(X)astheconvexhullofX.When=,thefollowingformulationyieldsanoptimalobjectivefunctionvalueequivalenttothatof( 3 ): maxx2Conv(X) (3a)s.t.)]TJ /F5 11.955 Tf 11.95 0 Td[(Z(x,)0, (3b)whereZ(x,)=min2(x,).(Observethat)]TJ /F7 11.955 Tf 12.52 0 Td[(min2(x,)0isequivalentto)]TJ /F3 11.955 Tf 11.99 0 Td[((x,)0,82.)DeneF( 3 )asthe(convex,polyhedral)feasibleregionof( 3 )given.Also,letEF( 3 )=f(qr,li,uqr,lij,qr,l):(q,r)2N,l=1,...,jEF( 3 )jgbethesetofextremepointstoF( 3 ).Notethatf=0isalwaysfeasibletotheinnerproblemof( 3 ),andthat(x,)jNj(jNj)]TJ /F7 11.955 Tf 19.11 0 Td[(1)=2;thus,( 3 )isfeasibleandbounded.Anequivalent 54

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formulationof( 3 )is(x,)=minl2EF( 3 )8<:X(q,r)2N0@qr,l+X(i,j)2Auqr,lijCij(x,)1A9=;. (3)Wenowprovethatthefeasibleregionof( 3 )isaconvexset.Foraxedu,,,,thefunctionminimizedin(x,)isaconcavefunctionofxbecause( 3 )minimizesanitenumber(jEF( 3 )j)ofconcavefunctionsofx.FunctionZ(x,)istheminimumofjjconcavefunctions,soZ(x,)isaconcavefunctionofx.Finally,because)]TJ /F5 11.955 Tf 9.3 0 Td[(Z(x,)isaconvexfunction,thefeasibleregionof( 3 )isaconvexset. 3.3.2ComputationofSampleSizeof-VectorsWenowutilizetheorypresentedin[ 19 ]and[ 40 ]todeterminethesamplesizejjof-vectorsneededtoprovideaprobabilisticguaranteeonthemaximumgapbetweentheoptimalobjectivefunctionvaluesto( 3 )and( 3 ).First,observethat( 3 )isoftheformofasampledconvexprogram(SCPN)in[ 40 ].TomaintainLipschitzcontinuityoftheconstraintfunction)]TJ /F5 11.955 Tf 12.4 0 Td[(Z(x,),weassumethatthesubgradient@Cij(x,)=@xisboundedwithaLipschitzconstantL,whosevaluewelaterspecify.KanamoriandTakeda[ 40 ]denethetailprobabilityontheworst-caseviolationatxoverasp(L,x)=PfZ(x,))]TJ /F3 11.955 Tf 11.96 0 Td[(L
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choiceofq()inthispaper.Theforegoingdevelopmentsallowustoapplythefollowingtheoremfrom[ 40 ]. Theorem3.1. SupposethatConv(X)isacompactset,theconstraintfunctionZ(x,)iscontinuousforx2Xand,andisacompactandconvexsetsuchthatthereexistsapositiveconstant>0andapoint2forwhichfy:jj)]TJ /F9 11.955 Tf 12.32 0 Td[(yjj0,thenthereexistsauniformlowerboundq(L)onthetailprobabilityp(L,x)over.NotethatwecanutilizeTheorem 3.1 inthecontextofourproblem,becauseConv(X)iscompact,Z(x,)iscontinuous,andisconvexandcompact.Next,letjj=min8<:W2N:2jNj)]TJ /F8 7.97 Tf 8.94 0 Td[(1Xi=0Wii(1)]TJ /F3 11.955 Tf 11.96 0 Td[()W)]TJ /F6 7.97 Tf 6.58 0 Td[(i9=;, (3)where2(0,1)isalevelparameter,2(0,1)isacondenceparameter,and2jNjisthedimensionofConv(X).Satisfyingthefollowingtheoremfrom[ 40 ]guaranteesthattheprobabilitythatthescenario-approximationobjectivefunctionvalueisnogreaterthanq)]TJ /F8 7.97 Tf 6.59 0 Td[(1()fromoptimalisatleast1)]TJ /F3 11.955 Tf 11.96 0 Td[(,whereq)]TJ /F8 7.97 Tf 6.59 0 Td[(1()=2LjNjRinthispaper. Theorem3.2. Supposethatthereexistsauniformlowerboundq(L)onthetailproba-bilityp(L,x)over.For,2(0,1),withjjatleastaslargeastheright-handsideof( 3 ),thefollowinginequalityholdsforanoptimalsolutionxof( 3 ):PZ(x,))]TJ /F7 11.955 Tf 22.7 0 Td[(maxx2Conv(X)Z(x,)q)]TJ /F8 7.97 Tf 6.59 0 Td[(1()1)]TJ /F3 11.955 Tf 11.96 0 Td[(. (3) 3.3.3Cutting-PlaneSolutionMethodsGivenasubsetofdisplacementvectors,weemployacutting-plane(CP)approachtondascenario-approximationsolutionto:maxx2Xmin2(x,). (3) 56

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WeexaminetwoCPalgorithms,onebasedongeneralizedBendersdecomposition[ 32 ]andtheotheradaptedfromKelley'smethod[ 13 ],inthefollowingsubsections.Werstdenetheinitialrelaxedmasterproblem(RMP) max (3a)s.t.x2X (3b)(v,x)2V, (3c)whereVisinitiallyempty,butwillbeiterativelydenedbyasetofafnevaluefunctioninequalitiesthatboundvintermsofx.(NotethatRMPcanbeboundedbyincorporatingtheconstraintjNj(jNj)]TJ /F7 11.955 Tf 18.87 0 Td[(1)=2into( 3c ).)WewilldynamicallygenerateCPsviathesolutionofsubproblems(SP)inthisprocess.Thesubproblemforeachdisplacementvectoristhethirdstageof( 3 ),whosedual(DSP)isgivenby( 3 ).ThestepsofourproposedCPalgorithmareasfollows. ALGORITHM-CP 1. Seta=0andLB=. 2. SolveRMP,obtaining^aand^xa;^aisanupperboundontheoptimalvalueof( 3 ). 3. SolveDSP,82.If^a>(^xa,),thenaddaCP(oftheformprovidedinSections 3.3.3.1 or 3.3.3.2 )to( 3c ).SetLB=maxfLB,min2(^xa,)g.IfanyCPsareadded,thenseta=a+1andrepeatStep 2 .Otherwise,terminatewith^xaasanoptimalsolution. AfterinitializingthealgorithminStep 1 ,wesolvetheRMPatiterationainStep 2 .InStep 3 ,weobtainjj(feasibleto( 3 ))solutions,eachrepresentingthemaximumobjectivefunctionvalueforagivenpair^xaand2.WeaddatleastoneCPperiterationofALGORITHM-CPtoRMPuntilweconvergetoanoptimalsolution.(Itisworthnotingthatpreliminarycomputationalexperimentsrevealthataddingeitheramost-violatedinequalityoranaggregatedinequality,ratherthaneveryviolated 57

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inequality,inStep 3 islesseffectivethanthecurrentprocedure.)ThisalgorithmperformsatmostjjjEDSPjiterations,whereEDSPisthesetofextremepointsforDSP. 3.3.3.1GeneralizedBendersDecompositionTheformofeachCPaddedtoALGORITHM-CPisX(q,r)2N0@^qr+X(i,j)2A^uqrijCij(x,)1A, (3)where,foraCPaddedduringiterationa,^qrand^uqrijrepresentanoptimalsolutionvectortoDSPwhenx=^xaand2.Thus,replacing( 3c )with( 3 )foreveryextremepointofDSPandforall2isequivalenttosolving( 3 ).Notethatbecausethedualfeasibleregionsdonotdependon,theextremepointstoeachsubproblemDSPareidentical.ConvergenceofthisalgorithmisguaranteedbecauseSP,82,isconvex[ 32 55 ].Now,supposethatwehavethefollowingconcavecapacityfunction(whichwecall`1-L):Cij(x,)=1)]TJ 13.16 8.81 Td[(jj(xi)]TJ /F21 11.955 Tf 11.95 0 Td[(i))]TJ /F7 11.955 Tf 11.95 0 Td[((xj)]TJ /F21 11.955 Tf 11.95 0 Td[(j)jj1)]TJ /F5 11.955 Tf 11.95 0 Td[(D1 D0)]TJ /F5 11.955 Tf 11.96 0 Td[(D1, (3)whereD0=L1+L2)]TJ /F7 11.955 Tf 13.28 0 Td[(2(DT)]TJ /F5 11.955 Tf 13.29 0 Td[(R)isthemaximumpossibledistancebetweenanytwosensorsi,j2N,andthedistanceatwhichCij(x,)=0.Observethatasthe`1-normin( 3 )increases,sodoesthesetoffeasiblesolutionsto( 3 ).Byincorporatingnonnegativevariables!ij,o,8i,j2N,o=1,2,wecanreplacethenonlinearfunctionjj(xi)]TJ /F21 11.955 Tf 13.36 0 Td[(i))]TJ /F7 11.955 Tf 13.36 0 Td[((xj)]TJ /F21 11.955 Tf 13.37 0 Td[(j)jj1in( 3 )withthelinearfunctionP2o=1((xi,o)]TJ /F3 11.955 Tf 11.96 0 Td[(i,o))]TJ /F7 11.955 Tf 11.96 0 Td[((xj,o)]TJ /F3 11.955 Tf 11.96 0 Td[(j,o)+!ij,0).Thus,wereformulatethethirdstageof( 3 )(whichwecall`1-L-SP)asfollowsforany^xand. maxX(q,r)2NXj2Nnfqgfqrqj (3a)s.t.Xj2FS(i)fqrij)]TJ /F12 11.955 Tf 26.19 11.36 Td[(Xl2RS(i)nfrgfqrli=08(q,r)2N,i2Nnfq,rg (3b) 58

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fqrij1)]TJ /F12 11.955 Tf 13.15 17.11 Td[(P2o=1((^xi,o)]TJ /F3 11.955 Tf 11.95 0 Td[(i,o))]TJ /F7 11.955 Tf 11.95 0 Td[((^xj,o)]TJ /F3 11.955 Tf 11.95 0 Td[(j,o)+!ij,o))]TJ /F5 11.955 Tf 11.96 0 Td[(D1 D0)]TJ /F5 11.955 Tf 11.96 0 Td[(D18(i,j)2A,(q,r)2N (3c)Xj2FS(q)fqrqj18(q,r)2N (3d))]TJ /F3 11.955 Tf 11.96 0 Td[(!ij,o2(^xi,o)]TJ /F3 11.955 Tf 11.96 0 Td[(i,o))]TJ /F7 11.955 Tf 11.96 0 Td[(2(^xj,o)]TJ /F3 11.955 Tf 11.95 0 Td[(j,o)8o=1,2,(i,j)2A (3e)fqrij0,8(i,j)2A,(q,r)2N (3f)!ij,o0,8o=1,2,(i,j)2A, (3g)where( 3e )and( 3g )ensurethat(^xi,o)]TJ /F3 11.955 Tf 11.96 0 Td[(i,o))]TJ /F7 11.955 Tf 11.96 0 Td[((^xj,o)]TJ /F3 11.955 Tf 11.96 0 Td[(j,o)+!ij,0j(^xi,o)]TJ /F3 11.955 Tf 11.96 0 Td[(i,o))]TJ /F7 11.955 Tf 11.95 0 Td[((^xj,o)]TJ /F3 11.955 Tf 11.96 0 Td[(j,o)j,8(i,j)2A,2,o=1,2. (3)(Notethattheright-handsideofconstraint( 3c )isanondecreasingfunctionofwij,o.Thus,iftheright-handshadowpriceofthisconstraintispositiveatoptimalityforsome(i,j)2A,(q,r)2N,thenjj(xi)]TJ /F21 11.955 Tf 9.49 0 Td[(i))]TJ /F7 11.955 Tf 9.49 0 Td[((xj)]TJ /F21 11.955 Tf 9.49 0 Td[(j)jj1=P2o=1((xi,o)]TJ /F3 11.955 Tf 11.95 0 Td[(i,o))]TJ /F7 11.955 Tf 11.96 0 Td[((xj,o)]TJ /F3 11.955 Tf 11.95 0 Td[(j,o)+!ij,0).)Associatingdualvariablesij,owithconstraints( 3e ),whereallotherdualvariables(,u,)from( 3 )areassociatedwiththeirpreviouscorrespondingconstraints,thedual(`1-L-DSP)of`1-L-SPis minX(q,r)2N0@qr+X(i,j)2Auqrij 1)]TJ /F12 11.955 Tf 13.15 17.12 Td[(P2o=1((^xi,o)]TJ /F3 11.955 Tf 11.96 0 Td[(i,o))]TJ /F7 11.955 Tf 11.96 0 Td[((^xj,o)]TJ /F3 11.955 Tf 11.96 0 Td[(j,o)))]TJ /F5 11.955 Tf 11.96 0 Td[(D1 D0)]TJ /F5 11.955 Tf 11.96 0 Td[(D1!1A+X(i,j)2A2Xo=1ij,o(2(^xi,o)]TJ /F3 11.955 Tf 11.96 0 Td[(i,o))]TJ /F7 11.955 Tf 11.96 0 Td[(2(^xj,o)]TJ /F3 11.955 Tf 11.96 0 Td[(j,o)) (3a)s.t.qri)]TJ /F3 11.955 Tf 11.95 0 Td[(qrj+uqrij08(q,r)2N,(i,j)2A,i2Nnfq,rg,j2Nnfqg (3b)qr)]TJ /F3 11.955 Tf 11.95 0 Td[(qrj+uqrqj18(q,r)2N,j2FS(q) (3c)X(q,r)2Nuqrij D0)]TJ /F5 11.955 Tf 11.96 0 Td[(D1)]TJ /F3 11.955 Tf 11.96 0 Td[(ij,o08o=1,2,(i,j)2A,i2Nnfrg,j2Nnfqg (3d)uqrij08(q,r)2N,(i,j)2A (3e) 59

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qr08(q,r)2N (3f)qriunrestricted8(q,r)2N,i2Nnfq,rg (3g)ij,o08o=1,2,(i,j)2A. (3h)Thus,theformofCPsaddedinALGORITHM-CPbecomesX(q,r)2N ^qr+X(i,j)2A^uqrij 1)]TJ /F12 11.955 Tf 13.15 17.11 Td[(P2o=1((xi,o)]TJ /F3 11.955 Tf 11.95 0 Td[(i,o))]TJ /F7 11.955 Tf 11.95 0 Td[((xj,o)]TJ /F3 11.955 Tf 11.95 0 Td[(j,o)))]TJ /F5 11.955 Tf 11.95 0 Td[(D1 D0)]TJ /F5 11.955 Tf 11.95 0 Td[(D1!!+X(i,j)2A2Xo=1^ij,o(2(xi,o)]TJ /F3 11.955 Tf 11.96 0 Td[(i,o))]TJ /F7 11.955 Tf 11.95 0 Td[(2(xj,o)]TJ /F3 11.955 Tf 11.95 0 Td[(j,o)). (3)ThestepsofthisCPalgorithmarethatofALGORITHM-CP,whereDSPand(x,)arereplacedwith`1-L-DSPandtheobjectivefunctionof( 3a ),respectively. 3.3.3.2Kelley'sMethodTocombatthepotentialnonlinearity(withrespecttox)oftheconcavecapacityfunctionCij(x,),8(i,j)2A,wecomparetheCPsgeneratedinALGORITHM-CPwithCPsbasedonKelley'smethod.DeningCgij(^xa,)asasubgradientwithrespecttoxofCij(x,)at^xaand@Cij(^xa,)asthesubdifferentialofCij(x,)at^xa,theCPsaddedinALGORITHM-CPareoftheformX(q,r)2N0@^qr+X(i,j)2A)]TJ /F7 11.955 Tf 5.73 -9.68 Td[(^uqrijCij(^xa,)+^uqrijCgij(^xa,)(x)]TJ /F7 11.955 Tf 11.81 0 Td[(^xa)1A, (3)where^qrand^uqraretheoptimalvaluesofqranduqrij,respectively,insolvingDSP.WedenotetheCPalgorithmbasedonKelley'smethodasALGORITHM-CP,whereallCPsaddedareoftheform( 3 ). 3.4ComputationalResultsInthissection,wecomparetheefcacyofusingtheCPsfromSections 3.3.3.1 and 3.3.3.2 inALGORITHM-CP.(Foreaseofdescription,werefertoCPsfromSection 3.3.3.1 asGBDCPsandthosefromSection 3.3.3.2 asKelley'sCPs.)AlloptimizationproblemsaresolvedwithCPLEX12.5usingConcert2.9TechnologyonanIBMSystem 60

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x3650withtwoIntelE5640Xeonprocessorsand24GBmemory,andareimplementedusingC++.Toevaluatetheeffectivenessofourproposedsolutionmethods,wegenerateasetoftestinstancesbyvaryingparametersjNj,jTj,DT,andD1,alongwithasetoftargetlocations.EachinstanceisinitializedwithL1=L2=1000,R=25,and==0.01.Werandomlygenerate25setsoftargetlocations.Notethattheparametervaluecombinationssuchthat(jNj=10,jTj=10,DT=200,andD1=25)or(jNj=10,jTj=12,DT=f100,200g)or(jNj=12,jTj=15,DT=f100,200g)aretestedoverall25targetlocationsets,whiletheremainingcombinationsaretestedoveronesetoftargetlocations.Alloutputexpressedinthesetablesareaveragesoverallruns(overthetargetlocationsets)foreachparametercombination.WesetA=f(i,j):i)]TJ /F7 11.955 Tf 13.16 0 Td[(1ji+1,j6=ig.Additionally,wesamplethe-vectorsfromtheuniformdistributionsuchthati,8i2N,ischosenrandomlyinR2withan`1-normmagnitudebetween0andR.Recallthatq)]TJ /F8 7.97 Tf 6.59 0 Td[(1()=2LjNjRistheinverseofthecumulativedistributionfunctionoftheuniformdistribution.SincewevaryjNj,wemoreaccuratelyrefertoq)]TJ /F8 7.97 Tf 6.59 0 Td[(1()usingasubscriptN(i.e.,q)]TJ /F8 7.97 Tf 6.59 0 Td[(1N()).(Notethatforeachinstancesize(intermsofjNj),thereisauniquejj-valueequaltothesmallestsamplesizeof-vectorsnecessaryfortheprobabilisticguarantee(detailedinSection 3.3.2 )ontheoptimalobjectivefunctionvalue.)Additionally,settheLipschitzconstantL=jNj(jNj)]TJ /F7 11.955 Tf 17.94 0 Td[(1)=(D0)]TJ /F5 11.955 Tf 11.96 0 Td[(D1)duetothefollowingsetofrelations:j(x,^))]TJ /F3 11.955 Tf 11.96 0 Td[((x,)j jj^)]TJ /F7 11.955 Tf 12.04 2.66 Td[(jj=X(q,r)2Ndqr D0)]TJ /F5 11.955 Tf 11.95 0 Td[(D1jNj(jNj)]TJ /F7 11.955 Tf 17.94 0 Td[(1) 2(D0)]TJ /F5 11.955 Tf 11.96 0 Td[(D1),wheredqr=j^)]TJ /F12 11.955 Tf 12.75 8.96 Td[(P(i,j)2A^uqrijCij(^x,^))]TJ /F12 11.955 Tf 12.74 13.27 Td[(qr)]TJ /F12 11.955 Tf 11.96 8.96 Td[(P(i,j)2AuqrijCij(x,)j,^qrand^uqrijareoptimalto(x,^),andqranduqrijareoptimalto(x,),andthelastinequalityfollowsbecausedqr1. 61

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ForinstancesinwhichjNj<15,thetablecolumnTimereferstotheCPUtimeinsecondsuntilALGORITHM-CPterminates(towithin0.01%ofoptimality).Therelativeoptimalitygap,referredtointhetablesasGap,fortheseinstancesis(q)]TJ /F8 7.97 Tf 6.59 0 Td[(1N(0.01)+0.0001)=(Bestpossibleobjectivevalueof( 3 ))(i.e.,themaximumpossiblegapbetweenanobjectivefunctionvalue(within0.01%ofoptimality)of( 3 )obtainedbyALGORITHM-CPandthatofanoptimalsolution).ForinstanceswherejNj=15,thevaluesinthecolumnTimearesecondsuntilasolutionisfoundwithin5%oftheoptimalobjectivefunctionvalueof( 3 ).ThevalueofthecolumnGapfortheseinstancesistheadditionoftherelativeoptimalitygapand(q)]TJ /F8 7.97 Tf 6.59 0 Td[(1N(0.01)+0.0001)=(Bestpossibleobjectivevalueof( 3 )). Table3-1. EffectofDTonSolutionQuality,D1=25 GBDKelley's jNjjTjDTjjCPsTimeGapCPsTimeGap 55100187425720150.001326932120.001355200187422656140.001122127140.001155300187418637120.001017352100.001010101003179612032820.0024423551680.002310102003179556792460.0021551451940.002110103003179551072450.0019635581810.001910121003179516062140.0024478391490.002410122003179565752790.0021559932020.002110123003179492661990.0019483831140.001912151003801692666040.0028709914270.002812152003801760726830.0025773135170.002512153003801894036540.0023880864040.002315151004412194687830.04272887010480.03411515200441249433310.0371137364300.040815153004412353157540.0271176823850.0271 Table 3-1 anditscorrespondingFigure 3-3 showtheeffectofvaryingDTonthesolutionquality.Amongtheseinstances,thoseusingGBDCPstakeapproximatelyoneminutelongerthanthoseusingKelley'sCPs.AsDTincreasesfrom100to200units,CPUtimedecreasesbyanaverageof21%forthosesolvedusingeithertypeofCPs.Ingeneral,thereisanegativecorrelation(morepronouncedusingKelley'sCPs)betweenDTandCPUtime.ThisresultmayinpartbeduetotheincreaseinfeasibleareawithwhichtoplaceamonitorasDTincreases.Inthelimit,asDTincreasestoaverylarge 62

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Figure3-3. CPUTimeComparisonofTable 3-1 Instances value(e.g.,DTmaxfL1,L2g),theproblemissolvedeasily(e.g.,anysolutionwithallsensorsplacedwithinD1)]TJ /F5 11.955 Tf 11.96 0 Td[(Rofeachotherisoptimal). Table3-2. EffectofD1onSolutionQuality,DT=200 GBDKelley's jNjjTjD1jjCPsTimeGapCPsTimeGap 5525187422656140.001122127140.00115550187422418170.001124833130.001155100187425213130.001225914120.00121010253179556792460.0021551451940.00211010503179651532490.0021454931550.002110101003179504033040.0021612021870.00221012253179565752790.0021559932020.00211012503179574442610.0021561232090.002110121003179538392510.0022527211240.00221215253801760726830.0025773135170.00251215503801759716530.0026758275510.002612151003801634258420.0026682945790.0026151525441249433310.0371137364300.04081515504412182356330.0325363597120.02681515100441249174580.042649172150.0426 Table 3-2 andFigure 3-4 comparethechangeintimeandnumberofCPsaddedasD1(i.e.,perfectcommunicationdistance)increases.Specically,thoseinstancesinvolvingGBDCPs(Kelley'sCPs)experiencea17%(21%)increaseinCPUtimeasD1increasesfrom25to50units.However,asD1increasesfrom50to100units,CPUtimeonlyincreasesby3%forthoseinstancesutilizingGBDCPs,whilethoseutilizingKelley'sCPsexperiencea32%decreaseintime.AsD1increases,thepotentialforalternative 63

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Figure3-4. CPUTimeComparisonofTable 3-2 Instances optimalsolutionstobecomeavailablemayincrease,becauseifD1(DT)]TJ /F5 11.955 Tf 10.91 0 Td[(R),thenanysensoratadistanceyD1fromanothersensorhasD1)]TJ /F5 11.955 Tf 12.16 0 Td[(yspacewithwhichtomoveinanydirectionwithoutaffectingcoverageorthecommunicationvaluebetweenthetwosensors.(IfD1=D0,theobjectivefunctionvalueismaximizedwheneverafeasiblesolutionto( 3 )isachieved.) Table3-3. EffectofCPTypeonSolutionQuality GBDKelley's jNjjTjjjCPsTimeGapCPsTimeGap 55187423250140.001222778110.001210103179590032670.0021576071860.002110123179502972120.0021515311510.002112153801753116880.0026730795030.002615154412206746760.0344241646460.0326 Table 3-3 andFigure 3-5 showaverageCPs,CPUtime,andGapcolumninformationforeachinstancesize(intermsofjNjandjTj).Withallotherparametersxed,increasingbothjNjandjTj,ingeneral,leadstoanincreaseinCPUtime.Onaverage,instancesutilizingKelley'sCPssolvefasterthanthoseusingGBDCPs;specically,39outof50instancessolvefasterwithKelley'sCPs(thelargestdifferencebeingjustundersixminutes)andtherestsolvingasfastorfasterwithGBDCPs(thelargestdifferencebeingjustoverfourminutes).Thisresultsuggeststhat,ingeneral, 64

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Figure3-5. CPUTimeComparisonofTable 3-3 Instances Kelley'sCPsaremoreeffectiveinsolvingtheseinstancesthanGBDCPswhenusingALGORITHM-CP.Allinstancesexecutedinunderhalfanhour,with21(18)minutesasthelargestCPUtimeusingGBDCPs(Kelley'sCPs).ItisworthnotingthatthereisnosignicantcorrelationbetweenchangeinCPUtimeandchangeinCPsasjNj,jTj,DT,orD1increases.(AsCPUtimeincreases,someinstancesexpressanincreaseinCPs,whileothersshowadecreaseinCPs.)Thisresultsuggeststhatthetimerequiredtosolvethesubproblemsisnotassignicantasthetimerequiredtosolvethemasterproblem. 65

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CHAPTER4ONTHECOMPLEXITYOFSPARSEMAXIMUMFLOWPROBLEMS 4.1MotivationandLiteratureReviewThemaximumowproblemseekstomaximizeowfromasourcenodeqtoasinknodetusingcapacity-constraineddirectedarcsinaownetworkG=(N,A).Avarietyofapplicationsexistforthemaximumowproblem,includingpipeline,road,telecommunication,andelectricitymodels[ 5 ].Theproblemiswell-knowntobesolvableinpolynomialtime([ 26 ]and[ 28 ]).However,asthemaximum-owproblemisadaptedtopracticalapplications,severaladditionaltypesof(sparsity-inducing)constraintsarisethatmaydisruptthiscomplexity.Forexample,theremayberestrictionsonthequantityofarcsthatcanreceiveow.Thistypeofconstraintiswrittenusingthe`0-normasjjxjj0s0,wherexisavectorthatdescribesowonarcsands02Z+isaboundonarcshavingpositiveow[ 21 ].Relevantapplicationsinvolvingthisrestrictionincludeshippingscenarios,where,forexample,owismaximallyroutedbetweenanoriginanddestinationthroughsomeboundedquantityofintermediateroadsegments.Asimilarapplicationinvolvesawatchguardrequiredforeacharcusedinatransportationnetwork,withalimitednumberoftotalwatchguardsavailable.Alternatively,thetotalsumofowthroughallarcsmaybeconstrained.Thisconstraintiswrittenusingthe`1-normasjjxjj1s1,wheres12N+isaboundonthetotalowthroughallarcs.Inwirelesssensornetworkproblemsseekingtomaximizeow,thisconstraintmayariseduetoenergyrestrictions,resultinginaboundonthemaximumamountofowallowedinthesystem[ 17 ].Anotherapplicationinvolvesdeterminingthemaximumowofgoodsthroughadistributionnetwork,wherenodesrepresentdistributioncentersandarcsrepresentmodesoftransportation,eachwithanassociatedcost,witharestrictionimposedontotaltransportationcost.Anadditionalapplicationdeterminesthemaximumowofdatapacketsbetweentwomachinesinacomputernetwork,withrestrictionsimposedontotaltransfercost[ 20 ].Observethatin 66

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compressedsensing,this`1-normconstraintistypicallyusedasamore(asymptotically)efcientwaytorecoverasignaloriginallyconstrainedusing`0-normconstraints[ 39 ].Additionalstudieshaveexaminedrestrictionsonthenumberofpathsusedtorouteow.Severalstudies[ 27 43 44 ]haveanalyzedtheunsplittableowproblem,whichconsidersacapacitateddirectedgraphalongwithasetofdemandsrequiredtoberoutedfromasetofsourcestoasetofsinks.Inthisproblem,eachdemandmustowalongasinglepathfromsourcetosinkwithoutviolatingcapacityconstraints.Otherstudies[ 10 22 ]haveexaminedthek-splittableowproblem,whichissimilartotheunsplittableowproblemwiththeexceptionthateachdemandmustowfromitssourcetoitssinkalongatmostkpaths.Similarly,MahjoubandMcCormick[ 47 ]analyzethecomplexityofseveralvariationsofintegermaximumowproblems(usingapath-owformulation)inwhichowisrestrictedtopathsofboundedlength.Theseauthorsprovethatifthenumberofarcsisgreaterthanthree,theproblemisNP-hard.However,thereisnotmuchintheliteratureexaminingarestrictiononthetotalquantityofowsplitsinmaximum-owapplications.Forexample,asingle-pathowconstraintisrelevantincomputernetworkswhen,inordertoproperlyconveyinformation(e.g.,amessage),datapacketscannotbesplitduringtravelfromanoriginmachinetoadestinationmachine.Otherapplicationsinvolveanyscenariowhereaparticular(unique)entitymusttravelwithowatalltimes.Alternatively,apartialrestrictiononsplittingoccurs,forexample,inshippingapplications,withnodesrepresentingdistributioncentersandarcsrepresentingvehicles.Whenshipmentsarriveatadistributioncenter,theymustbeunloaded.Aseachshipmentisloadedtocontinuetothenextdestination,anadditionalvehicle-personnelcostisincurredperadditional(>1)vehicleused,withalimitonthetotalvehiclepersonnelcost.Inthiswork,weperformananalysisonthecomplexityofthemaximumowproblemwithcertainsparsityconstraints.Eachsparsity-inducingrestrictionisauniquecombinationofthenumberofowsplitsandtheinclusionofeitherarestrictionon 67

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thetotalamountofowonallarcsorthetotalnumberofarcsusedinthenetwork.WeanalyzeatotaloftwelvepreviouslyunclassiedproblemsandclassifyeachasNP-hardorpolynomial-timesolvable,providingpolynomial-timealgorithmsorconvexhullrepresentationsifpolynomial-timesolvable.Thechapterproceedsasfollows.InSection 4.2 ,wepresenttheinitialmaximumowformulationandadditionalconstraintsthatweconsider.InSection 4.3 ,weexploretheeffectsoncomplexityofaddingtheseconstraintstothemaximumowproblem. 4.2ProblemsofInterestGivenagraphG(N,A)withn=jNjnodesandm=jAjarcs,asourcenode(1)andasinknode(n)inN,themaximumowproblemcanbestatedasalinearprogrammingproblem[ 5 ]: maxV (4a)s.t.Xj2FS(1)x1j=V (4b)Xj2FS(i)xij)]TJ /F12 11.955 Tf 18.99 11.36 Td[(Xk2RS(i)xki=08i2N (4c)xijCij8(i,j)2A (4d)xij08(i,j)2A (4e)Herexijdenotestheowalongarc(i,j)2A,Vistheowquantityleavingnode1,Cij0isthecapacityofarc(i,j)2A,FS(i)=fk2N:(i,k)2Agistheforwardstarofnodei2N,andRS(i)=fk2N:(k,i)2Agisthereversestarofnodei2N.(Throughoutthispaper,weassumethatGisstronglyconnected.)Weanalyzethreedifferenttypesofsparsityconstraints.Thersttypeisan`0-normconstraintjjxjj0s0,thusboundingthequantityoftotalarcsreceivingpositiveow.Thesecondtypeconsidersan`1-normconstraintjjxjj1s1,thusboundingthetotalquantityowingthroughthenetworkoverallarcs.Sinceallowisnonnegative,wecanrewrite 68

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thisconstraintasX(i,j)2Axijs1. (4)DeningthetotalquantityofsplitsinasetofowsasS(x)=PjNji=1maxfai)]TJ /F7 11.955 Tf 12.84 0 Td[(1,0g,whereaiisthenumberofarcsleavingnodei2Nwithpositiveow,thethirdtypeofsparsityconstraintinvolvesarestrictiononthesplitquantity.Throughoutthepaper,wedenoteeachproblemthatweconsiderasMFfg,wherelistedinsidethebracketsareadditionalconstraintsaddedtothemaximumowproblem( 4 ). 4.3ComplexityAnalysisandPolynomial-TimeAlgorithmsInSection 4.3.1 ,weexaminetheeffectsofaddingthe`0-normconstraintjjxjj0s0(therebyboundingthenumberofarcswithnonzeroow)andcomparethistoinsteadaddingthe`1-normconstraintjjxjj1s1(thusboundingthetotalowoverallarcs).WethenexamineinSection 4.3.2 thecasewhere,inadditiontoeitheran`0-or`1-normconstraint,owsplittingisprohibited,andanalyzehowthisaffectsthecomplexityofthemaximumowproblem.InSection 4.3.3 ,weconsidersimilarproblemsforthecasewhereowsplittingispartially-restrictedratherthanfully-restricted,andanalyzetheeffectofthis(partial)ow-splittingrestrictiononthecomplexityofthemaximumowproblem. 4.3.1UnrestrictedFlowSplittingWerstexaminethecomplexityoftheproblemMFfjjxjj0s0g,whichisaspecialcaseoftheNP-hardbudget-restrictedmaxowproblem[ 29 ].Theorem 4.1 belowshowsthatproblemMFfjjxjj0s0gisNP-hardingeneral.(EiseltandFrajer[ 29 ]showthataspecialcaseofthisprobleminwhichallcapacitiesaresettooneissolvableinpolynomialtime.) Theorem4.1. Thefollowingdecisionproblem,denotedbyD-MFfjjxjj0s0gisNP-complete:GivengraphG(N,A),witht2Nreachablefromq2N,doesthereexistasetoffeasibleowssummingtoatleastfleavingnodeqthatusess0orfewerarcs? 69

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Proof. First,weproveD-MFfjjxjj0s0gisinNP.Givenasetoffeasibleowsonarcs,ittakesO(n)timetocomputeowleavingnodeqandO(m)timetoverifyowenteringandleavingallothernodes.Todetermineifthenumberofarcsusedisnomorethans0takesO(m)time.Thus,thetotaltimetoverifyasolutionisO(m).Therefore,D-MFfjjxjj0s0g2NP.WenowprovethatD-MFfjjxjj0s0gisNP-hardbyreducingfrom3-SAT.Letf=v+cands0=3v+2c,wherevisthenumberofvariables,eachvariableisrepresentedbytwoliteralsiandi,8i=1,...,v,andcisthenumberofclauses.Thereductionprocedurerstcreatesafour-columngraphwithn=3v+c+2nodesandm=5v+4carcsasfollows. 1. Createtwonodes,labeledqandt,astherstandfourthcolumnofnodes,respectively. 2. Createnodeslabelediandi,foralli=1,...,v,asthesecondcolumnofnodes. 3. Createnodeslabeledii,8i=1,...,v,andnodeslabeledbytheircorrespondingformulaforeachofthecclausesasthethirdcolumnofnodes. 4. Createdirectedarcs(withcapacity1)fromqtoeverysecondcolumnnode. 5. Createdirectedarcs(withcapacityone)fromeverysecond-columnnodetoeverythird-columnnodeifthelabelofthesecond-columnnodeisaliteralinthelabelofthethird-columnnode. 6. Createdirectedarcs(withcapacity1)fromeverythirdcolumnnodetot.ThetotalcomplexityoftheprocedureusedtocreatethegraphaboveisO(1)+O(v)+O(c)+O(v)+O(vc)+O(v+c)=O(vc), (4)whereeachtermintheleft-handsideof( 4 )isthecomplexityofeachstepoftheaboveprocedure,respectively.AnillustrationofanexampleofthisreductiongraphisshowninFigure 4-1 .Wenextprovethatifaninstanceof3-SATissatisable,thenthereexistsacorrespondinginstanceofD-MFfjjxjj0s0gthatcontainsasetofowssummingto 70

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123 1 1 77 '' 234 1 1 134 1 2 EE @@ 234 1 && q 1 BB 1 FF 1 BB 1 99 1 // 1 %% 1 1 1 2 77 1341 // t3 GG EE 77 11 1 88 3 JJ EE '' 22 1 @@ 4 JJ II GG EE '' 33 1 EE 4 // 44 1 HH Figure4-1. Reductiongraphwitharccapacities(capacityoneonunlabeledarcs)forthe3-SATformula123^234^134^234^134(4variablesand5clauses). atleastv+cusing3v+2corfewerarcs.Givenasetofsatisfyingassignmentstoa3-SATformula,wecomputeowsonarcsinthereductiongraphasfollows. 1. CreateasetKofthecclausesandtwovcmatricesDandDofliteral-clauserelationshipssuchthatDij=1(Dij=1)ifi(i)isTRUE,and0otherwise. 2. Foreachi=1,...,v,ifi(i)isTRUE,thenowoneunitonarc(q,i)((q,i))andoneunitonarc(i,ii)((i,ii)). 3. Foreachi=1,...,v,andj2K,ifi(i)receivespositiveowfromnodeqandDij=1(Dij=1),thenaddoneunitofowon(q,i)((q,i))andoneunitofowfromi(i)toclausej,settingK=Knfjg,andexitingthisstepwhenK=;. 4. Flowoneunitalongeacharcfromthethirdtothefourthcolumn. 71

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Theforegoingprocedureresultsinaowofv+cusing3v+2carcs.Thetotaltimetoconverttheinstanceof3-SATtoaninstanceofD-MFfjjxjj0s0gisO(v)+O(c)(tocreatethegraphusedinthereduction)+O(vc)(togenerateowsonarcs)=O(vc).WenowprovethatifaninstanceofD-MFfjjxjj0s0gcontainsasetofowssummingtoatleastf=v+cusings0=3v+2corfewerarcs,thenthecorrespondinginstanceof3-SATissatisable.Inordertoobtainfv+c,theremustbev+cpositive-owarcsfromthethirdcolumnofnodestotbecauseeacharcfromthethirdcolumntothasacapacityofone.Byconservationofowandcapacityconstraints,theremustbev+carcswithoneunitofowfromthesecondtothethirdcolumn.Atotalof2v+2carcshavebeenusedthusfar,leavingamaximumofvarcsavailablefromqtothesecondcolumn.Iffewerthanvarcshaveaowof1fromqtothesecondcolumn,thensomenodeii,i2N,inthethirdcolumndoesnotreceiveaunitofow(sinceowfromthesecondcolumntonodeiicanonlycomefromeitherthesecond-columnnodelabelediori).Therefore,varcsfromqtothesecondcolumnhavepositiveow.Weobtainasatisfyingassignmentbysettingior(i),respectively,toTRUEforsecond-columnnodelabels(i.e.,literals)correspondingtonodesreceivingpositiveowfromq.Suppose,bycontradiction,both(q,i)and(q,i)receivepositiveowforsomei2N.Sincewecanonlyusevarcsfromqtothesecondcolumn,thereissomenodejj,j=1,...,v,inthethirdcolumnthatcannotreceivepositiveowfromthesecondcolumn,resultinginamaxowofonlyv+c)]TJ /F7 11.955 Tf 11.99 0 Td[(1possibletot,whichisacontradictiontoanoverallowofatleastv+c.Thus,thereexistspositiveowoneither(q,i)or(q,i),8i=1,...,v.Therefore,D-MFfjjxjj0s0gisNP-complete. Settingt=vandq=1,Theorem 4.1 impliesMFfjjxjj0s0g(i.e.,theoptimizationversionofD-MFfjjxjj0s0g)isNP-hard.Theforegoinganalysisremainsunchangedifxisrestrictedtobeintegral.Ifweconsiderthecasewhereinsteadofboundingthetotalnumberofarcsreceivingpositiveow,weinsteadboundthetotalquantityofowthroughthenetwork 72

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overallarcs,weobtainMFfjjxjj1s1g,wherejjxjj1s1isexpressedas( 4 ).Sinceitcanbewrittenasalinearprogram,MFfjjxjj1s1gissolvableinpolynomialtime.SupposeowsinMFfjjxjj1s1garerestrictedtobeintegral,obtainingMFfxinteger,jjxjj1s1g.Relaxingintegralityconstraints,thisproblemisaspecialcaseofthelinear-costconstrainedmaximumowproblem(LCCMFP),withcapacitycostseachsettoone[ 6 ].Sincethecapacity-scalingalgorithmdetailedin[ 6 ]guaranteestheidenticationofanintegeroptimalsolutiontoLCCMFPinpolynomialtime,thissolutionisalsooptimaltoMFfxinteger,jjxjj1s1g.Thus,MFfxinteger,jjxjj1s1gissolvableinpolynomialtime. 4.3.2Fully-RestrictedFlowSplittingWenowconsiderthecasewhereowisrestrictedtoasinglepath;i.e.,atmostonepositive-owarcleavesnodei,8i2N.WerstanalyzethecomplexityofMFfS(x)0,jjxjj0s0g.Foranyc,a>0,werstdenetheprocedureEXPAND(G,c,a),which Figure4-2. ExampleGraphwithn=6. transformsagraphG(N,A)intoanexpandednetworkGE(NE,AE)suchthatNEconsistsofeachnodei2Nduplicatedateachoftheanodelevels,denedassetsofduplicatesofthenodesN,withtheexceptionthattherstlevelcontainsonlythesourcenodeandthelastlevelcontainsonlythesinknode.Foranynodei2Natlevell,8l=1,...,L)]TJ /F7 11.955 Tf 12.06 0 Td[(1,ofGE,anarc(withcapacityCij)tonodej2Natlevell+1existsinAEif(i,j)2Aand 73

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Cijc.Thus,thecollectionofnodesNEconsistsofL)]TJ /F7 11.955 Tf 12.09 0 Td[(2duplicatesofeachnodeinN,andL)]TJ /F7 11.955 Tf 11.95 0 Td[(1duplicatesofthesourceandsinknodes. EXPAND(G,c,a) 1. Createa)]TJ /F7 11.955 Tf 12.55 0 Td[(2columnsofnnodes(eachlabeledasauniquei2N).Createtwomorecolumns,oneforthesourcenode(1)andsinknode(n). 2. Createanarc(withcapacityCij)fromnodei2Nincolumnk,k=1,...,a)]TJ /F7 11.955 Tf 12.25 0 Td[(1,tonodejincolumnk+1ifarc(i,j)2AandCijc.(Alsocreatearcs(withinnitecapacity)fromnode1atlevelktonode1atlevelk+1andfromnodenatlevelktonodenatlevelk+1,8k=1,...,a)]TJ /F7 11.955 Tf 11.95 0 Td[(1.) Figure 4-3 illustratesanexampleofanexpandednetworkofFigure 4-2 Figure4-3. ExampleExpandedNetworkforGraphinFigure 4-2 (s0=4andL=minf6,4+1g=5).EachsolidarcisduplicatedfromFigure 4-2 andeachdashedarcisadditionalarcwithinnitecapacity. Assumingthatthetotalnumberofpositiveowarcsisbounded,wenowdeneaformulationtosolveMFfS(x)0,jjxjj0s0g.DeneaconstantL=minfn,s0+1g 74

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tobethemaximumnumberofnodesadjacenttoarcsreceivingpositiveowinanoptimalsolution.(Notethatweseekthesource-to-sinkpathwhosesmallestcapacityislargestamongallfeasiblepathsusingatmostL)]TJ /F7 11.955 Tf 12.75 0 Td[(1arcs.)DeneTasthesetofuniquecapacitiesCij,(i,j)2A.Anelementt2TwillbedenotedbyPt.CreatetheexpandednetworkGE(NE,AE)usingEXPAND(G,c=0,a=L).Denebinaryvariableszt,8t2T,whichequal1ifanoptimalpathhasobjectivePt,and0otherwise.Also,denevariablesflijt=1iftheoptimalpathusesarc(i,j)fromlevelltol+1andPtistheoptimalobjectivefunctionvalue,and0otherwise.DeneFS(i,l)andRS(i,l)astheforwardstarandreversestar,respectively,ofnodei2Natlevell=1,...,L.Assumeflijt=0,8(i,j)2A,l=1,...,L,andt2TifCij
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sideofconstraints( 4c )and( 4d )consistofone1andtherest0s.Thus,thereexistsanoptimalsolutioninwhichfisbinary.Toprovezisbinary,supposebycontradictionthatzisfractionalinsomeoptimalsolution(whereowisf).Thisimplieszh1,...,zhrareeachpositiveforhi2T,i=1,...,r,forsome1rjTj,andtheoptimalobjectivefunctionvalueisPh1++Phr.SinceeachPt,8t2T,isunique,supposethatPhr
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Thus,weachievethedesiredcontradictionasPt2TPtzt=Pri=1Phrzhr
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ALGORITHMC-0-1(G(N,A),s1) 1: fort2Tdo 2: intnumArcs=minfbs1=Ptc,n)]TJ /F7 11.955 Tf 11.95 0 Td[(1g 3: EXPAND(G,Pt,numArcs+1) 4: boolPath=RunBFS(GE) 5: ifPath=1then 6: MaxMinFlow=Pt 7: Breakoutofallforloops 8: else 9: fornumArcs=minfbs1=Ptc,n)]TJ /F7 11.955 Tf 11.95 0 Td[(1g+1tominfbs1=Pt+1c,n)]TJ /F7 11.955 Tf 11.96 0 Td[(1gdo 10: if(s1=numArcs)=Pt+1then 11: Breakoutofinnerforloop 12: endif 13: Expand(G,Pt,numArcs+1) 14: boolPath=RunBFS(GE) 15: ifPath=1then 16: MaxMinFlow=s1=numArcs 17: Breakoutofallforloops 18: endif 19: endfor 20: endif 21: endfor 78

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EachiterationoftheinnerforlooptakesO(mn)timetocreatetheexpandednetworkinline 13 andperformbreadth-rstsearch(themostcomplexstepsintheinnerforloop).Similarly,therestofthealgorithmrunsinO(m2n)timeduetothecreationofatmostmexpandednetworks(line 3 )andtheexecutionofatmostmbreadth-rstsearches(line 4 ).Thus,ALGORITHMC-0-1exhibitsaworst-caserunningtimeofO(m2n).IfweadditionallyrestrictowtobeintegralinMFfS(x)0,jjxjj1s1g,weobtainMFfxinteger,S(x)0,jjxjj1s1g.WecanprovethatthisproblemissolvableinpolynomialtimeusingthesameargumentforproblemMFfS(x)0,jjxjj1s1g,withtheexceptionthatwereplacewiths1=numArcsinline 16 ofALGORITHMC-0-1withbs1=numArcscandallinstancesofPtwithbPtc. 4.3.3Partially-RestrictedFlowSplittingSupposethatinsteadofrestrictingowtoasinglepath,werestrictthemaximumnumberofsplitstobenomorethansomeinteger0
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asetoffeasibleowssummingtoatleastfrequiringatmostlsplitsandutilizingatotalofatmosts1unitsofow(summedoverallarcsinA)?. Proof. First,weprovethatD-MFfS(x)l,jjxjj1s1gisinNP.Givenasetoffeasibleowsonarcs,ittakesO(n)timetoverifythatfunitsofowleavenodeq.Determiningifthenumberofsplitsisatmostlandifthetotalowisatmosts1takesO(m)time.Thus,thetotaltimeisO(m)toverifyasolution.Therefore,D-MFfS(x)l,jjxjj1s1g2NP.WenowprovethatD-MFfS(x)l,jjxjj1s1gisNP-hardbyreducingfrom3-dimensionalmatching.Letf=lands1l.DeneW,X,YasnitedisjointsetseachwithlelementsanddeneZf(wi,xj,yk):wi2W,xj2X,yk2Y,i,j,k=1,...,lg. (4)Thereductionprocedurerstcreatesave-columngraphwith3l+2nodesand2l2+2larcsasfollows. 1. Createtwonodes,labeledq(source)andt(sink),astherstandfthcolumnofnodes,respectively. 2. Createnodeslabeledwi,xi,andyi,8i=1,...,l,inthesecond,third,andfourthcolumns,respectively. 3. Createarcswithcapacityonefromnodeqtoeverynodeinthesecondcolumnandfromeverynodeinthefourthcolumntonodet. 4. Foreachtriple(wi,xj,yk)2Z,i,j,k=1,...,l,createarcsfromwiinthesecondcolumntoxjinthethirdcolumnandfromxjtoykinthefourthcolumn.TheoverallcomplexityoftheprocedureusedtocreatethetransformedgraphisO(1)+O(l)+O(l)+O(l2),whereeachtermcorrespondstothecomplexityofthestepsintheproceduregivenabove.TheillustrationinFigure 4-4 showsthisreductiongraph,includingamatching(consistingofacollectionofeachsetofcirclednodes)whenl=3andZ=f(w1,x2,y2),(w2,x1,y1),(w2,x2,y2),(w3,x2,y2),(w3,x3,y3)g.WenextprovethatifthereisamatchingMZofltriplessuchthateachelementofW,X,andYiscontainedinexactlyonetriple,thenthereexistsacorresponding 80

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Figure4-4. ExampleReductionGraphfor3-dimensionalmatchingtoD-MFfS(x)l,jjxjj1s1g(withcircledelementsofamatching). solutiontoD-MFfS(x)l,jjxjj1s1gthatcontainsasetofowssummingtoatleastf=l,requiringnomorethanlsplits,andatmosts1=4lunitsofow(summedoverallarcsinA).Bydenitionofa3-dimensionalmatching,wi16=wi2,xj16=xj2,andyk16=yk2foranytwodistincttriples(wi1,xj1,yk1),(wi2,xj2,yk2)2M.Foreach(wi,xj,yk)2M,pushoneunitofowonarcs(wi,xj)and(xj,yk).(Sinceoneunitofowentersandexitseachnodeinthesecondthroughfourthcolumns,anyowpathfromthesecondcolumntothelastcolumndoesnotaccountforanyadditionalsplits.)Pushoneunitofowoneacharc(q,wi)and(yi,t),8i=1,...,l.Thus,lunitsowfromqtot,withlsplits,utilizingatotalof4lunitsofavailablecapacity.Thetotaltimetoconverttheinstanceof3-dimensionalmatchingtoaninstanceofD-MFfS(x)l,jjxjj1s1gisO(l2).WenowprovethatifaninstanceofD-MFfS(x)l,jjxjj1s1gcontainsasetofowssummingtoatleastf=lrequiringnomorethanlsplitsandutilizingatotalofatmosts1=4lunitsofow,thenacorrespondingsolutionto3-dimensionalmatchingexistswithasetMZWxXxYofltriplessuchthateachelementofW,X,andYiscontainedinexactlyonetriple.Notethatoneunitowsfromqtoeachnodeinthesecondcolumn(elementofW)becauseowisatleastlunits,resultinginatotalofl 81

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splits.Sinceowcannotbesplitthroughtherestofthenetwork,thereisonepathfromeachsecondcolumnnodetonodet.Letthecollectionofnodes(excludingqandt)oneachpathfromqtotformatriple.SinceeachelementofW,X,andYiscontainedinexactlyonetriple,thecollectionoftriplesrepresentsamatchingforaninstanceof3-dimensionalmatching.Therefore,D-MFfS(x)l,jjxjj1s1gisNP-complete. Settingt=nandq=1,Theorem 4.3 impliesMFfS(x)l,jjxjj1s1g(i.e.,theoptimizationversionofD-MFfS(x)l,jjxjj1s1g)isNP-hard.Usingthesameargument,butinsteadrestrictingxtobeintegral,problemMFfxinteger,S(x)l,jjxjj1s1gisalsoNP-hard. 82

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CHAPTER5CONCLUSIONSANDFUTURERESEARCHInChapter 2 ,weprescribedalternativesolutionmethodsfortheproblemofefcientlyplacingstationarymonitorsinanareaofinteresttominimizeanadversary'smaximumevasionprobability.Wemodeledtheproblemasatwo-stageMINLPinanetworksetting.Onesolutionmethod,ExactAlg,isanitelyconvergentalgorithmthatdeterminesoptimalmonitorlocationsoverthecontinuousspace.ComputationalexperimentsdemonstratethatExactAlgconvergestooptimalityoninstancesinvolvinggridsofrelativelysmallsizewithinreasonablecomputationallimits.Anothersolutionmethod,MidcolAlg,utilizesExactAlgbutrestrictsmonitorstomidcolumnlocations,exploitingtheresultthatmanyinstanceshaveoptimalsolutionsexistingatmidcolumnlocations.Thismidcolumn-restrictedversionofthealgorithmreducessolutiontimewhilestillobtainingprovablynear-optimalsolutions.Inanefforttofurtherdecreasesolutiontime,whilestillobtaininganear-optimalsolution,weimplementathirdapproach(FDF),whichsolvesamixed-integerlinearprogramthatfullydiscretizesthefeasiblemonitorlocations.Forinstancestakingplaceonlargergrids,weobtainprovablynear-optimalsolutionswithFDFinwellunderonehour.Becauseadaptabilityofamodeltodifferentscenariosiscrucialtoitssuccessinreal-worldimplementation,oneimportantnoteisthatwhilewemodeltheevasionprobabilityasaconcavefunctionofthedistancefromamonitortoanarc,anynondecreasingfunctioncaneasilysubstitutedwithoutalteringoursolutionschemes.Onenoteworthyextensioninvolvesdevelopingamethodthatiterativelydeterminesthesetoffeasiblemonitorlocationsbasedonapriorsolutionratherthaninitiallyxingthisset.Thischangemayfurtherimprovetheeffectivenessofourapproachbyallocatingbinaryvariablesmoreefciently.Additionally,amajorreasonthattheexactalgorithmwasnotabletosolvelargerprobleminstancesinvolvestheincorporationofbig-Mconstantsintheconstraints,whichweakenthemixed-integerprogrammingformulations. 83

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Findingarelaxationmoresuitabletointegerprogrammingsolutionapproachesisanotherpromisingfutureresearchdirection.InChapter 3 ,westudiedasensor-placementproblemseekingtoplaceasetofsensorsinlocationssuchthat,afteraworst-casedisplacement,thesensorscoveraxedsetoftargetsandintra-sensorcommunicationismaximized.Weformulatedthismodelasathree-stageprogram,wheretherststageplacessensorsinitially,thesecondstagedisplacessensors,andthethirdstagedeterminesintra-sensorcommunication.Sincetheinnerproblemisalinearprogram,wereformulatedthisproblemasatwo-stageproblem.Duetothenonconvexityofthesecondstage,weutilizedascenario-approximationapproachtosolvingtheproblem,producingaguaranteedworst-caseboundfromoptimality.Wedevelopedacutting-planealgorithmutilizingoneoftwotypeofcuttingplanes:onebasedongeneralizedBendersdecomposition,andtheotherbasedonKelley'smethod.Computationalexperimentssuggestthat,ingeneral,theutilizationofcuttingplanesbasedonKelley'smethodoutperformstheuseofcuttingplanesbasedongeneralizedBendersdecomposition.Anavenueforfutureresearchinvolvesutilizingabundle-typesolutionmethod,suchastheacceleratedprox-level(APL)method,whichhasbeenshowntoconvergefaster,ingeneral,thanKelley'smethod[ 46 ].TheAPLmethodusesabundle,composedofthefunctionvalueandasubgradientatasolutioncoordinate,inproceedingtoanupdatedsolutionvalueduringaniterate.Othermethods,suchasageneralizedversionofNesterov'smethod[ 51 ],forbothsmoothandnon-smoothconvexproblemsproduceoptimalsolutions,butaregreatlyaffectedbycertainfunctionparameterssuchastheLipschitzconstant.However,APLoperatesindependentofanysmoothnessinformation.Moreover,APLusuallyconvergeslinearlyinpractice,evenfornon-smoothconvexproblems,andutilizesnon-Euclideanprox-functionstoproducearateofconvergence 84

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thatisdimension-independentofthefeasiblesetofsolutions,thuspreventingeachiterationfrombecomingtoocomputationallyexpensive.InChapter 4 ,weanalyzedthecomplexityoftwelvesparsemaximumowproblems,developedwithacombinationofrestrictionsonowsplitting,totalpositive-owarcs,andtotalowoverallarcs.TheresultsofthisworkaresummarizedinTable 5-1 ,wherePdenotesthattheproblemissolvableinpolynomialtime.Inparticular,wefoundthateachsparsemaximumowproblemrequiringowfromsourcetosinktobealongonepathissolvableinpolynomialtime.TheotherproblemsanalyzedwereproventobeNP-hard,withtheexceptionoftwoproblemsinvolvingunconstrainedowsplittingbutconstrainedtotalow,bothproventobesolvableinpolynomialtime.Foreachproblemsolvableinpolynomialtime,wedevelopedeitherapolynomial-timealgorithmoraconvexhullrepresentation.AnoteworthyextensionofthisresearchincludesexploringappropriatesolutionmethodsforsolvingtheproblemsproventobeNP-hard. Table5-1. ComplexityResults xcontinuousxinteger jjxjj0s0jjxjj1s1jjxjj0s0jjxjj1s1 S(x)0PPPPS(x)lNP-hardNP-hardNP-hardNP-hardS(x)<1NP-hardPNP-hardP 85

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APPENDIXAPROOFSThisappendixincludestheproofsforTheorems 2.2 2.4 ,and 2.5 .First,westatetwolemmasthatwillbeusedintheseproofs. Lemma2. Let:R!Rbeaconcave,monotonicallynondecreasingfunction.Forscalarsd10andd2>0,wehaveminf(1+2))]TJ /F3 11.955 Tf 11.95 0 Td[((1):1+2d1+d2,2d2,10g=(d1+d2))]TJ /F3 11.955 Tf 11.96 0 Td[((d1). (A) Proof. For1and2suchthat1+2d1+d2,and2d2,itfollowsthat1d1.Wehave(1+2))]TJ /F3 11.955 Tf 11.96 0 Td[((1) 2(d1+d2))]TJ /F3 11.955 Tf 11.95 0 Td[((1) d1+d2)]TJ /F3 11.955 Tf 11.95 0 Td[(1 (A)(d1+d2))]TJ /F3 11.955 Tf 11.95 0 Td[((d1) d2, (A)where( A )and( A )followbecauseisconcave.Thus,wehave(1+2))]TJ /F3 11.955 Tf 11.96 0 Td[((1) 22(1+2))]TJ /F3 11.955 Tf 11.95 0 Td[((1) 2d2 (A)(d1+d2))]TJ /F3 11.955 Tf 11.96 0 Td[((d1) d2d2, (A)where( A )followsfromthefactthat2d2,and( A )followsfromtheresultprovenby( A )and( A ).Equalityin( A )holdsbysetting1=d1and2=d2. Lemma3. Let:R!Rbeaconcave,monotonicallynondecreasingfunction.Forscalarsd10,andd2>0,wehavemaxf(1+2))]TJ /F3 11.955 Tf 11.95 0 Td[((1):1d1,0<2d2g=(d1+d2))]TJ /F3 11.955 Tf 11.96 0 Td[((d1). (A) Proof. First,notethatbecauseisnondecreasing,(1+d2)(1+2)forany1,andforany2>0.Hence,theoptimizationproblemin( A )ismaximized 86

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bysetting2=d2.Thefactthatisconcaveandnondecreasingthenimpliesthatmaxf(1+d2))]TJ /F3 11.955 Tf 11.96 0 Td[((1):1d1gisoptimizedat1=d1.Thiscompletestheproof. ProofofTheorem 2.2 .Supposethatduringthea-thiterationofExactAlg,GAPgap,x=^x,=^,w=^w,andf=_f.WeknowthatGAPsijk,b,agap jSj(nc)]TJ /F8 7.97 Tf 6.58 0 Td[(1)forsomemonitor-arccombinationsijk,andsomebsuchthat^wsijk,b,a=1anddsijk,b+1,aR,whereGAPsijk,b,aisthevalueofGAPsijk,biniterationa(andsimilardenitionsapplyfordsijk,b,a,^wsijk,b,a,Bsijk,a,(^x,^)sijk,b,a,andpsijk,b,a).Hence,atleastonedistanceintervalforsijkisrenediniterationa.Thus,dening=e (jj(^xs)]TJ /F8 7.97 Tf 6.59 0 Td[(cijk)jj),werstnotethatGAPsijk,b,a=e (jj(^xs)]TJ /F8 7.97 Tf 6.58 0 Td[(cijk)jj)+ln(vijk))]TJ /F5 11.955 Tf 11.96 0 Td[(e(^x,^)sijk,b,a))]TJ /F16 11.955 Tf 11.96 0 Td[(GAPsijk,b,a=e(^x,^)sijk,b,a. (A)Using( A ),weobtain psijk,b+1,a+1)]TJ /F5 11.955 Tf 11.96 0 Td[(psijk,b,a+1> (jj(^xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj))]TJ /F3 11.955 Tf 11.95 0 Td[((^x,^)sijk,b,a (Aa)=ln())]TJ /F7 11.955 Tf 11.95 0 Td[(ln()]TJ /F16 11.955 Tf 11.95 0 Td[(GAPsijk,b,a) (Ab))]TJ /F7 11.955 Tf 23.91 0 Td[(ln1)]TJ /F3 11.955 Tf 32.28 8.09 Td[(gap jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1), (Ac)where( Aa )followsbecausepsijk,b+1,a+1= (jj(^xs)]TJ /F9 11.955 Tf 13.39 0 Td[(cijk)jj)andpsijk,b,a+1<(^x,^)sijk,b,abecausepsijk,b,a+1=psijk,b,a,notingthatln()isstrictlyincreasing,( Ab )followsbydenitionofandGAPsijk,b,a,and( Ac )followsbyLemma 2 ,with()=ln(),d1=1)]TJ /F3 11.955 Tf 12.48 0 Td[(gap(jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)))]TJ /F8 7.97 Tf 6.58 0 Td[(1,d2=gap(jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)))]TJ /F8 7.97 Tf 6.58 0 Td[(1,d1+d2,andGAPsijk,b,ad2.Atanyiterationaofthealgorithm,denep0=epsijk,b0,apforsomeb02Bsijk,asuchthat^wsijk,b0,a=1,wherep0istheendpointofaninterval[epsijk,b0,a+1,p0]or[p0,epsijk,b0+1,a+1]createdforiterationa+1,andwhereepsijk,b0,a+1andepsijk,b0+1,a+1aretheleftandrightendpoints,respectively,fortheformerandlatterprobabilityintervals,respectively.Foreaseofnotation,wedenotepL=epsijk,b0,a+1intheformercaseandpR=epsijk,b0+1,a+1in 87

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thelattercase.Notethatifp0istheright(left)endpointof[pL,p0]([p0,pR]),thenpL6=p0(pR6=p0)becauseGAP-valuesarezeroatendpoints.First,considertheinterval[p0,pR]anddenel=pR)]TJ /F5 11.955 Tf 12.38 0 Td[(p0.Weseekanle)]TJ /F8 7.97 Tf 7.99 0 Td[(ln1)]TJ /F38 5.978 Tf 15.42 4.68 Td[(gap jSj(nc)]TJ /F17 5.978 Tf 5.76 0 Td[(1)=1)]TJ /F3 11.955 Tf 32.28 8.09 Td[(gap jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F8 7.97 Tf 6.59 0 Td[(1by( A )andthedenitionsofp0andl.Rearrangingterms,weobtainl>p01)]TJ /F3 11.955 Tf 32.28 8.09 Td[(gap jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(p0 (A)gapp jSj(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(gap, (A)sincep0p.Lettinglbetheright-handsideof( A )andusingthefactthatpR=p0+l,weobtainl0,wehave)]TJ /F7 11.955 Tf 11.29 0 Td[(ln1)]TJ /F3 11.955 Tf 32.28 8.08 Td[(gap jSj(nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)=ln(p+l))]TJ /F7 11.955 Tf 11.95 0 Td[(ln(p) (A)>l(ln(p0))]TJ /F7 11.955 Tf 11.96 0 Td[(ln(p)) p0)]TJ /F3 11.955 Tf 11.96 0 Td[(p (A)l(ln(p0))]TJ /F7 11.955 Tf 11.95 0 Td[(ln(p00)) p0)]TJ /F5 11.955 Tf 11.96 0 Td[(p00 (A)>ln(p0)+(p0)]TJ /F5 11.955 Tf 11.96 0 Td[(pL)(ln(p0))]TJ /F7 11.955 Tf 11.95 0 Td[(ln(p00)) p0)]TJ /F5 11.955 Tf 11.96 0 Td[(p00)]TJ /F7 11.955 Tf 11.96 0 Td[(ln(p0) (A)ln(pL)+(p0)]TJ /F5 11.955 Tf 11.96 0 Td[(pL)(ln(p0))]TJ /F7 11.955 Tf 11.96 0 Td[(ln(p00)) p0)]TJ /F5 11.955 Tf 11.96 0 Td[(p00)]TJ /F7 11.955 Tf 11.96 0 Td[(ln(p0) (A) 88

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=ln(pL))]TJ /F3 11.955 Tf 11.96 0 Td[((x,)sijk,b0,a+1 (A)where( A )followsbysubstitutinglby( A ),( A )followsbecausep+l
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G(xm,fm))]TJ /F5 11.955 Tf 11.95 0 Td[(G(x,fm) (A)=Xs2SXi2InfncgXj2JXk2J (jj(xms)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj))]TJ /F3 11.955 Tf 11.95 0 Td[( (jj(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj)fmijk, (A)where( A )followsbecauseG(x,f)G(x,fm)bydenitionofxandf.Becauseweemploythe`1-norminthispaper,eachmonitorshiftaffectsthedetectionprobabilitiesofatmostQarcsoneithersideofthemonitoraffectedbyitsdetectionradius;thus,atmost2Q+1arcs,indexedbyq2f)]TJ /F5 11.955 Tf 28.66 0 Td[(Q,...,0,...,Qg,areaffectedintotal.Dene(ijk)sq,8s2S,q2f)]TJ /F5 11.955 Tf 28.47 0 Td[(Q,...,0,...,Qg,asthearcmidpointlocatedqmidcolumnsintheoppositedirectionoftheshiftofmonitors,wherefm(ijk)sq=1.Forinstance,ifformonitors2Swehavexs,1=3.8,theshiftistotheleft.Hence,q=0referstoanarcwithcolumnmidpoint3.5,q=1referstoanarcwithcolumnmidpoint4.5,andq=)]TJ /F7 11.955 Tf 9.29 0 Td[(1referstoanarcwithcolumnmidpoint2.5,i.e.,inthesamedirectionoftheshift.Wenowboundtheright-handsideof( A )asfollows.Xs2SXi2InfncgXj2JXk2J[ (jj(xms)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj))]TJ /F3 11.955 Tf 11.95 0 Td[( (jj(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj)]fmijk=Xs2SQXq=)]TJ /F6 7.97 Tf 6.59 0 Td[(Q )]TJ 5.48 .48 Td[((xms)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq))]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ 5.48 .48 Td[((xs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq) (A)Xs2SQXq=0 )]TJ 5.48 .48 Td[((xms)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)sq))]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ 5.48 .48 Td[((xs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq), (A)wherewetake )]TJ 5.48 .48 Td[((xs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq)=0foranyqcorrespondingtoacolumnmidpointoutsidetherange[1.5,nc)]TJ /F7 11.955 Tf 11.67 0 Td[(0.5].Notethat( A )followsduetomonotonicityof ()andthefactthat )]TJ 5.48 .48 Td[((xms)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)sq))]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ 5.48 .48 Td[((xs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq)0,8q=)]TJ /F5 11.955 Tf 9.3 0 Td[(Q,...,)]TJ /F7 11.955 Tf 9.29 0 Td[(1,s2S. 90

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UsingLemmas 2 and 3 ,wecanboundtheright-handsideof( A )asfollows.Xs2SQXq=0 )]TJ 5.48 .48 Td[((xms)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq))]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ 5.48 .48 Td[((xs)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)sq)Xs2S" )]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jj(xms)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)s0)jj)]TJ /F3 11.955 Tf 11.96 0 Td[( )]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)s0)jj+QXq=1 qL nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[( (q)]TJ /F7 11.955 Tf 11.96 0 Td[(0.5)L nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1# (A)Xs2S" (H))]TJ /F3 11.955 Tf 11.96 0 Td[( H+L 2(nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)+QXq=1 qL nc)]TJ /F7 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[( (q)]TJ /F7 11.955 Tf 11.96 0 Td[(0.5)L nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1#, (A)where( A )followsfromLemma 3 with()= (),d1=(q)]TJ /F7 11.955 Tf 12.43 0 Td[(0.5)L(nc)]TJ /F7 11.955 Tf 12.43 0 Td[(1))]TJ /F8 7.97 Tf 6.58 0 Td[(1,andd2=0.5L(nc)]TJ /F7 11.955 Tf 12.41 0 Td[(1))]TJ /F8 7.97 Tf 6.58 0 Td[(1,andthefactthat(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq)d1and(xms)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)sq))]TJ /F12 11.955 Tf -442.76 -13.75 Td[((xs)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)sq)=j(xms)]TJ /F9 11.955 Tf 12.26 0 Td[(c(ijk)sq)1j)-247(j(xs)]TJ /F9 11.955 Tf 12.25 0 Td[(c(ijk)sq)1jd2,8q=1,...,Q.Similarly,( A )followsfromLemma 2 with()= (),d1=H,andd2=0.5L(nc)]TJ /F7 11.955 Tf 12.41 0 Td[(1))]TJ /F8 7.97 Tf 6.59 0 Td[(1,andthefactthatjj(xs)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)s0)jjd1+d2andjj(xs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)s0)jj)-278(jj(xms)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)s0)jj=j(xs)]TJ /F9 11.955 Tf 12.08 0 Td[(c(ijk)s0)1j)-233(j(xms)]TJ /F9 11.955 Tf 12.09 0 Td[(c(ijk)s0)1jd2.Combining( A )( A ),weobtaintherstinequalityof( 2 ).2 ProofofTheorem 2.5 .Thelastinequalityof( 2 )followsfromthefactthatFDFisarestrictionofthemidcolumn-restrictedversionof( 2 ).Toshowtherstinequality,consideranoptimalsolution(x,m,x,c,f,m)tothemidcolumn-restrictedversionof( 2 )andsupposethatweshifteachmonitors2Sfromlocationx,mstoitsnearestlocationxFDFspertainingtoafeasiblesolutiontoFDF,breakingtiesarbitrarily,withobjectivefunctionvalueyFDF1.SupposefFDFistheoptimaladversarypathcorrespondingtoxFDF.Ourproofvalidatesthattherstinequalityof( 2 )holdsevenify,FDF1isreplacedwithyFDF1,whichestablishestheclaim.Wehavey,FDF1)]TJ /F5 11.955 Tf 11.95 0 Td[(G(x,m,f,m)yFDF1)]TJ /F5 11.955 Tf 11.95 0 Td[(G(x,m,f,m) (A) 91

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yFDF1)]TJ /F5 11.955 Tf 11.95 0 Td[(G(x,m,fFDF) (A)=Xs2SXi2InfncgXj2JXk2J )]TJ 5.48 .48 Td[((xFDFs)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk))]TJ /F3 11.955 Tf 11.96 0 Td[( (jj(x,ms)]TJ /F9 11.955 Tf 11.96 0 Td[(cijk)jj)fFDFijk, (A)where( A )followsbecauseG(x,m,fFDF)G(x,m,f,m)bydenitionofx,mandfFDF.Forthisproof,sinceeachmonitorshiftiswithinthesamecolumn,wedene(ijk)sqthesameasinTheorem 2.4 withtheexceptionthatpositive(negative)qreferstoanarcwithcolumnmidpointqcolumnstotheright(left)ofthemonitor.Becauseweemploythe`1-norminthispaper,eachmonitorshiftaffectsthedetectionprobabilitiesofatmost2Qmd+1arcs.UsingLemma 3 ,wenowboundtheright-handsideof( A )asfollows.Xs2SXi2InfncgXj2JXk2J)]TJ /F3 11.955 Tf 5.48 -9.68 Td[( )]TJ 5.48 .48 Td[((xFDFs)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk))]TJ /F3 11.955 Tf 11.95 0 Td[( (jj(x,ms)]TJ /F9 11.955 Tf 11.95 0 Td[(cijk)jj)fFDFijk=Xs2SXq2Q)]TJ /F3 11.955 Tf 5.48 -9.68 Td[( )]TJ 5.48 .48 Td[((xFDFs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq))]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ 5.48 .48 Td[((x,ms)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq) (A)Xs2S )]TJ 5.48 .48 Td[((xFDFs)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)s0))]TJ /F3 11.955 Tf 11.95 0 Td[( )]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jj(x,ms)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)s0)jj+Xq2Qnf0g H 2(N)]TJ /F7 11.955 Tf 11.95 0 Td[(1)+qL nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[( qL nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1! (A)Xs2S H 2(N)]TJ /F7 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[( (0)+Xq2Qnf0g H 2(N)]TJ /F7 11.955 Tf 11.95 0 Td[(1)+qL nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[( qL nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1!, (A)where( A )followsfromLemma 3 with()= (),d1=qL(nc)]TJ /F7 11.955 Tf 12.71 0 Td[(1))]TJ /F8 7.97 Tf 6.59 0 Td[(1,andd2=0.5H(N)]TJ /F7 11.955 Tf 12.61 0 Td[(1))]TJ /F8 7.97 Tf 6.59 0 Td[(1,andthefactthat(x,ms)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)sq)d1and(xFDFs)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)sq))]TJ /F12 11.955 Tf -435.74 -13.74 Td[((x,ms)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)sq)=j(xFDFs)]TJ /F9 11.955 Tf 10.17 0 Td[(c(ijk)sq)2j)-278(j(x,ms)]TJ /F9 11.955 Tf 10.17 0 Td[(c(ijk)sq)2jd2,8q2Qnf0g.Similarly,inequality( A )followsfromLemma 3 with()= (),d1=0,andd2=0.5H(N)]TJ /F7 11.955 Tf 9.96 0 Td[(1))]TJ /F8 7.97 Tf 6.59 0 Td[(1,andthefactthatjj(x,ms)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)s0)jjd1and(xFDFs)]TJ /F9 11.955 Tf 11.96 0 Td[(c(ijk)s0))-226(jj(x,ms)]TJ /F9 11.955 Tf 11.95 0 Td[(c(ijk)s0)jj= 92

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j(xFDFs)]TJ /F9 11.955 Tf 12.12 0 Td[(c(ijk)s0)2j)-278(j(x,ms)]TJ /F9 11.955 Tf 12.12 0 Td[(c(ijk)s0)2jd2.Combining( A )( A ),weobtaintherstinequalityof( 2 ).2 93

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APPENDIXBEXTENDEDCOMPUTATIONALRESULTS TableB-1. MLIMEffectivenessinExactAlg;nc=f4,7g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)g,andp=f0.01,0.25g NoMLIMMLIM ncnrRAFpIterationsTimeIterationsTime 45100a0.258781045100b0.25222245100c0.257872345100a0.011960196645100b0.011335133945100c0.01503225024245417a0.25152021520945417b0.251152115745417c0.25976912245417a0.0145417b0.01333173230945417c0.015210415380247100a0.2575075347100b0.25241247100c0.25131647100a0.013401947100b0.0144144347100c0.011723089347417a0.25162581626347417b0.25171591412547417c0.25132181329847417a0.0147417b0.0139134239135247417c0.0132316131207175100a0.2544144475100b0.2532232575100c0.2533712275100a0.01195251953275100b0.019153915675100c0.016166512075208a0.2544644975208b0.25161775208c0.2535134775208a0.0131133331134675208b0.01102021020375208c0.011880914548 94

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TableB-2. MLIMEffectivenessinExactAlg;nc=f7,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)g,andp=f0.01,0.25g NoMLIMMLIM ncnrRAFpIterationsTimeIterationsTime 77100a0.254146311877100b0.25929812777100c0.2510595524177100a0.0118118077100b0.011123166134877100c0.016132177208a0.25116191161677208b0.25521413877208c0.258414744277208a0.0177208b0.0177208c0.01105100a0.25125127105100b0.25357115105100c0.2552055290105100a0.0122382230105100b0.013170146105100c0.01105139a0.25274132105139b0.25117118105139c0.2575417586105139a0.01122018122031105139b0.014333158105139c0.01107100a0.2556623395107100b0.2522732277107100c0.2549034850107100a0.01107100b0.01107100c0.01107139a0.2557485750107139b0.25614535905107139c0.258172971645107139a0.01107139b0.01107139c0.01 95

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TableB-3. EffectofponExactAlgwithnc=f4,10g,nr=f5,7g,jSj=2,R=f100,1.25L=(nc)]TJ /F7 11.955 Tf 11.95 0 Td[(1)g,andp=f0.01,0.25,0.75g(Full) ncnrRAFpTimencnrRAFpTime 45100a0.751105100a0.751345100b0.751105100b0.75845100c0.750105100c0.751245100a0.2510105100a0.252745100b0.252105100b0.251545100c0.2523105100c0.2529045100a0.0166105100a0.0123045100b0.0139105100b0.014645100c0.01242105100c0.0145417a0.755105139a0.751145417b0.751105139b0.75645417c0.751105139c0.751245417a0.25209105139a0.253245417b0.2557105139b0.251845417c0.25122105139c0.2558645417a0.01105139a0.01203145417b0.01309105139b0.015845417c0.01802105139c0.0147100a0.755107100a0.753547100b0.752107100b0.752447100c0.752107100c0.753047100a0.2553107100a0.2539547100b0.252107100b0.2527747100c0.256107100c0.2585047100a0.019107100a0.0147100b0.0143107100b0.0147100c0.0193107100c0.0147417a0.755107139a0.753547417b0.757107139b0.752147417c0.751107139c0.753647417a0.25263107139a0.2575047417b0.25125107139b0.2590547417c0.25298107139c0.25164547417a0.01107139a0.0147417b0.011352107139b0.0147417c0.012071107139c0.01 96

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TableB-4. EffectofNonFDFSolutionwithnc=f20,40g,nr=f10,15g,jSj=f2,4g,R=200,andp=0.75(Full) ncnrjSjAFNTimeTheoGap 20102a660.0698820102a10110.0555520102b650.0581020102b1080.0512320102c630.0617720102c1030.0445420104a6140.0058420104a10240.0041820104b6140.0043520104b10270.0041020104c650.0045620104c1060.0032820152a6170.1642720152a10330.1377420152b6150.1702320152b10340.1357520152c6140.1310720152c1090.1128120154a6400.0290920154a101310.0268720154b6890.0356020154b101290.0283220154c6210.0175520154c10180.0161540102a6300.0049540102a10680.0043940102b6330.0039840102b10580.0033640102c6100.0061940102c10100.0053140104a61390.0000340104a102560.0000240104b61590.0000240104b101530.0000140104c6200.0000340104c10190.0000340152a6890.0371840152a103040.0369740152b61590.0340040152b102780.0273540152c6420.0260940152c10490.0241640154a65680.0012340154a107720.0010340154b63480.0007240154b1021500.0006040154c6930.0005540154c10850.00047 97

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TableB-5. FDFSolutionQualityonLargeInstanceswithnc=f80,100g,nr=15,jSj=f2,4g,R=f100,200g,p=f0.75,0.95g,andN=6(Full) ncjSjRAFpTimeTheoGap 802100a0.955030.0030663802100b0.954660.00260039802100c0.955280.0019125802100a0.756270.0033594802100b0.7512730.00338824802100c0.756530.0020939802200a0.955400.0029610802200b0.955580.00243187802200c0.955190.0017038802200a0.7511290.0015674802200b0.7513540.0012782802200c0.7510850.0008352804100a0.9510970.0000233804100b0.9514190.0000166804100c0.9511080.0000091804100a0.7518590.0000104804100b0.75804100c0.7516530.0000033804200a0.9512310.0000157804200b0.9516250.0000102804200c0.9510900.0000052804200a0.75804200b0.75804200c0.7523620.00000031002100a0.958500.00096321002100b0.959630.00069581002100c0.957980.00045861002100a0.7514150.00090651002100b0.751002100c0.7512460.00041061002200a0.958330.00087101002200b0.959370.00059041002200c0.958190.00037761002200a0.7516220.00029191002200b0.751002200c0.7517830.00011941004100a0.9523470.00000191004100b0.9529410.00000101004100c0.9522700.00000041004100a0.751004100b0.751004100c0.751004200a0.9524700.00000111004200b0.9534170.00000051004200c0.9522310.00000021004200a0.751004200b0.751004200c0.75 98

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BIOGRAPHICALSKETCH AndrewRomichhasspentseveralyearsinanumberofstates,includingMaryland,Colorado,andFlorida.HeisaformerDepartmentofHomelandSecurityFellowwhoseresearchhasinvolvednationalsecuritynetworkoptimizationandinterdictionproblems.HehasbeenaFloridaGatorfornearlyadecade,havinggraduatedSummacumLaudewithaB.S.andM.S.inIndustrialandSystemsEngineering,andwillreceivehisPh.D.fromthesamedepartmentinAugust2013. 104