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Two-Level System Interdiction

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Title: Two-Level System Interdiction
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Sullivan, Kelly M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

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Subjects / Keywords: discrete -- interdiction -- network -- nuclear -- 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

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Abstract: We consider a number of interdiction problems with applications in homeland security. First, we study a stochastic interdiction model formulated by Morton et al. (IIE Transactions, 39:3-14,2007) that aims to locate radiation sensors at border crossings in order to detect and help prevent the smuggling of nuclear material. In this model, an interdictor places sensors at customs checkpoints to minimize a potential smuggler's maximum probability of crossing a border undetected. Our contribution focuses on a version of this model in which the interdictor has different, and likely more accurate, perceptions of the system's parameters than the smuggler does. We develop a class of valid inequalities along with a corresponding separation procedure that can be used within a cutting-plane approach to reduce computational effort. We then extend this work by performing a polyhedral study for the deterministic special case. For this case, we develop the minimal convex hull representation for the polytope linking the interdictor's variables with the smuggler's. In the process, we expose an exponential class of easily-separable inequalities that generalize all of those developed so far for this class of problems. We argue that some instances of the stochastic model have facets corresponding to the solution of NP-hard problems. Our computational results show that the cutting planes developed in this chapter may strengthen the linear programming relaxation of the stochastic model by as much as 25 percent. Next, we consider the interdiction of a capacitated network that exists in Euclidean space. Nodes in this network exist at a point in space and (directed) arcs connect node pairs in a straight line. An opponent wishes to maximize flow from a source node to a sink node across the network, while an interdictor seeks to minimize the opponent's maximum flow by choosing multiple locations to attack. In this problem, attacks are made at points in space. Damage is inflicted on each arc by reducing its capacity as a function of the distance from the midpoint of the arc to an attack.  We provide mathematical programming-based approaches for solving this problem.
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 Kelly M Sullivan.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Smith, Jonathan.

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Permanent Link: http://ufdc.ufl.edu/UFE0044471/00001

Material Information

Title: Two-Level System Interdiction
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Sullivan, Kelly M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: discrete -- interdiction -- network -- nuclear -- 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 consider a number of interdiction problems with applications in homeland security. First, we study a stochastic interdiction model formulated by Morton et al. (IIE Transactions, 39:3-14,2007) that aims to locate radiation sensors at border crossings in order to detect and help prevent the smuggling of nuclear material. In this model, an interdictor places sensors at customs checkpoints to minimize a potential smuggler's maximum probability of crossing a border undetected. Our contribution focuses on a version of this model in which the interdictor has different, and likely more accurate, perceptions of the system's parameters than the smuggler does. We develop a class of valid inequalities along with a corresponding separation procedure that can be used within a cutting-plane approach to reduce computational effort. We then extend this work by performing a polyhedral study for the deterministic special case. For this case, we develop the minimal convex hull representation for the polytope linking the interdictor's variables with the smuggler's. In the process, we expose an exponential class of easily-separable inequalities that generalize all of those developed so far for this class of problems. We argue that some instances of the stochastic model have facets corresponding to the solution of NP-hard problems. Our computational results show that the cutting planes developed in this chapter may strengthen the linear programming relaxation of the stochastic model by as much as 25 percent. Next, we consider the interdiction of a capacitated network that exists in Euclidean space. Nodes in this network exist at a point in space and (directed) arcs connect node pairs in a straight line. An opponent wishes to maximize flow from a source node to a sink node across the network, while an interdictor seeks to minimize the opponent's maximum flow by choosing multiple locations to attack. In this problem, attacks are made at points in space. Damage is inflicted on each arc by reducing its capacity as a function of the distance from the midpoint of the arc to an attack.  We provide mathematical programming-based approaches for solving this problem.
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 Kelly M Sullivan.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Smith, Jonathan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044471:00001


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TWO-LEVELSYSTEMINTERDICTIONByKELLYM.SULLIVANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012KellyM.Sullivan 2

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Tomyparents,grandparents,andthreewonderfulsisters. 3

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ACKNOWLEDGMENTS First,Iwouldliketothankmyadvisor,Dr.J.ColeSmith,formentoringmeduringmystudiesattheUniversityofFlorida.Hehastaughtmemuchaboutresearchandteachingandprovidedmewithaconstantsourceofencouragement.IthankDr.Smithforansweringeverysingleoneofmyemails,texts,andcallsoverthelastfouryears,nomatterthetimeofdayornight.Iamthankfulforthefeedbackofthreeveryhelpfulcommitteemembers:Dr.JosephHartman,Dr.JeanPhilippeRichard,andDr.MyThai.Theircommentshavegreatlyimprovedtheresearchpresentedinthisdocument.ThanksalsotoDaveMortonandFengPanwhomadesignicantcontributionstothenuclearinterdictionresearchpresentedinthisdissertation.IalsoowethankstoDr.C.RichardCassadyforhisguidanceandfriendshipaswell.Hetaughtmyrstengineeringclass,servedasmyrstresearchadvisor,andisthereasonIampursuingacareerintheeld.AmongthemanyfriendsthatIhavemadeinGainesville,IwanttoexpresssincerestthankstoShantihSpantonwhowasthereformanygoodtimesandsomereallytoughonestoo.IalsowanttothankmydearfriendsSiqianShenandSoheilHemmati,aswellasmybestfriend,KimCarlson.Finally,Iammostgratefulfortheloveandsupportofmyparents,MikeandCarol,andtheworld'sbestsiblings,Susan,Michelle,Staci,andKevin.Withoutmyfamily,Iwouldbelost. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 1.1MotivationandSignicance .......................... 12 1.2LiteratureSurvey ................................ 13 2SECURINGABORDERUNDERASYMMETRICINFORMATION ........ 16 2.1Background ................................... 16 2.2RevisedBiPSNIPFormulation ......................... 18 2.3StepInequalitiesandSeparation ....................... 24 2.4ComputationalResults ............................. 34 3CONVEXHULLREPRESENTATIONFORTHEDETERMINISTICBORDERMONITORINGPROBLEM .............................. 42 3.1Background ................................... 42 3.2BasicFormulation ............................... 47 3.3Reformulation .................................. 49 3.3.1ConvexHullFormulation ........................ 49 3.3.2Obtainingconv(P)throughSSRLT .................. 60 3.3.3GeneralizedStepInequalities ..................... 63 3.4ExtensionstotheMultiple-OpponentProblem ................ 67 3.5ComputationalResults ............................. 71 4GEOGRAPHICALINTERDICTIONOFAMAXIMUMFLOWNETWORK .... 73 4.1Background ................................... 73 4.2MaximumFlowInterdictionModels ...................... 75 4.2.1GenericSingle-StageFormulation .................. 75 4.2.2SpecicationtoE-MFNIP ....................... 78 4.3SolvingoveraGeneralCapacityFunction .................. 81 4.4DiscreteModels ................................ 86 4.4.1DiscretizedE-MFNIPModel ...................... 88 4.4.2UsingDE-MFNIPtoProvideaLowerBoundfor?E ......... 93 4.4.3Discretize-and-ReneSolutionMethodology ............. 99 4.5ComputationalResults ............................. 100 5

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5CONCLUSION .................................... 110 REFERENCES ....................................... 113 BIOGRAPHICALSKETCH ................................ 118 6

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LISTOFTABLES Table page 2-1Comparisonofalgorithmsasasymmetryparameterdisvaried. ........ 36 2-2Comparisonofalgorithmsassensoreffectivenessisvaried. ......... 37 2-3ComparisonofalgorithmsasbudgetBisvaried. ................. 37 3-1Deriveddataforexampleproblem. ......................... 45 3-2Comparisonofcontinuousrelaxationsunderdifferentcut-addingstrategies. .. 72 4-1Randomnetworkgenerationproles. ....................... 104 4-2ComputationalresultsfornetworksAthroughD. ................. 108 4-3ComputationalresultsfornetworksEthroughH. ................. 109 7

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LISTOFFIGURES Figure page 2-1Computationtime(inseconds)vs.BforMexico-Canada-U.S.instance. .... 38 2-2Optimalitygap(in%)vs.BforMexico-Canada-U.S.instance. .......... 39 2-3Numberofbranch-and-boundnodesvs.BforMexico-Canada-U.S.instance. 39 2-4Numberofstepinequalitiesaddedvs.BforMexico-Canada-U.S.instance. .. 40 2-5Optimalvalue(ratio)vs.BfortheinstanceMexico-Canada-U.S.instance. ... 41 2-6Plotofsensor-placementlocationsvs.BfortheMexico-Canada-U.S.instance. 41 4-1Examplenetworkwithonlyonepossiblecutset. ................. 82 4-2f-functionandlower-boundingg-functionsandrenementafterpartitioningZ. 84 4-3Examplestepfunctionfandlower-boundingconvexg-functions. ........ 84 4-4Networkforpartition-and-reneexample. ..................... 87 4-5Renementofg-functionsforpartition-and-reneexample. ........... 87 4-6Functionfij(d)lower-boundedbyrelaxedfunctionfij[maxf0,d)]TJ /F4 11.955 Tf 11.95 0 Td[(ag]. ...... 93 4-7Testnetwork. ..................................... 101 4-8Comparisonofobjectivevaluesasbisvaried. .................. 102 4-9Plotofattacklocationsascapacityparameterbisvaried. ............ 103 4-10Upperandlowerboundsobtainedateachiterationofdiscretize-and-reneforFnetworkwithK=5attacks. ............................ 106 4-11Solutiontimesforthreesubroutinesateachiterationofdiscretize-and-rene. 106 8

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTWO-LEVELSYSTEMINTERDICTIONByKellyM.SullivanAugust2012Chair:J.ColeSmithMajor:IndustrialandSystemsEngineeringWeconsideranumberofinterdictionproblemswithapplicationsinhomelandsecurity.First,westudyastochasticinterdictionmodelformulatedbyMortonetal.(IIETransactions,39:3,2007)thataimstolocateradiationsensorsatbordercrossingsinordertodetectandhelppreventthesmugglingofnuclearmaterial.Inthismodel,aninterdictorplacessensorsatcustomscheckpointstominimizeapotentialsmuggler'smaximumprobabilityofcrossingaborderundetected.Ourcontributionfocusesonaversionofthismodelinwhichtheinterdictorhasdifferent,andlikelymoreaccurate,perceptionsofthesystem'sparametersthanthesmugglerdoes.Wedevelopaclassofvalidinequalitiesalongwithacorrespondingseparationprocedurethatcanbeusedwithinacutting-planeapproachtoreducecomputationaleffort.Wethenextendthisworkbyperformingapolyhedralstudyforthedeterministicspecialcase.Forthiscase,wedeveloptheminimalconvexhullrepresentationforthepolytopelinkingtheinterdictor'svariableswiththesmuggler's.Intheprocess,weexposeanexponentialclassofeasily-separableinequalitiesthatgeneralizeallofthosedevelopedsofarforthisclassofproblems.WearguethatsomeinstancesofthestochasticmodelhavefacetscorrespondingtothesolutionofNP-hardproblems.Ourcomputationalresultsshowthatthecuttingplanesdevelopedinthischaptermaystrengthenthelinearprogrammingrelaxationofthestochasticmodelbyasmuchas25percent. 9

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Next,weconsidertheinterdictionofacapacitatednetworkthatexistsinEuclideanspace.Nodesinthisnetworkexistatapointinspaceand(directed)arcsconnectnodepairsinastraightline.Anopponentwishestomaximizeowfromasourcenodetoasinknodeacrossthenetwork,whileaninterdictorseekstominimizetheopponent'smaximumowbychoosingmultiplelocationstoattack.Inthisproblem,attacksaremadeatpointsinspace.Damageisinictedoneacharcbyreducingitscapacityasafunctionofthedistancefromthemidpointofthearctoanattack.Weprovidemathematicalprogramming-basedapproachesforsolvingthisproblem. 10

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CHAPTER1INTRODUCTIONOperationsresearchisabroadeldthatisconcernedwtihthedevelopmentofquantitativetechniquestosupportdecisionmaking.Typically,operationsresearchisconcernedwiththedesignoroperationofasystemtoachievemaximumfunctionality.Withinoperationsresearch,theeldofinterdictionseekstoidentifycombinationsofdisruptionsthathavethemostdetrimentaleffecttoasystem'sfunctionality.Manysystemshavethecapabilitytoreacttodisruptionbyreroutingsuppliesorreassigningdemand,forinstance,soidentifyingmosteffectiveinterdictionsisnottrivial.Theearliestinterdictionstudies(see,e.g.,[ 1 2 ])weremilitaryinnature,seekingstrategicaladvancesthatwoulddisruptanenemy'ssupplyand/orcommunicationinfrastructure.Since,interdictionresearchhasexpandedtoconsidertheeffectsofmanytypesofdisruptions,bothintentionalandrandom,leadingtoanincreasedunderstandingofvulnerabilityinmanytypesofsystemsinavarietyofindustries.Interdictionproblemstypicallyconsideradefenderwhoattemptstominimizecostsviatheusageofanunderlyingsystem,andanattackerwhoactstoinhibitthedefender'suseofthesystemviaselectinginterdictionactions.Mostinterdictionliteraturefallsintothecategoryoftwo-levelmodels.Intwo-levelmodels,theattackeranddefendereachmakeonemove:Theattackeractsrsttodamagethedefender'ssystem,thedefenderthendevelopsanoperationalplanforhis/hersystemthatminimizescostgiventheattacker'saction.Theattacker'sgoalistomaximizethedefender'sminimumcost,andtherefore,two-levelmodelsexhibitthestructureofaStackelberggame,whicharenaturallyformulatedasmin-maxmathematicalprograms.Theseoptimizationproblemsareusuallyverydifculttosolvebecauseevaluatingtheobjectivefunctionrequiressolvingthedefender'soptimizationproblem.Someattentionhasalsobeengiventohigher-ordernestedinterdictiongames,aswell.Inthree-levelmodels,thedefenderactsrsttoconstruct,fortify,orprotectcomponentsofhis/hersystembeforetheattacker 11

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solvesantwo-levelproblemproblem.Theabilitytosolvethree-level(orhigher)problemscouldbeextremelyvaluableindesigningrobustsystems,butdoingsoisimpossibleuntilefcientmethodsareavailablefortwo-levelproblems.Inthisdissertation,wefocuson(i)advancingstate-of-the-arttheoryrelatingtosolvingexistingdiscretetwo-levelinterdictionmodels,and(ii)discoveringanewunderstandingofsystemvulnerabilitythroughthedevelopmentofnewdiscretetwo-levelinterdictionmodels.MotivationfortheproblemsconsideredinChapters 2 4 ispresentedinSection 1.1 followedbyasurveyofrelevantliteratureinSection 1.2 1.1MotivationandSignicanceTheresearchinChapters 2 and 3 ofthisdissertationfocusesoninterdictionmodelswithapplicationtothepreventionofnuclearsmuggling.Chapter 4 developsanewclassofinterdictionmodelsaimedatexaminggeographicalpressure-pointsinasupplynetwork.Weprovidefurtherbackgroundandmotivationtotheseapplicationareasbelow.Nuclearmaterialishighlyregulatedworldwidewithsome190nationspartytotheNuclearNon-ProliferationTreaty(NPT)[ 3 ].Thistreatyaimstopreventthespreadofnuclearweaponstechnologyandultimatelyachieveglobalnucleardisarmament.Unfortunately,securitybreachesatnuclearfacilities,particularlythoseintheFormerSovietUnion(FSU),haveresultedinthetheftandtrafckingofradioactivematerialthatcouldbeusedbyaterroristclanorNPTnon-partytoconstructnuclearweapons.Onewaytoreducethethreatofnuclearproliferationistoinstallradiationsensorsalongthoroughfaresthatmightbepopularroutesfortrafckers.PerformingsuchinterdictionsispreciselytheaimoftheSecondLineofDefenseprogramoftheUnitedStatesDepartmentofEnergy,whopromoteandregulatetheimplementationofnuclearsensorsintheFSUandaroundtheworld.Sensorinstallationshavedualbenet:(1)deterrenceofpotentialsmugglers,and(2)detectionandpreventionofactualsmugglers.However,installationofsensorsiscostlyandtypicallyperformedincrementally;thus,a 12

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bestimplementationstrategyisnotobvious.Moreover,inmanycases,covertinstallationofsensorsisimpossibleorimpractical,andthus,potentialsmugglersarecapableofreroutingtoavoidthistechnology.Chapters 2 and 3 relatetothedevelopmentofcost-effectivesensor-installationstrategiesminimizeapotentialsmuggler'schanceofsuccessfullytrafckingnuclearmaterialacrossaborder.ThemotivatingapplicationsofChapter 4 areimportantaswellbutdiffergreatlyfromthoseofChapters 2 and 3 .Manycriticalsystems(e.g.,transportationnetworksorpowergrids)well-denedgeographicalcharacteristics.Moreover,thesesystemsaresubjecttoanumberofdisruptions(e.g.,earthquakes,explosiveweapons,orradiationsensors)thathaveeffectoveranarea.TheresearchpresentedinChapter 4 aimstomodelthesegeometricrelationshipsforasimplemaximum-ownetwork(andhopefullypavethewayforfuturegeographicalinterdictionmodels). 1.2LiteratureSurveyInterdictionliteraturedatesbacktoWollmer's1964paper[ 1 ]thatposesthefollowingmaximumownetworkinterdictionproblem(MFNIP):Findthenarcsinamaximum-ownetworkthat,whenremoved,resultinthesmallestmaximum-ow.Anumberofextensionstothisworkweredevelopedinthefollowingyears.Luboreetal.[ 4 ]developsanefcientalgorithmforsolvingtheprobleminthecaseofasinglearcremoval.Ghareetal.[ 5 ]developanimplicitenumerationalgorithmforsolvingtheproblemforgeneraln.Ratliffetal.[ 6 ]developanetwork-modicationmethodforsolvingMFNIP.McMastersandMustin[ 2 ]consideravariantofMFNIPinwhichpartialinterdictionofarcsispermitted.Corley[ 7 ]showsthatMFNIPcanbeadaptedtoidentifythemostcriticalnodesinamaximumownetwork.MFNIPisknowntobeNP-hardeitherundertheassumptionthateacharcmustbeeithercompletelydestroyedorunaffected[ 8 ]orifarcscanbepartiallyinterdicted[ 9 ].Wood[ 8 ]providesasingle-levelmixed-integerprogramming(MIP)formulationofMFNIPusingadualization-linearizationtechniquethathasprovenusefulinmodelingothermin-maxproblemsaswell,and 13

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whichformsthebasisofmostcontemporaryinterdictionresearch.Roysetetal.[ 10 ]giveabi-objectivemodelbasedonWood'sMIP,andAltneretal.[ 11 ]developvalidinequalitiesthatstrengthenthemodelof[ 8 ].FulkersonandHarding[ 12 ]andGolden[ 13 ]studytheproblemofcontinuouslyinterdictingashortestpathsubjecttoabudgetconstraint.Thediscreteshortestpathnetworkinterdictionproblem(SPNIP)(see,e.g.,[ 14 16 ])seekstondthenarcswhoseremovalmaximizesthelengthofanetwork'sshortestpath.ThisproblemisknowntobeNP-hardingeneral[ 17 ],butKhachiyanetal.[ 18 ]giveapolynomialalgorithmforthecaseinwhichaxednumberofoutgoingarcsmustberemovedfromeachnode.IsraeliandWood[ 19 ]formulateSPNIPasamixedintegerprogram(MIP)anddetaildecompositionalgorithmstosolvetheproblem.Interdictionofamoregeneralminimum-cost-ownetwork[ 20 ]andofamulticommodity-ownetwork[ 21 22 ]hasalsobeenstudied.Whilethemajorityofinterdictionresearchcentersaroundnetwork-relatedsecond-stageproblems,interdictionoffacilitylocation-allocationproblems[ 23 27 ]hasalsoreceivedasignicantamountofattention.Recentcontributionsinthisareaconsiderproblemsinwhichinterdictionshaverandomeffectsonthesystem[ 28 29 ],andinwhichtheotherplayer'sbehaviorisuncertain[ 25 30 33 ].Baileyetal.[ 34 ]modelaproblemthathasbothofthesetypesofuncertainty.Whilemoststudiesofinterdictionunderuncertaintyuseanobjectivefunctionbasedonexpectation,[ 35 36 ]maximizetheprobabilitythatthedisruptioncausedisinexcessofathreshold.Informationasymmetrybetweenattackeranddefenderhasbeenstudied[ 32 37 39 ]ashavemultiple-objectiveproblems[ 10 39 40 ].Applicationsofinterdictionmodelsvarywidely,includinginfectioncontrol[ 41 ],naturaldisastermitigation[ 42 ],militarytacticdevelopment[ 43 ],naturalresourceandinfrastructuresecurity[ 38 44 46 ],andpreventionofnuclearproliferation[ 32 33 47 ].Manyapproacheshavebeendevelopedtomodelandsolvetwo-levelinterdictionproblems,butmostareadoptedfromeither(i)thetransformationofamin-max 14

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problemintoasinglestageminimizationproblem,asdemonstratedin[ 8 ],or(ii)outer-approximatingdecompositionapproachessuchasBendersdecomposition[ 48 ](see[ 19 33 ]forexamples).Approaches(i)and(ii)aredependentuponthelower-levelmaximizationproblemhavingtractablestructure,asisthecasewithnetworkowproblemsaswellasotherlinearprograms.Difcultiesareassociatedwithbothapproaches:Approach(i)typicallyresultsinanonlinearformulationwhichmustbelinearizedbyintroducingadditionalvariablesandconstraints.Approach(ii)mayprovedifcultbecausetheattacker'sdecisionistypicallycombinatorial,meaningthattheBendersmasterproblem,whichmustbesolvedandre-solvediteratively,maybeanintegerprogram.Three-levelinterdictionproblemsarecommonlyapproached(asin[ 49 50 ])byreformulatingthelowertwolevelsviaapproach(i)andsolvingtheresultingmodelasasubroutineinanalgorithmthatimplicitlyenumeratesrst-levelsolutions. 15

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CHAPTER2SECURINGABORDERUNDERASYMMETRICINFORMATION 2.1BackgroundInthischapter,weconsiderastochasticinterdictionproblemformulatedbyMortonetal.[ 32 ].Inthisproblem,aninterdictorattemptstominimizeanuclearsmuggler'sprobabilityoftraversingaborderundetectedbymonitoring(i.e.,installingradiationsensorsat)alimitednumberofcustomscheckpoints.Inordertocrosstheborder,thesmugglermustpassthroughasinglecheckpoint.Associatedwitheachcheckpointareevasionprobabilitiesthatdependonwhetherthecheckpointismonitored.Thecharacteristicsofthesmuggler,includingthenatureofthematerialbeingsmuggled,thesmuggler'sstartinglocationandnaldestination,etc.,areuncertain,butareassumedtofollowaknowndiscretedistribution.Therealizations,orscenarios,ofthisdiscretedistributionaremanifestedintheevasionprobabilitiesateachcheckpoint.Theinterdictor'sgoalistoidentifyasubsetofcheckpointstomonitorthatminimizesthesmuggler'sunconditionalprobabilityofevasion,i.e.,theweightedsumofevasionprobabilitiesoverallscenarios.Thetimingofeventsandimbalanceofinformationarekeyfeaturesintheproblemwestudy.Theinterdictormustchoosewhichcheckpointstomonitorbeforethescenariocharacterizingthesmugglerisrevealed.Giventhelocationsofthesensors,thesmugglerthentraversesthecheckpointhavingthegreatestevasionprobability.Thesmugglerisassumedtohaveperfectknowledgeofwherethesensorsareplaced,andthus,theproblemexhibitsthestructureofaStackelberggame.However,incontrasttomostresearchonthisclassofproblems,weconsiderthecaseinwhichthesmuggler'sperceptionofcheckpoint-evasionprobabilities,bothwithandwithoutsensorinstallation,donotnecessarilymatchtheinterdictor'sperception.Akeyassumptionisthattheinterdictorknowsthesmuggler'sperceptionoftheseprobabilities,or,moregenerally,thattheinterdictoriswillingtoassumeaprobabilitydistributionoverthesmuggler's 16

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perception.Thus,theinterdictormayusedeception(e.g.,notmonitoringacheckpointforwhichthesmuggleroverestimatestheevasionprobability)tohisadvantage.Inwhatfollows,werefertotheinterdictor'sperceptionoftheevasionprobabilitiesasbeingthetrueevasionprobabilities.Theproblemwestudyiscloselyrelatedtoaclassofnetworkinterdictionproblemsinwhichthesmugglerseeksamaximum-reliabilitypaththroughanetworkG(N,A)andsensorsmaybeplacedonanyarcinthenetwork.Mortonetal.[ 32 ]describethisstochasticnetworkinterdictionproblem(SNIP)alongwithanumberofrelatedproblems,includingtheproblemwestudyinthischapter.ForthecaseofSNIPinwhichsensorsmayonlybeinstalledonapredenedcut-setofarcs(i.e.,thosearcs(i,j)2Asuchthati2Tandj2NnTforsomeTN),SNIPreducestoaninterdictionproblemdenedonabipartitenetworkandisthereforedesignatedasBiSNIP.TheproblemwestudyisexactlyBiSNIP(whereeachcut-setarccorrespondstocrossingthroughacheckpoint),butwithouttherestrictionthatthesmugglermustknowthetrueevasionprobabilities.Sincethesmuggler'sperceptionoftheseprobabilitiesneednotmatchthetrueprobabilities,ourproblemisageneralizationofBiSNIPreferredtoasBiPSNIP.TherelationshipbetweenSNIPandBiPSNIPisdescribedin[ 32 ],andassuch,weomitthedetailsofSNIPinfavorofdeningBiPSNIPdirectly.Mortonetal.[ 32 ]deneBiSNIP,BiPSNIP,andtheircounterpartswhensensorinstallationisnotrestrictedtoacut-set.Inaddition,[ 32 ]formulatesbothBiSNIPandBiPSNIPasMIPsanddetailsvalidinequalitiesforBiSNIP(butnotforBiPSNIP)andacorrespondingseparationprocedurethatimprovescomputationtimessignicantly.BiSNIPisproventobeNP-hardin[ 51 ],whichalsoestablishestheresultforthemoregeneralBiPSNIP.Inthischapter,wemakethefollowingcontributionstotheanalysisofBiPSNIP.WerevisetheMIPformulationforBiPSNIPgivenin[ 32 ]toobtainaformulationthatistighterwhileusingfewerconstraints.WethendenestepinequalitiesforBiPSNIPthat 17

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generalizetheinequalitiesgivenforBiSNIPin[ 32 ].Wedevotesignicantattentiontotheanalysisoftheseinequalities,leadingto(1)characterizationofnecessaryandsufcientconditionsfortheseinequalitiestodenefacetstotheconvexhullofinteger-feasiblesolutions,and(2)aprocedurethatidentiesamost-violatedstepinequalityinpolynomialtime.Ourcomputationalresultssuggestthataddingthestepinequalitiesinbranch-and-cutfashionimprovessolutiontimesoversolvingtheproblemwithcommercially-availableMIPsolvers.Weorganizetheremainderofthischapterasfollows.Section 2.2 summarizesourformulationofBiPSNIPanditsrelationtothatin[ 32 ].WedeneournewstepinequalitiesinSection 2.3 ,characterizetheseinequalities,anddescribeanefcientseparationroutine.WedemonstratetheeffectivenessofourtechniquesonasetoftestinstancesinSection 2.4 ,andexaminecharacteristicsofoptimalsolutionsprescribedbyourmodelsinthecontextofaUnitedStatesbordersecurityprobleminvolvingtheroadnetwork. 2.2RevisedBiPSNIPFormulationInthissection,wedeneBiPSNIPanddevelopthenotationtoformulateBiPSNIPasaMIP.WethenrevisetheBiPSNIPformulationgivenin[ 32 ]withonethatistighterandrequiresfewerconstraints.InBiPSNIP,weconsideraninterdictorwhoinstallssensorsonasubsetofcheckpointswiththeaimofthwartingasmuggler'sattempttocrossaborder.Letdenoteanitesetofscenarios,whereeachelement!2speciesthetrueandperceivedevasionprobabilitiesforapotentialsmuggler.Inthismanner,wemodelthecaseinwhichtheinterdictorisunsureofthesmuggler'scharacteristics,andhencethesmuggler'sevasionprobabilitiesbothtrueandperceivedthrougheachcheckpoint.Weassumetheinterdictoriswillingtomodelthisuncertaintyviaadiscretedistributionontheseevasionprobabilities.Letf!,!2,betheprobabilitythatscenario!is 18

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realized.Intuitively,wevieweachscenario!asauniquesmuggler,andhencewewillrefertosmuggler!.LetK=f1,...,ngdenotethesetofcheckpoints.Eachsmugglerischaracterizedbythefollowingdata,denedforeach!2andk2K: p!k:smuggler!'strueevasionprobabilitythroughcheckpointkifkisnotmoni-tored, q!k:smuggler!'strueevasionprobabilitythroughcheckpointkifkismonitored, p!k:smuggler!'sperceivedevasionprobabilitythroughcheckpointkifkisnotmonitored, q!k:smuggler!'sperceivedevasionprobabilitythroughcheckpointkifkismoni-tored.Theasymmetryoftheproblemhingesontheassumptionthatsmuggler!knowsonlythep!-andq!-values,aswellasthesensorlocations,whenselectingacheckpointtotraverse.Bycontrast,theinterdictorhasfullknowledgeofboththetrueprobabilities(p!andq!)andtheperceivedprobabilities(p!andq!)whendecidingwhichcheckpointstomonitor.Iftheinterdictorisunsureofthesmuggler'sperception,butiswillingtopositaprobabilitydistributiononthatperception,thesamemodelholdswith!2denedtoincludethisuncertaintyinthesmuggler'scharacteristics.Forsimplicityinexposition,weassumethatallp-andq-valuesaredistinctandpositive,andthatp!k>q!k,8!2,k2K.Aftersensorshavebeenplaced,eachsmuggler'sproblemamountstoselectingthecheckpointwiththegreatestperceivedprobability.Thatis,let^xk,k2Kindicatewhether(1)ornot(0)checkpointkismonitored.Eachsmuggler!traversesthecheckpointk!thatmaximizesp!k(1)]TJ /F3 11.955 Tf 12.28 0 Td[(^xk)+q!k^xkoverallcheckpointsk2K.Theresultingtrueevasionprobabilityforsmuggler!isp!k!if^xk!=0andq!k!otherwise.Theinterdictorseekstominimizethesumofallpossiblesmugglers'trueevasionprobabilities,weightedbytheprobabilitythatscenario!isrealized. 19

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Foreach!2,letq!=maxk2Kfq!kgandletk!?bethecheckpointforwhichq!=q!k!?.Notethatifp!kq!gtobethesetofcheckpointsthatwithoutsensorsinstalledsmuggler!perceivesasallowinganevasionprobabilityabovethatofafullymonitoredborder.Deneanauxiliarycheckpointn+1,onwhichnosensorcanbeplaced,wherep!n+1=q!k!?andp!n+1=q!k!?.Checkpointn+1correspondstothesmuggler'schoiceifallcheckpointsinK!aremonitored.LettingK!=K![fn+1g,wedeneAasthose(!,k)pairsinf1,...,n+1gsuchthatk2K!.Smuggler!'sproblemcannowbestatedasfollows:Selectthecheckpointk2K!havingthelargestp!k-valueamongallcheckpointsthatarenotmonitored.(Note:Becausep!k>p!n+1forallk2K!andcheckpointn+1cannotbemonitored,checkpointn+1issmuggler!'sselectionifandonlyifallofthecheckpointsinK!aremonitored.)Aconsequenceofthissimplicationisthatsensorinstallationnoweffectivelyremovesacheckpointandasaresult,theproblemnolongerdependsuponqandq.InordertorepresenttheBiPSNIPmodel,wedeneadditionalnotation,andsummarizenotationdenedpreviously,asfollows.DATA :setofscenarios,i.e.,setofpotentialsmugglers f!:probabilityofrealizingscenario!2 K=f1,...,ng:setofcheckpoints B:budgetforinstallingsensors bk:costtoinstallasensoratcheckpointk(0bkB,8k2K) K!K:setofcheckpointsthatarecandidatesforselectionbysmuggler!2 n+1:auxiliarycheckpoint,chosenbysmuggler!2ifeachcheckpointinK!receivesasensor K!=K![fn+1g 20

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A=f(!,k):!2,k2K!g p!k:trueevasionprobabilityforsmuggler!throughcheckpointk2K! p!k:perceivedevasionprobabilityforsmuggler!throughcheckpointk2K!(p!k>p!n+1>0,8k2K!)VARIABLES xk=1ifasensorisinstalledatcheckpointk0otherwise !=smuggler!'strueevasionprobability !=smuggler!'sperceivedevasionprobability y!k=1ifsmuggler!choosescheckpointk0otherwiseModel( 2 )belowisarevisedversionoftheBiPSNIPformulationgivenby[ 32 ],whereX=x2f0,1gjKjjPk2KbkxkB. MinX!2f!!, (2a)subjectto!p!ky!k,(!,k)2A, (2b) !p!k(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xk),(!,k)2A, (2c) !=Xk:(!,k)2Ap!ky!k,!2, (2d)Xk:(!,k)2Ay!k=1,!2, (2e)y!k1)]TJ /F4 11.955 Tf 11.95 0 Td[(xk,(!,k)2A, (2f)y!k0,(!,k)2A, (2g)x2X. (2h)Theobjectivefunction( 2a )minimizesthetrueevasionprobability.Constraints( 2b )boundtheconditionaltrueevasionprobabilities,conditionedoneachsmuggler!,dependingonwhichcheckpointsaremonitoredandwhichcheckpointischosenby 21

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smuggler!.Foragivenbinaryx=^x,smuggler!solves minf !: !p!k(1)]TJ /F3 11.955 Tf 12.14 0 Td[(^xk),8k2K!g.(2)Constraints( 2c )( 2g )representoptimalityconditionsfor( 2 )intheformofprimalfeasibility( 2c ),dualfeasibility( 2e )( 2g ),andstrongduality( 2d ).Constraints( 2h )ensurethatsensorinstallationisdiscreteandfeasiblewithinbudgetrestrictions.Weletxn+10andremoveitfromthemodelsothatconstraints( 2c )and( 2f )haveright-handsidep!n+1and1,respectively,for(!,n+1). Remark2.1. IfXisdenedbyacardinalityconstraint,i.e.,bk=18k2KandB2Z,thenModel( 2 )canbesimplied.Smuggler!'schoiceislimitedtotheB+1checkpointsk2Kwiththelargestp!k-values.Accordingly,constraints( 2b )and( 2c )correspondingtocheckpointsthatwillnotbechosencanberemoved,andcorrespondingy-variablescanbexedtozero.Thisnotionextendstohandlenon-unitbk-valuesandnon-integerB.Wetraversesmuggler!'scheckpointskfromlargesttosmallestp!k,summingbkuntilthesumexceedsB.TheremainingcheckpointscanberemovedfromK!,andhenceA.2Wenowreformulatemodel( 2 )inordertotightentheformulation.Again,letk!denotethecheckpointthatsmuggler!traversessothaty!k!=1andy!k=0fork6=k!.Forxed!,thismeansthat( 2b )reducesto!0forallk6=k!suchthat(!,k)2A,andthek!-thconstraintisasimplenonnegativelowerboundon!.Sinceweminimize!,itisoptimaltoset!equaltothelowerboundgivenbythek!-thconstraint.Thisconditioncanequivalentlybegivenby !=Xk:(!,k)2Ap!ky!k,!2.(2) 22

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Updating( 2 ),weobtainthevalidMIPformulation MinX!2f!!, (2a)subjecttox2X,!=Xk:(!,k)2Ap!ky!k,!2, (2b)( 2c )through( 2g ).Replacing!intheobjectivefunction( 2a )withtheright-handsideof( 2b )andeliminating( 2b )isalsovalid,butweprefertheformofmodel( 2 )foreaseofexpositionintheremainderofthischapter.Model( 2 )canalsobesimpliedaccordingtoRemark 2.1 .Model( 2 )hasjAj)-307(jjfewerconstraints,asweremovedjAjconstraintsoftheform( 2b ),andaddedjjconstraintsoftheform( 2b ).Moreover,model( 2 )istighterthanmodel( 2 )inthesensethatsolutionsthatarefeasibletothelinearprogrammingrelaxationof( 2 )arefeasibletothelinearprogrammingrelaxationof( 2 ),buttheconversedoesnothold.Wenowformalizethesestatements. Proposition2.1. LetP1denotethelinearprogrammingrelaxationof( 2 )andP2thelinearprogrammingrelaxationof( 2 ),i.e.,therelaxationsfromreplacingXin( 2 )and( 2 )withX=x2[0,1]jKjjPk2KbkxkB.Let(^x,^,^ ,^y)beafeasiblesolutiontoP2.Then(^x,^,^ ,^y)isalsofeasibletoP1. Proof. Weneedonlyshowthat(^x,^,^ ,^y)satises( 2b )sincetheremainderoftheconstraintsin( 2 )arein( 2 )aswell.Byhypothesis, ^!=Xk:(!,k)2Ap!k^y!k,8!2.(2)Sinceallofthetermsin( 2 )arenonnegative,wehavethat^!p!k^y!kforeach(!,k)2A.So(^x,^,^ ,^y)satises( 2b )andmustbefeasibletoP1. 23

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Remark2.2. TheconverseofProposition 2.1 isfalse:Feasibilityof(^x,^,^ ,^y)toproblemP1doesnotensurethat(^x,^,^ ,^y)isfeasibletoP2,eveninthe(nontrivial)caseinwhichoneoftheconstraints( 2b )istightforeach!2.Asanexample,supposethatK!=f1,2g,8!2,andallp-valuesarestrictlypositive.Further,considerthevector^x=(0.5,0.5)2Xandlet^y!1=^y!2=0.5foreach!2.Observethat^xand^ysatisfyconstraints( 2e )and( 2f )( 2h ).Dene ^ !=Xk:(!,k)2Ap!k^y!k=0.5p!1+0.5p!2,(2)tosatisfy( 2d ).Sincep!k0forall(!,k)2A,( 2 )implies^ !0.5p!1and^ !0.5p!2foreach!2,i.e.,( 2c )issatised.Under^y,constraints( 2b )reduceto!0.5p!k,k2f1,2g,!2.Foreach!2,letk!2argmaxk2f1,2gp!k,andset^!=0.5p!k!sothatthischoiceof(^x,^,^ ,^y)satises( 2b ),andisfeasibletoP1.Toassesswhether(^x,^,^ ,^y)isfeasibletoP2itsufcestoexamineconstraint( 2b )sinceallotherconstraintsofP2arealreadyveried.Constraint( 2b )reducesto!=0.5p!1+0.5p!2,!2,inthiscase.However,^!=0.5p!k!<0.5p!1+0.5p!2,giventhatp!1andp!2arepositive.Thus,(^x,^,^ ,^y)isnotfeasibletoP2.2 2.3StepInequalitiesandSeparationInthissectionwederiveaclassofvalidinequalitiesthatboundstheevasionprobability!foragivenscenario!2.Following[ 32 ],werefertoourinequalitiesasstepinequalitiesbecausetheboundfor!iscomputedbysubtractingstepswhichmaybetakenornot,dependingonthevalueofxfromaconstant.OurvalidinequalitiesforBiPSNIPgeneralizetheBiSNIPinequalitiesof[ 32 ].Inaddition,wecharacterizenecessaryandsufcientconditionsunderwhichourinequalitiesarefacet-dening,andweidentifyaprocedurethatseparatesamost-violatedBiPSNIPstepinequalityinpolynomialtime.OurfacetresultandseparationprocedurefortheBiPSNIPinequalitiesgeneralizeanalogousresults,givenin[ 51 ]and[ 32 ]respectively,fortheBiSNIPinequalities. 24

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Withoutlossofgenerality,assumethatthecheckpointsinK!areuniquelyindexedsuchthatK!=f1,...,n0g,where p!1>>p!n0>q!.(2)Withtheseassumptions,smuggler!willtraversethelowest-indexedcheckpointinK!=K![fn+1gthatdoesnothaveasensor.Recallthatauxiliarycheckpointn+1,whichneverhasasensor,correspondstothesmuggler'schoiceifallcheckpointsinK!haveasensor.TheinequalitiesproposedinthissectionareassociatedwithsubsetsofK!.Hence,weintroducethefollowingnotationcorrespondingtothesesubsets. H=fh1,...,hmgK!,whereh1<i(`i)]TJ /F5 11.955 Tf 11.96 0 Td[(i)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xhi),(2)where`m+1m(=`m)sothattheterm(m)]TJ /F5 11.955 Tf 12.07 0 Td[(`m+1)xhm=0.However,asthefollowingexampleshows,inequalities( 2 )areonlysometimesvalid,dependingonourchoiceofH. 25

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Example.Supposen=5and,asindicatedin( 2 ),supposethesmugglerpreferscheckpointsintheorder1,2,3,4,5,providedtheydonothavesensors.Thatis,thesmugglerchoosesthelowest-indexedcheckpointwithoutasensoror,ifcheckpoints15allreceivesensors,thesmugglerchoosestheauxiliarycheckpoint6.Forthisscenario,let(p!1,...,p!6)=(0.82,0.8,0.9,0.65,0.7,0.55).Smuggler!'sorderingofthecheckpointsdiffersfromthetrueordering,indicatinginformationasymmetry.Forexample,smuggler!preferscheckpoint1tocheckpoint3eventhoughp!1
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x2f0,1gjKjjPk2KbkxkB,issufcientlylarge,Hinducesavalidinequalityoftheform( 2 )onlyifconditions(a)and(b)aresatised. Proof. First,supposethatconditions(a)and(b)hold.Let(^x,^!)beasubvectorofafeasiblesolution(^x,^,^ ,^y)tomodel( 2 ).Let^xn+10,andletk0bethesmallestindexinK!suchthat^xk0=0.Weshowthattheright-handsideof( 2 )isatmostp!k0.Tothisend,foreachk2H,denei(k)astheelementofI(H)satisfyinghi(k)=k.Sincethetermsof( 2 )correspondingtoi2I(H):hi>k0areallnonpositive,andtermscorrespondingtoi2I(H):`i>i,hii(k0), (2a)p!k0`1)]TJ /F28 10.909 Tf 39.2 10.37 Td[(Xi2I(H):`i=i,hii(k0),correspondingto( 2a ).Notethattheterms`1)]TJ /F14 11.955 Tf 11.76 8.96 Td[(Pi2I(H):`i=i,hihm.Ifk0hm,thenp!k0mbyhypothesis(b),whichsatises( 2b ).Finally,supposethatk02Hand`i(k0)=i(k0)incase2.Theright-handsideof( 2b )collapsestoi(k0)=p!k0,andso( 2b )holdsagain.Next,supposethatcondition(a)isviolated.Inthiscase,thereexistssomek0,k002K!forwhichk02Gk00,k0=2H,andk002H.Inparticular,letk002Hagainbethesmallestindexgreaterthank0suchthat`i(k00)=i(k00).Then,considerthesolutioninwhich ^xk=18k=1,...,k0)]TJ /F3 11.955 Tf 11.96 0 Td[(1, 27

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^xk0=0, ^xhi=08i2I(H):`i=i,hi>k0, ^xhi=18i2I(H):`i>i,hi>k0, ^xk=0(arbitrarily)8k2K!nH,k>k0.Notethat^x2XwhenBissufcientlylarge.Forthissolution,weshouldndthat^!=p!k0.Buttheright-handsidetermsof( 2 )arezeroforeachtermi2I(H)forwhichhi>k0,andforeachtermi2I(H)inwhichbothhii.Thus,( 2 )becomes !`1)]TJ /F14 11.955 Tf 39.79 11.36 Td[(Xi2I(H):`i=i,hip!k0.Thesolutioninwhich^xk=1,8k=1,...,k0)]TJ /F3 11.955 Tf 12.5 0 Td[(1,and^xk=0,8k=k0,...,n0,n+1,causesalltermscorrespondingtoi2I(H),`i>iof( 2 )toequalzero,andtheremainingtermstocollapsetom.Sincem>p!k0,( 2 )isnotvalid.Thiscompletestheproof. Notethatinourexampleabove,thechoicesofHusedtogenerate( 2 )and( 2 )satisfytheconditionforvalidityfromTheorem 2.1 ,butthechoiceofHusedtogenerate( 2 )doesnot:Inthatcase,32H,but1=2Hand12G3.Ifthesmuggler'sperceivedevasionprobabilitiesmatchthetrueevasionprobabilities,i.e.,p=p,then( 2 )isvalidforanyHthatincludesn+1.Thisspecialcasetakestheformofthestepinequalitiesderivedin[ 32 ]forthesituationwithsymmetricinformation.Inthefollowingtheorem,weestablishanotherimportantparallelbetweenthesymmetricandasymmetriccases.First,weestablishthedimensionalityofthefeasibleregionlinking!andthex-variables.Fortheremainderofthissection,weassumeforsimplicitythatjK!j=n0=n.ThecaseinwhichjK!j
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Lemma2.1. LetY!=f(^!,^x):9(x,, ,y)feasibletomodel( 2 )with^!!and^x=xg.Then,thedimensionofY!isn+1,8!2. Proof. Wex!2andreorderthecheckpointindicessothatcondition( 2 )issatised.Wegenerateck2Y!,k=1,...,n+2,whichdenoten+2afnelyindependentpointsoftheform(^!,^x)asfollows.Letc1=(p!2,e1)andck=(p!1,ek),k=2,...,n,whereekdenotesthen-dimensionalunitvectorwitha1inthek-thcomponent.Letcn+1=(p!1,0)andcn+2=(p!1+1,0).Notethatthe^x-componentofeachvectorsatises^x2XbytheassumptionthatbkB,8k2K.The^-componentsatisesconstraint( 2b )byconstructionandbycondition( 2 ).Formthe(n+1)(n+1)matrixCinwhichrowkcorrespondsto(ck)]TJ /F4 11.955 Tf 12.61 0 Td[(cn+2),fork=1,...,n+1.WeprovethattheonlysolutiontoTC=0isthetrivialsolutionk=0,8k=1,...,n+1.Columns2,...,n+1ofCareidentitycolumns,implyingthat1==n=0.Column1nowimpliesthatn+1=0,asrequired. Wenowestablishconditionsunderwhich( 2 )inducesafacettotheconvexhullofinteger-feasiblesolutionstothebudget-relaxedBiPSNIP. Theorem2.2. AssumethatthehypothesesofLemma 2.1 hold,andthatthecheckpointindicesareorderedaccordingto( 2 ).Fix!2andletY!beidenticaltoY!butwithouttherestrictionthatPk2KbkxkB.Then,avalidinequality( 2 )satisfyingtheassumptionsofTheorem 2.1 inducesafacettoY!ifandonlyifindex1isanelementofH. Proof. First,supposethat1=2H.Ifp!1<`1,thenbyTheorem 2.1 ,( 2 )wouldnotbevalid.Else,wewouldhavethatp!1>`1.Considerlifting( 2 )byadding(1)]TJ /F4 11.955 Tf 12.12 0 Td[(x1)toitsright-handside.Ifx1=1,theinequalityisvalidforanyvalueof.Ifx1=0,thencantakeonavalueofp!1)]TJ /F5 11.955 Tf 12.29 0 Td[(`1>0.Sincethisvalueispositive,( 2 )canbestrengthenedandthusdoesnotdeneafacet. 29

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Nowsupposethatcheckpoint1isanelementofH.Wegenerateafnelyindependentpoints,ck2Y!,k=1,...,n+1,oftheform(^!,^x)thatarebindingon( 2 ).DeneI?=fi2I(H):`i>ignf1g.Fork2H,lettherstcomponentofckbep!k,thenextk)]TJ /F3 11.955 Tf 12.18 0 Td[(1componentsbe1,andallothercomponentsbe0,exceptforthosecorrespondingtovariablesxhifori2I?suchthathi>k,whichtakeonvaluesof1.Fork=2H,kn,letckbegivenexactlyasc1,exceptfora1inthecomponentcorrespondingtoxk.Ifn+1=2H,thendenecn+1exactlyasc1,exceptfora1inthecomponentcorrespondingtoxhm.(Thiscomponentmusthavebeenzeroinallpointsthusfar,andsocn+1isdistinctfromc1,...,cn.)Observethatc1,...,cn+1areallelementsofY!,andallarebindingon( 2 ).DeneCasann(n+1)matrixwhoserowsarecomposedofthe(ck)]TJ /F4 11.955 Tf 11.25 0 Td[(c1)vectorsfork=2,...,n+1,suchthatrowkofCcorrespondsto(ck+1)]TJ /F4 11.955 Tf 12.67 0 Td[(c1).Weprovethatc1,...,cn+1areafnelyindependentbyshowingthatChasrankn.Tothisend,weshowthatcolumns2,...,n+1arelinearlyindependent.LetCdenotethematrixcomposedofthelastncolumnsofC.WeprovethattheonlysolutiontoC=0,isthetrivialsolutionk=0,8k=1,...,n.Fork2K!nH,k6=n+1,Ck)]TJ /F12 7.97 Tf 6.59 0 Td[(1=0impliesthatk=0,whereCk)]TJ /F12 7.97 Tf 6.58 0 Td[(1isthe(k)]TJ /F3 11.955 Tf 12.55 0 Td[(1)-strowofC,correspondingtothelastncomponentsof(ck)]TJ /F4 11.955 Tf 12.54 0 Td[(c1).Wecanthusremovethek-thcolumnofCalongwiththerowofCcorrespondingto(ck)]TJ /F4 11.955 Tf 12.88 0 Td[(c1).Ifn+12K!nH,thenweperformthesameoperations,inferhm=0andeliminateitscorrespondingrowandcolumninC.Fork=hi,i2I?,equationsChi)]TJ /F12 7.97 Tf 6.59 .01 Td[(1=0andCh`i)]TJ /F12 7.97 Tf 6.58 .01 Td[(1=0togetherimplythatk=0,andwecaneliminatefromCcolumnkandtherowcorrespondingto(ck)]TJ /F4 11.955 Tf 12.43 0 Td[(c1).Let~CdenotethereducedC-matrixandobservethat~Cisalowertriangularmatrixwithnonzerodiagonals.Thus,~Chasfullrank,i.e.,theremainingkmustallbezero.ThisimpliesthattherankofCisn,asisthedimensionofthefaceintersectedby( 2 ). Notethatforthecaseinwhichp=p,thestepinequality( 2 )inducesafacettoY!wheneverHincludescheckpoint1,provingthatthestepinequalitiesusedin 30

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[ 32 ]areindeedfacetstotheconvexhullofinteger-feasiblesolutionsfortheBiSNIPproblem.ThisspecialcasewasprovenbyPan[ 51 ],aresultthatisnowgeneralizedbyTheorem 2.2 .Whilestepinequalitieshelptotightenthecontinuousrelaxationofmodel( 2 ),theyarenotsufcienttorepresenttheconvexhulloffeasiblesolutionsto( 2 ),eveninthecasewherethereisonlyonescenarioandprobabilitiesaresymmetric.Asanexample,considerascenario!withcheckpointsK!=f1,2,3gand(p!1,p!2,p!3)=(p!1,p!2,p!3)=(1,0.4,0).Supposeb1=b2=1andB=1.Considerafractionalinterdictionvector,(^x1,^x2)=(0.75,0.25),andobservethat(^y!1,^y!2,^y!3)=(0.25,0.5,0.25)combinedwith^!=^ !=0.45completesafeasiblesolutiontothecontinuousrelaxationofmodel( 2 ).Thereareonlytwostepinequalitiesforthisscenario,givenas!1)]TJ /F4 11.955 Tf 12.45 0 Td[(x1and!1)]TJ /F3 11.955 Tf 12.16 0 Td[(0.6x1)]TJ /F3 11.955 Tf 12.16 0 Td[(0.4x2.Under^x,thesereduceto!0.25and!0.45,respectively,neitherofwhichcutoff(^x,^y,^ ,^).Inthisproblem,thesetXconsistsofonlythreesolutions,X=f(0,0),(0,1),(1,0)g,whichresultin!1,!1,!0.4,respectively,foranyfeasiblesolutionto( 2 ).Theonlywaytoobtain^xasaconvexcombinationofthepointsinXis0.75(1,0)+0.25(0,1)whichwouldimply!0.75(0.4)+0.25(1)=0.55if(^x,^y,^ ,^)wereintheconvexhulloffeasiblesolutionsto( 2 ).Since^!=0.45,(^x,^y,^ ,^)residesoutsidetheconvexhulloffeasiblesolutions,yetcannotbecutoffbyastepinequality.Inthisexample,thestepinequalitieswereaccountingforasolution(x1=x2=1and!=0)thatisinfeasiblebecauseofthebudgetconstraint.(ThisisacasewheretheassumptioninTheorem 2.2 aboutBbeingsufcientlylargefails.)Wenowdescribeamodiedstepinequalitythattakesintoaccounttherestrictionsimposedbythebudgetconstraint.Forsimplicity,weconsideronlythecaseinwhichXisgovernedbyacardinalityconstraint,i.e.,bk=18k2K,althoughasdiscussedinRemark 2.1 thistypeofanalysisextendsinastraightforwardwaytohandleageneralknapsackconstraint. 31

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Remark2.3. WhenXisdenedbythecardinalityconstraintPk2KxkBandcheckpointsareorderedaccordingto( 2 ),smuggler!willchoosethelowest-indexedcheckpointinthesetf1,...,B+1gthatisnotmonitored.AsdescribedinRemark 2.1 ,model( 2 )canbesimpliedbyremoving(!,k)fromAforallk>B+1.Stepinequalityconstructionismodiedaswell:Generating( 2 )nowentailsselectingHfromasmallerK!=f1,...,B+1g.Whenstepinequalitiesaregeneratedinthisway,theassumptionthatBissufcientlylargeisnotnecessaryforTheorems 2.1 and 2.2 tohold(becauseallpointsusedintheirproofssatisfyPk2KxkB).2Wenowdescribeaseparationprocedurethattakesasinputasolution^xtothelinearprogrammingrelaxationofmodel( 2 ).Inpolynomialtime,theseparationprocedureeitheridentiesamost-violatedstepinequalityorshowsthattherearenoviolatedstepinequalities.Theprocedureaccomplishesthisbymaximizingtheright-handsideof( 2 )withrespecttoallcheckpointsubsetsHcorrespondingtovalidfacet-deningstepinequalities.Thismaximizationiscarriedoutbysolvingalongest-pathproblemonanacyclicnetwork,G(V,E).LetthenodesetV=K![fs,tg,wheresdenotesadummystartingnodeandtadummyterminationnode.WeconstructthearcsetEasfollows:Createarc(s,k)foreachk2K!satisfyingp!1p!k,includingk=1.Similarly,createarc(k,t)foreachk2K!satisfyingp!kp!k0,8k0=k+1,...,n+1.Finally,createarc(k,k0)fork,k02K!ifk
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Givenasolution^xtothelinearprogrammingrelaxationofmodel( 2 ),wedenethearclengths,ckk0,(k,k0)2E,asfollows: csk0=p!k0)]TJ /F14 11.955 Tf 13.96 11.36 Td[(Xj2Gk0(p!k0)]TJ /F4 11.955 Tf 11.95 0 Td[(p!j)(1)]TJ /F3 11.955 Tf 12.14 0 Td[(^xj),8(s,k0)2E, (2a) ckk0=)]TJ /F3 11.955 Tf 9.3 0 Td[((p!k)]TJ /F4 11.955 Tf 11.96 0 Td[(p!k0)^xk)]TJ /F14 11.955 Tf 21.09 11.35 Td[(Xj2Gk0nGk(p!k0)]TJ /F4 11.955 Tf 11.95 0 Td[(p!j)(1)]TJ /F3 11.955 Tf 12.13 0 Td[(^xj),8(k,k0)2E,k,k02K!, (2b) ckt=0,8(k,t)2E. (2c) Considerthelengthofans-tpathgiventhearclengths( 2 ).Choosing(s,k0)initializesourconstructionoftheright-handsideof( 2 )viaarclengthcsk0in( 2a ).Anarc(k,k0)inthes-tpathwithk,k02K!contributesckk0from( 2b ),whichaccountsforoverlapinGkandGk0withthecontributioncapturingtheadditionofcheckpoints(Gk0[fk0g)nGktoH.Finally,thereisnoadditionalcontributionfromthearcthatterminatesint.RecallthatTheorem 2.1 speciestwoconditionsthatcharacterizeasetHthatyieldsavalidstepinequalityandTheorem 2.2 speciesonefurtherconditionthatensuresHyieldsafacet.Ourseparationprocedureyieldsavalidinequality,sincetherstconditionofTheorem 2.1 issatisedbyallowingtheselectionofcheckpointk2K!withinHonlyifallcheckpointsinGkarealsoinH,andthesecondconditionofTheorem 2.1 issatisedbyourconstructionofthe(k,t)arcsinE.Furthermore,theseparationprocedureyieldsafacet-deninginequalityduetoourconstructionofthearcs(s,k)2E,whichensurethatcheckpoint1belongstoH.Moreover,anyfacet-deningvalidstepinequalityinducedbyasetHcanberecoveredbychoosingapaththroughnodeshifori2I(H)suchthat`i=i.Becausethereisaone-to-onecorrespondencebetweens-tpathsinournetworkandfacet-deningstepinequalities,ourseparationroutineyieldsamost-violatedstepinequality. 33

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2.4ComputationalResultsInthissection,werstpresentcomputationalresultsobtainedfromsolvingrandomlygeneratedinstancesofmodel( 2 )viaabranch-and-cutalgorithmunderanumberofdifferentcut-generationstrategies.WethenanalyzeaprobleminstancederivedfromtheroadnetworkinCanada,Mexico,andtheUnitedStates,andprovideinsightintoourmodel'ssolutions.AllcomputationswereperformedusingaDellPoweredgeT610machinecontainingtwosix-corehyperthreading3.33GHzXeonprocessors.EachMIPproblemwassolvedusingCPLEX12.1usingcallbackfunctionsfromtheConcertTechnology2.9C++librarytoimplementtheseparationprocedure.WenowdescribetheprocedureusedtogenerateBiPNSIPinstancesgivenasetofcheckpointsKandscenarios.Werstrandomlygeneratedp!k(thetrueevasionprobabilities)uniformlyontheinterval[0.25,0.75]foreach(!,k)2K.Weassumetheeffectivenessofasensorisgovernedbyaparameter2[0,1),sothatq!k=p!k.Valuesp!k(orp!k)foreachk2Kcanbeusedtodeduceapreferenceorderofcheckpointsthekwiththegreatestp!kispreferredrst,thesecond-greatestp!kispreferredsecond,andsoon.Ifeachsmuggler!'spreferenceorder(i.e.,theorderobtainedfromthep!k-values)matchesthepreferenceorderobtainedfromthep!k-values,theproblemissymmetricandcouldbesolvedusingthesimplerBiSNIPmodelandstepinequalitiesfrom[ 32 ].Ourhypothesisisthattheproblemismoredifculttosolvewhenthep!k-andp!k-valuesleadtopreferenceordersthatdonotmatchclosely.Tocontrolthedegreetowhicheachsmuggler'spreferenceorderresemblesthepreferenceorderobtainedfromp!k,wespecifyd0andrandomlygeneratep!kuniformlyintheinterval[p!k)]TJ /F4 11.955 Tf 12.91 0 Td[(d,p!k+d].Naturally,dmustbechosensothatallofthep!k-valuesfallinsidetherange[0,1].Whend=0,p!kequalsp!k,resultinginthesymmetricmodel.Theparameterisassumedtobeknowntoeachsmuggler,sothatq!k=p!k.Underlarger,fewercheckpointsarerelevanttoeachsmuggler's 34

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optimizationproblem(i.e.,jK!jissmaller),andhence,theproblemshouldbeeasiertosolve.WeconstructsetsK!foreach!2asdescribedinSection 2.2 ,excludinganycheckpointsthatwouldneverbechosenbysmuggler!,evenwhenallothercheckpointshavesensors.Weassumethatbk=1andBisinteger-valuedsothatx2XstatesthatatmostBsensorsmaybeplaced,andwegeneratestepinequalitiesasinRemark 2.3 .Tosolvemodel( 2 ),weemployabranch-and-cutalgorithmwithparameters(D,R).Inthisalgorithm,weaddstepinequalitiesonlyintherstDlevelsofthebranch-and-boundtree,andweaddatmostRcutsperscenarioateachnode.Wetestthefollowingalgorithms:Nostepinequalitygeneration(0,0),aggressivestepinequalitygeneration(100,100),andtwohybridmethods(10,2)and(20,1).(Wealsoimplementedvariationsofthesealgorithmsinwhichstepinequalitiesareaddedexhaustivelyattherootnode.Ourexperiencesuggeststhatthisresultsinlongersolutiontimes,andresultsarethusomitted.)WeusecallbackfunctionsfromtheConcertTechnologylibrarytoaddstepinequalitiesasneededinalgorithms(100,100),(10,2),and(20,1).DoingsodisablessomefeaturesofCPLEXthatcouldbeusedinsolvingthe(0,0)implementation.Inordertoensureafaircomparison,wedisablethesameCPLEXfeaturesinthe(0,0)implementationbyrunninganemptycallback.Weinitiallysetparametersd=0.1and=0.75,andgeneratedveinstanceseachhavingsizejKj=jj=30withB=15,andjKj=jj=50withB=25.Wethengeneratedmodiedinstancesbyvaryingoneoftheparametersandholdingtherestconstant.Weconsideredd2f0,0.1,0.25g,2f0.5,0.75,0.85g,B2f7,15,22g(forthejKj=jj=30experiments),andB2f12,25,37g(forthejKj=jj=50experiments).Wesolvedeachtestcaseusingthebranch-and-cutalgorithmsexplainedabove,andreportSpearman'srankcorrelation(columnS)fortheB+1greatestp!k-andp!k-values(averagedacrossallscenarios),theaveragesolutiontime(inseconds,givenbyT(s))foreachofthefouralgorithms,theaverageoptimalitygap(columnG(%),whichisthe 35

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Table2-1. Comparisonofalgorithmsasasymmetryparameterdisvaried. (0,0)(10,2)(20,1)(100,100) jjjKjBdST(s)G(%)UT(s)G(%)UT(s)G(%)UT(s)G(%)U 3030150.7501.0020.00.004.10.004.20.004.60.000.10.8662.60.006.30.005.20.004.00.000.250.788.50.001.20.001.40.001.40.005050250.7501.0036001.9525790.1130070.2327440.430.10.8636003.4519500.1119250.1120040.210.250.7936002.352530.002930.004660.00 absoluteoptimalitygapdividedbytheobjectivelowerbound,multipliedby100)afteronehourofcomputation,andthenumberofinstancesunnished(columnU)afteronehourofcomputation.Tables 2-1 2-3 showthatthe(D,R)=(10,2),(20,1),and(100,100)implementationsconsistentlyoutperformthepurebranch-and-boundimplementation,suggestingthatthestepinequalitiesandcorrespondingseparationprocedurehelpinsolvingBiPSNIPinstances.Implementation(100,100)performedthebestinonlyonegroupofexperiments(rowtwoineachofTables 2-1 2-3 )anddidsobyonlyaslightmargin.Implementations(10,2)and(20,1)wereusuallythetwobestimplementationsofthefour,with(10,2)typicallyoutperforming(20,1)byaslightmargin.WethereforeuseD=10andR=2intheremainderofourcomputations.Table 2-1 displaystheresultsasdisvaried.Asseeninthistable,algorithms(10,2)and(20,1)consistentlyoutperform(0,0)andusuallyoutperform(100,100).ThissuggeststhatwhilestepinequalitieshelpinsolvingBiPSNIP,theyshouldnotbeaddedtooaggressively.Perhapscounter-intuitively,theproblemsseemtobecomemucheasierasdincreases,i.e.,astheproblem'sasymmetryincreases.Oneexplanationforthisrelationshiparisesfromascenario!inwhichthesmugglerseverelyoverestimatesthetrueevasionprobabilityp!kthroughaparticularcheckpointk.Therefore,werequirefewersensorstoreducetheevasionprobabilityforscenario!top!kandsimplersolutionsensue.Variationof(seeTable 2-2 )showsamoreintuitiveresult:Lesseffectivesensors(i.e.,largervaluesof)meansthatfewercheckpointsarerelevantineachscenarioand 36

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Table2-2. Comparisonofalgorithmsassensoreffectivenessisvaried. (0,0)(10,2)(20,1)(100,100) jjjKjBdST(s)G(%)UT(s)G(%)UT(s)G(%)UT(s)G(%)U 3030150.10.50.78116.10.0020.30.0019.20.0023.00.000.750.8662.60.006.30.005.20.004.00.000.850.874.10.000.50.000.50.000.60.005050250.10.50.7736004.8536001.4536002.0536002.450.750.8636003.4519500.1119250.1120040.210.850.8725240.31480.00610.00520.00 Table2-3. ComparisonofalgorithmsasbudgetBisvaried. (0,0)(10,2)(20,1)(100,100) jjjKjdBST(s)G(%)UT(s)G(%)UT(s)G(%)UT(s)G(%)U 30300.750.170.556.50.000.30.000.30.000.30.00150.8662.60.006.30.005.20.004.00.00220.9111.00.003.60.004.20.004.70.0050500.750.1120.5634580.33180.00220.00240.00250.8636003.4519500.1119250.1120040.21370.9136003.1520570.0018870.0028690.83 theproblemiseasiertosolve.Again,alessaggressivecut-generationstrategyseemstoperformfavorablyintheseinstances.Inparticular,when=0.85,werequireanaverageof0.5secondstosolveourprobleminstanceswhenjj=jKj=30,andanaverageof48secondstosolvetheinstancesinwhichjj=jKj=50.ChangesinB(seeTable 2-3 )alsoseemtohaveapronouncedeffectonsolutiontime.InstancesinwhichBisaboutjKj=2requiredthemostcomputationandtherequiredcomputationaleffortisnotsymmetricaboutjKj=2.Low-budgetinstancessolvedinafractionofthetimeofhigh-budgetinstances,againbecauseofthenumberofrelevantcheckpointsinthesetsK!.Intheremainderofthissection,wepresentcomputationalresultsbasedondataderivedfromtheroadnetworkinMexico,Canada,andthecontiguous48U.S.states.Thedatainclude113checkpointscorrespondingtoportsofentryforroadcrossingsfromMexicoandCanadaintotheU.S.Thereare140scenariosarisingfromeachcombinationofsevenoriginsinMexico,sevenoriginsinCanada,and10destinationsintheUnitedStates.Evasionprobabilitiesp!karegivenforeach(!,k)wherean!withorigininCanadahasaccesstoallkontheU.S.-Canadaborderandanoriginin 37

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Figure2-1. Computationtime(inseconds)vs.BforMexico-Canada-U.S.instance. MexicosimilarlyhasaccesstoallkontheU.S.-Mexicoborder.Werandomlygenerateprobabilitiesp!kuniformlyintherange[p!k)]TJ /F4 11.955 Tf 11.69 0 Td[(p!k(1)]TJ /F4 11.955 Tf 11.69 0 Td[(p!k),p!k+p!k(1)]TJ /F4 11.955 Tf 11.69 0 Td[(p!k)],andweassumesensorstohaveeffectiveness=0.5.Thisinstancewassolved114times,correspondingtoeachvalueofB=0,1,...,113.Allbut17ofthe114runsnishedwithinthreehours,andtheremainderwereterminatedwithin2.4%ofoptimality(seeFigures 2-1 and 2-2 ).Statisticsonthenumberofnodesgeneratedfromthebranch-and-boundtreeandthenumberofstepinequalitiesgeneratedwithinthetimelimitareshowninFigures 2-3 and 2-4 .(Becausetheprocedureterminatesafterthreehours,wedonotknowtheactualnumberofbranch-and-boundnodesandstepinequalitiesneededtosolvethemostdifcultinstances.)Computationtimeincreasessharplyforthemedium-levelbudgetvalues,withthealgorithmfailingtoterminatewithinthethree-hourtimelimitwhenB=56andB=58,59,...,73.AsshowninFigure 2-5 ,sensorinstallationappearstohavediminishingreturns:Theincrementalimprovementinoptimalobjectivefunctionvalueislargerfortherstsensorthanfor,say,the50-th.Figure 2-6 illustratesthesolutiontoeachofthe 38

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Figure2-2. Optimalitygap(in%)vs.BforMexico-Canada-U.S.instance. Figure2-3. Numberofbranch-and-boundnodesvs.BforMexico-Canada-U.S.instance. 39

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Figure2-4. Numberofstepinequalitiesaddedvs.BforMexico-Canada-U.S.instance. 114problems,withrowscorrespondingtocheckpointsandcolumnscorrespondingtothe114instances.Foreachinstance,shadedpixelsindicatewhichcheckpointsreceivedsensors.AsillustratedinFigure 2-6 ,thesolutionstoinstancesB=1throughB=71largelyexhibitanestedstructure:Eachsuccessivesolutionbuildsupontheonebefore,movingonlyafewsensorstonewcheckpoints.AsBincreasesfrom1to71,theMexicanborderiscoveredrst,followedbytheeasternCanadianborder.Then,asBchangesfrom71to72,thesolutionobtainedshiftsthebulkofsensorsfromtheeasternCanadianbordertothewesternCanadianborder.AsBincreasesbeyond72,thesolutionsagainexhibitthenestedstructure,graduallyrestrengtheningtheeasternCanadianborder. 40

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Figure2-5. Optimalvaluevs.BfortheinstanceMexico-Canada-U.S.instance.TheoptimalvalueisreportedasaratiooftheoptimalobjectivefunctionvalueforbudgetlevelBtothatwhenB=0. Figure2-6. Plotofsensor-placementlocations(shadedpixels)vs.BfortheMexico-Canada-U.S.instance.Pixelcolorindicatesthegeographicalregioninwhicheachsensorislocated,asindexedintheguretotheright. 41

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CHAPTER3CONVEXHULLREPRESENTATIONFORTHEDETERMINISTICBORDERMONITORINGPROBLEM 3.1BackgroundWeconsidertheasymmetricbipartitestochasticnetworkinterdictionproblem(BiPSNIP),whichhasapplicationsincounteringnuclearsmuggling.Inthischapter,webrieysummarizeBiPSNIPinthecontextofagameplayedbyaleaderagainstanumberofopponents.Thegameisplayedoverasetofalternativesfromwhicheachopponentmustselectone.Theleadermayremovealimitednumberofthealternativesinordertoinuenceeachopponent'sselection.Theleaderseekstoselectalternativestoremovesuchthatthetotalcostoftheresultingopponent-alternativepairingsisminimized.Theopponentsandalternativesofthischapterplaytheroleofthesmugglersandcustomscheckpoints,respectively,ofChapter 2 .Weusemoregeneralterminologyinthischapterinordertoemphasizethat(i)BiPSNIPhasthestructureofaverysimplegameand(ii)BiPSNIPmayhavemanyapplicationsoutsidethescopeofnuclearinterdiction.Foramorein-depthintroductiontothisproblem,wereferthereaderto[ 32 ]andChapter 2 ofthisdissertation.Considerasetofalternativesf1,...,ngsimultaneouslyavailabletoasetofopponents.InBiPSNIP,theleadermayremoveeachalternativek2f1,...,ngatacostofbk,andthetotalcostofremovalsmustnotexceedB.WeconsiderthespecialcaseofBiPSNIPinwhichbk=1,8k2f1,...,ngandB2Z.Werefertothisversionasthecardinality-constrainedBiPSNIP,orC-BiPSNIP.Eachopponent!2inC-BiPSNIPvaluesthealternativesdifferentlyandseekstochoosethealternativethathasthegreatestperceivedvalue.Opponent!2perceivesthevalueofanalternativeitobe0ifiisinterdictedandp!iotherwise.For!2denethesequencefk!ig`i=1forindicesi2fi:p!k!i>0gsuchthat p!k!1>p!k!2>>p!k!`,(3) 42

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anddenen0=minf`,Bg.(Note:`andn0maytakedifferentvaluesforeachopponent!2,butwesuppresssubscripts!onthesevaluesfornotationalconvenience.)Thus,opponent!2selectsthealternativek!icorrespondingtothelowestindexi=1,...,n0thatisnotinterdicted;ifallareinterdicted,opponent!selectsalastresortalternative.Letn+1beadummyalternativerepresentingeachopponent'slastresort,andletk!n0+1=n+1,8!2.If`B+1,denep!n+1=p!k!B+1;else,denep!n+1=,where>0isanarbitrarilysmallconstant.Weassumethatalternativen+1cannotbeinterdicted. Remark3.1. Thisnotationmayalsobeusedtomodelthemoregeneralcaseinwhichopponent!2perceivesthevalueofalternativektobe^q!kifkisinterdictedand^p!k>^q!kotherwise(aspresentedinChapter 2 ).Inthiscase,deneq!=maxnk=1f^q!kgandp!k=^p!k)]TJ /F3 11.955 Tf 11.43 1.69 Td[(q!,8!2,k2f1,...,ngandproceedasbefore(computing`,n0,andthek!i-indicesusingthemodiedp-values).2 Remark3.2. AlthoughweconsiderC-BiPSNIP,manyoftheresultscontainedinthischapterarealsovalidforthemoregeneralBiPSNIP.Notably,allofthevalidinequalitiesinthischaptercanalsobeextendedtoBiPSNIP.However,theconvexhullresult(Theorem 3.5 )holdsonlyforC-BiPSNIP.2Fori=1,...,n,letxiequaloneifalternativeiisinterdictedandzerootherwise.Fori=1,...,n0+1,letbinaryvariabley0!iindicatewhetheralternativek!iischosenbyopponent!.Accordingly,foraxedinterdictionvectorx,eachopponent!2seeksasolutionto y!2argmaxn0+1Xi=1p!k!i(1)]TJ /F3 11.955 Tf 12.14 0 Td[(xk!i)y0!i, (3a)s.t.n0+1Xi=1y0!i=1, (3b)y0!i0,8i=1,...,n0+1, (3c)wherexk!n0+1=xn+1=0,8!2. 43

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Theleaderscoreseachalternative-opponentpairingaswell:Fori=1,...,n0+1,letp!k!idenotetheleader'scostincurredduetoopponent!2selectingalternativek!i.Theleaderseeksaninterdictionvectorxthatminimizesthesumofp-valuescorrespondingtoeachopponent'schoice,weightedbyf!>0: MinX!2f!!, (3a)s.t.!=n0Xi=1p!k!iy!i,8!2, (3b)y!isoptimalfor( 3 ),8!2, (3c)nXi=1xiB, (3d)x2f0,1gn. (3e)InthecontextofthenuclearsmugglinginterpretationofBiPSNIPdescribedinChapter 2 ,^p(^q)fromRemark 3.1 representseachsmuggler'sassessmentoftheprobabilityofsuccessfulevasionthroughacustomscheckpointwithout(with)asensor.Thep-valuesof( 3 )representthetrue-evasionprobabilities,knownonlybytheinterdictor,associatedwiththesamecustomscheckpoints.Wenowillustratemodels( 3 )and( 3 )usinganexample. Example.Supposen=6,B=4,=f1,2g,f1=f2=0.5,and(followingRemark 3.1 )^q!kand^p!karegivenforeachk2f1,...,6gand!2f1,2g: Opponent1:^p1=[0.3,0.55,0.4,0.6,0.8,0.2],^q1=[0.05,0.28,0.15,0.25,0.2,0.02]; Opponent2:^p2=[0.65,0.45,0.25,0.72,0.35,0.9],^q2=[0.25,0.3,0.17,0.45,0.07,0.55].ProceedingasinRemark 3.1 ,q,p,`,andn0arecomputedforeachopponentinTable 3.1 .Givenx2f0,1g6,constraint( 3c )requiresthatopponent1chooseeither(a)therstalternativekin(5,4,2,3),movingfromlefttoright,suchthatxk=0,or(b) 44

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Table3-1. Deriveddataforexampleproblem. !=1!=2 q!0.280.55p![0.02,0.27,0.12,0.32,0.52,)]TJ /F3 11.955 Tf 9.3 0 Td[(0.08][0.1,)]TJ /F3 11.955 Tf 9.3 0 Td[(0.1,)]TJ /F3 11.955 Tf 9.3 0 Td[(0.3,0.17,)]TJ /F3 11.955 Tf 9.3 0 Td[(0.2,0.35]`53n043(k!1,...,k!n0+1)(5,4,2,3,7)(6,4,1,7)(p!k!1,...,p!k!n0+1)(0.52,0.32,0.27,0.12,0.02)(0.35,0.17,0.1,) alternative7ifx5=x4=x2=x3=1.Likewise,opponent2mustchoosetherstalternativekin(6,4,1)suchthatxk=0oralternative7ifx6=x4=x1=1.Differentp-vectorsresultindifferentoptimalsolutionstomodel( 3 ).Forinstance,ifp1=p1andp2=p2(thesymmetriccase),thesolutionx=(0,1,0,1,1,1)forcesy14=y23=1(i.e.,opponents1and2selectalternativesk14=3andk23=1respectively)andisoptimalfor( 3 )withvalue0.5(0.12)+0.5(0.1)=0.11.Supposeinsteadp1andp2aregivenexactlyasp1andp2,exceptp14=p24=0.09.Inthiscase,bothopponentshaveseverelyovervaluedalternative4,andtheinterdictorcantakeadvantageusing,e.g.,x=(0,0,0,0,1,1).Constraint( 3c )forceseachopponenttoselectalternative4,resultinginan(optimal)objectivevalueof0.09inmodel( 3 ).2In[ 32 ](andChapter 2 ofthisdissertation),( 3c )isenforcedbyimposingtheKarush-Kuhn-Tucker(KKT)optimalityconditionsof( 3 ),whicharenecessaryandsufcientsince( 3 )isalinearprogram.Thecomplementaryslacknessconditions,alongwiththefactthatallp-valuesarepositive,implythatxk!iy!i=0forall!2,i=1,...,n0+1.Theresultingproblemisthusamixed-integerlinearprogrammingproblem,albeitonewhosecontinuousrelaxationtendstobefairlyweak.Thegoalinthischapteristodevelopamodelthatexplicitlycapturestherelationshipsbetweenx-andy-variables,makinguseofthek!i-orderingsratherthanthep!i-values,inordertoobtainatighterformulation.Ultimately,weareinterestedinstudyingtheset 45

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P=(x,y)2Xf0,1g(n0+1)jjy!i=xk!1xk!i)]TJ /F19 5.978 Tf 5.76 0 Td[(1(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xk!i),8!2,i=1,...,n0+1,(3)wherexk!n0+1=xn+1=0isxedasaconstantandX=fx2f0,1gn:Pni=1xiBg.Dueto( 3 ),if(x,y)2P,thenyrepresentsasolutionto( 3 )underinterdictionx.Model( 3 )iseasytosolvewhen=f1gisasingletonset,becausetheinterdictorneedonlycomparethen0+1solutionsresultingfrominterdictingallofcheckpointsk11,...,k1h)]TJ /F12 7.97 Tf 6.59 0 Td[(1andnotinterdictingcheckpointk1h,foreachh=1,...,n0+1.ThissuggeststhatwecanlikelyndatightformulationforPf1g.Asimilarformulationexistsforthecaseinwhichp1k11>>p1k1n0+1(i.e.,theinterdictorandopponentagreeuponarankingofalternatives)andthePixiBconstraintisremoved.Forthiscase,MillerandWolsey[ 52 ]provethatthestepinequalitiesof[ 32 ]convexifytheset 8><>:(1,x)9y1suchthat(1,x,y1)satises( 3b ),( 3c ),and( 3e )9>=>;,(3)correspondingto=f1g.However,inclusionofPixiBsignicantlychangesthefeasibleregionandinfactleadstoanexponentialclassoffacetstoconv[Pf1g].WecontributeinSection 3.3 afullcharacterizationofconv[Pf1g],andshowthatourrepresentationisminimal.Themotivationforobtainingthisconvexhullisintighteningthelinearprogrammingrelaxationsfor(stochastic,multi-scenario)BiPSNIPproblems.Partialconvexhullrepresentations,suchasthosegeneratedviatheintersectionofsubproblemconvexhulls,havebeenusedsuccessfullyinpreviousresearch(see,e.g.,thegeneraldiscussionsin[ 53 54 ])todevelopstrongcuttingplanesfordifcultproblems.Wethustesttheeffectivenessofvalidinequalitiesthatarefacet-deningtotheseone-scenarioconvexhullsinsolvingC-BiPSNIPinstancesinthischapter.Theremainderofthischapterisorganizedasfollows.InSection 3.2 ,wesimplifynotationfortheone-opponentproblemandcharacterizeitssolutionsusinganintuitive 46

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linearformulationwithbinaryvariables.WedevelopinequalitiesandacorrespondingseparationproceduretotightenthisformulationinSection 3.3 ,resultinginaminimally-representedidealformulation.InSection 3.4 ,wediscusstheextensionofourresultstothemultiple-opponentproblem.Wereportcomputationalresultsfromanumberofmultiple-opponentinstancesinSection 3.5 solvedwithandwithoutourinequalities. 3.2BasicFormulationForthemajorityofthischapter,weconsiderasingleopponent(andthusremovethe!-index)who,withoutlossofgenerality,prefersthealternativesinincreasingorder,i.e.,ki=i,8i=1,...,n0,and p1>p2>>pn0>pn0+1.(3)Accordingly,theopponentwillselectthelowest-indexedalternativeamongf1,...,n0gthatisnotinterdicted,orthedummyalternativen0+1ifalloff1,...,n0gareinterdicted.DeneP,theone-opponentfeasibilitysetanalogousto( 3 ),as P=8><>:(x,y)2Xf0,1gn0+1yi=x1xi)]TJ /F12 7.97 Tf 6.59 0 Td[(1(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi),8i=1,...,n0yn0+1=x1xn09>=>;.(3)Tocharacterizethepolyhedronconv(P),werstdevelopalinearrepresentationofP.LetPdenotethesetofall(x,y)2RnRn0+1satisfyingthefollowingconstraints: nXj=1xjB, (3a)n0+1Xi=1yi=1, (3b)y1=1)]TJ /F4 11.955 Tf 11.95 0 Td[(x1, (3c)yi1)]TJ /F4 11.955 Tf 11.96 0 Td[(xi,8i=2,...,n0, (3d)yixj,8i=2,...,n0+1,8j=1,...,i)]TJ /F3 11.955 Tf 11.95 0 Td[(1, (3e)yi0,8i=1,...,n0+1, (3f)0xj1,8j=n0+1,...,n. (3g) 47

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Constraint( 3a )isvalidforPsincex2X,8(x,y)2P.Followingfrom( 3 ), n0Xi=1yi=1)]TJ /F4 11.955 Tf 11.96 0 Td[(x1x2xn0=1)]TJ /F4 11.955 Tf 11.96 0 Td[(yn0+1,(3)andhence,( 3b )isvalidaswell.Constraint( 3c )issimplythedenition( 3 )ofy1.Theremainingy-variablesarelinearizedinthestandardwayusing( 3d )and( 3e ).(Thetypicallower-boundinglinearizationconstraints,e.g.,yi(1)]TJ /F4 11.955 Tf 12.28 0 Td[(xi))]TJ /F14 11.955 Tf 12.29 8.97 Td[(Pi)]TJ /F12 7.97 Tf 6.58 0 Td[(1j=1(1)]TJ /F4 11.955 Tf 12.29 0 Td[(xj)areimpliedby( 3b )( 3e )andarethusomitted.)Constraints( 3f )( 3g )arevalidbecausexandyarebinaryvectors.Constraints0xi1foreachi=1,...,n0areimpliedby( 3b )( 3e )and( 3f ),andarethereforeomitted.Wenowarguethat( 3 )isavalidformulationofP,i.e.,P\(ZnZn0+1)=P.Let(x,y)beanybinarysolutioninP.Ifxi=1,8i=1,...,n0,then( 3b )( 3d )and( 3f )implythatyn0+1=1,i.e.,(x,y)2P.Otherwise,letk?denotethesmallestindexi2f1,...,n0gsuchthatxi=0.From( 3c ),( 3d ),and( 3f ),yi=0forallik?)]TJ /F3 11.955 Tf 12.12 0 Td[(1.From( 3e ),yi=0forallik?+1andthus( 3b )impliesyk?=1.Therefore,(x,y)2P,provingthat( 3 )isavalidformulationofP.Althoughformulation( 3 )iscorrect,itisnotidealinthesensethatsomeofitsverticescontainfractionalvariablevalues.Forinstance,whenn=4andB=n0=2,thesolutionx=(1=2,1=2,0,1)togetherwithy=(1=2,0,1=2)isanextremepointofP.Inthefollowingsection,wedevelopinequalitiesthatarevalidforP,provethattheydenefacetstoconv(P),andshowthataddingthemto( 3 )resultsinanidealformulation.Towardsthisend,werstestablishthedimensionalityofP. Lemma3.1. ThesetPhasdimensionn+n0)]TJ /F3 11.955 Tf 11.96 0 Td[(1. Proof. BecausePRnRn0+1andeach(x,y)2Pmustsatisfythelinearlyindependentequalities( 3b )and( 3c ),itfollowsthat dim(P)(n+n0+1))]TJ /F3 11.955 Tf 11.96 0 Td[(2=n+n0)]TJ /F3 11.955 Tf 11.96 0 Td[(1.(3) 48

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Weshowthatdim(P)n+n0)]TJ /F3 11.955 Tf 12.34 0 Td[(1byprovidingn+n0afnelyindependentpointsc1,...,cn+n02Poftheform(x1,...,xn,y1,...,yn0+1).Letc1equalzeroinallcomponentsexceptforthey1-component,whichequalsone.Fori=2,...,n,letcibegivenexactlyasc1exceptforthexi-component,whichequalsone.Fori=1,...,n0,letcn+iequaloneincomponentsx1,...,xi,andyi+1,andzeroelsewhere.Notingthatn0B,eachofthepointsc1,...,cn+n0areelementsofP.LetCbethe(n+n0)]TJ /F3 11.955 Tf 12.57 0 Td[(1)(n+n0+1)matrixsuchthatrowi=1,...,n+n0)]TJ /F3 11.955 Tf 12.99 0 Td[(1isformedbyci+1)]TJ /F4 11.955 Tf 12.99 0 Td[(c1.DeneeCasthe(n+n0)]TJ /F3 11.955 Tf 12.37 0 Td[(1)(n+n0)]TJ /F3 11.955 Tf 12.37 0 Td[(1)matrixobtainedbydeletingthex1-andy1-columnsfromC.BecauseeCislower-triangularwithnonzerodiagonals,itfollowsthattherowsofeCandCarelinearlyindependent.Therefore,thepointsc1,...cn+n0areafnelyindependent,anddim(P)n+n0)]TJ /F3 11.955 Tf 11.96 0 Td[(1. 3.3ReformulationInthissection,webeginbyndingallclassesoffacet-deninginequalitiestothedeterministic(one-scenario)C-BiPSNIPformulationinSection 3.3.1 .WethenshowhowthisconvexhullcanbecapturedviatheSpecialStructuresReformulation-LinearizationTechnique(SSRLT)[ 55 ]inSection 3.3.2 .Finally,inSection 3.3.3 ,weshowthatthefacet-deninginequalitiesderivedinSection 3.3.1 actuallygeneralizethemoretraditionalstepinequalitiesthathavebeenproposedforBiPSNIP. 3.3.1ConvexHullFormulationBeginningwithmodel( 3 )fromtheprevioussection,wenowdevelopinequalitiesthatleadtoanidealformulation.Ourrstclassofinequalitiesexploittherelationshipbetweeny-variablestostrengthenthelinearizationconstraints( 3e ).Wenowdescribetheseinequalitiesandshowthattheyarebothvalidandfacet-deningforconv(P). Theorem3.1. Forj=2,...,n0,theinequality n0+1Xi=j+1yixj,(3)isvalidandfacet-deningforP. 49

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Proof. Werstshowthat( 3 )isvalid.Notethat( 3 )isimpliedby( 3b )whenxj=1,andisaconsequenceof( 3e )fori=j+1,...,n0+1whenxj=0.Toshowthat( 3 )isfacet-dening,wedemonstratethatthedimensionofthefaceofPinducedby( 3 )isn+n0)]TJ /F3 11.955 Tf 12.12 0 Td[(2.Wethusconstructafnelyindependentpointsc1,...,cn+n0)]TJ /F12 7.97 Tf 6.58 0 Td[(12Poftheform(x1,...,xn,y1,...,yn0+1)thatarebindingon( 3 ).Letc1equaloneinthey1-componentandzeroelsewhere.Fori=2,...,j)]TJ /F3 11.955 Tf 12.32 0 Td[(1,letcibegivenexactlyasc1withthexi-componentchangedfromzerotoone.Fori=j,...,n)]TJ /F3 11.955 Tf 11.53 0 Td[(1,deneciexactlyasc1,changingthexi+1-componentfromzerotoone.Fori=1,...,n0,letcn+i)]TJ /F12 7.97 Tf 6.59 0 Td[(1equaloneincomponentsx1,...,xi,andyi+1,andzeroelsewhere.LetCdenotethe(n+n0)]TJ /F3 11.955 Tf 12.91 0 Td[(2)(n+n0+1)matrixconstructedsuchthatrowiofCisgivenbyci+1)]TJ /F4 11.955 Tf 12.32 0 Td[(c1.LeteCbethematrixobtainedbyremovingthex1-column,thexj-column,andthey1-columnfromC,andobservethateCislower-triangularwithnonzerodiagonals.Thus,eCisnonsingularandc1,...,cn+n0)]TJ /F12 7.97 Tf 6.58 0 Td[(1areafnelyindependent,asrequired. Remark3.3. Inequality( 3 )isvalidforj=1aswell;however,itisimplied(inthecontinuoussense)by( 3b )and( 3c ),andisthereforenotfacet-dening.2Wenowcharacterizesomeinequalitiesthatcapturesubtlerelationshipsamongthex-andy-variablescausedbyinclusionofthecardinalityconstraint( 3a ).Let=f1,...,jjgand=f 1,..., jjgbesubsetsoff2,...,n0gandfn0+1,...,ng,respectively,suchthat 1<
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underwhich( 3 )isvalidandfacet-dening,andthenshowhowtoseparate( 3 )inpolynomialtime. Theorem3.2. Letf2,...,n0gandfn0+1,...,ngsatisfy( 3 ).Then( 3 )isvalidforP. Proof. Let(x,y)2P,andletk?betheindexforwhichyk?=1.Notethatk?exists,andyi=0foralli6=k?,by( 3b ).Ifjj+jj
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mustsatisfyx1=0.Becausec1hasjj+jjB)]TJ /F3 11.955 Tf 12.31 0 Td[(1x-variablesequalto1,c1isanelementofP.Furthermore,becausej1j=0andjj+jjB)]TJ /F3 11.955 Tf 12.02 0 Td[(1,theright-handsideof( 3 )isjj+jj,andtheleft-handsideof( 3 )matchesthisvaluebecausealljj+jjvariablesontheleft-handsideof( 3 )equalone.Fori=2,...,nsuchthati=2[,letcibedenedexactlyasc1exceptforaoneincomponentxi.Bythesameargumentasbefore,eachofthesepointsisinPandeachisonthefacedenedby( 3 ).Fori=2,...,nsuchthati2[,weagaindeneciexactlyasc1,exceptthatcomponentxiequalszeroinsteadofone,componentsx1,...,xBjjjj+1equalone,componenty1iszero,andcomponentyBjjjj+2equalsone.(NotethatB)]TJ -426.52 -23.9 Td[(jj)-277(jj+1n0bycondition(a).)Toseethateachsuchpointisfeasible,observethatf1,...,B)-326(jj)-327(jj+2g\([)=;byassumption(b).BecausetherstB)-299(jj)-299(jj+1x-variablesequaloneinthispoint,whilexBjjjj+2equalszero,wehavethat( 3b )( 3g )impliesyBjjjj+2=1asrequired.Also,allbutoneofthevariablescorrespondingtoindicesin[equalone,andsoatotalof(B)-245(jj)-246(jj+1)+(jj+jj)]TJ /F3 11.955 Tf 18.49 0 Td[(1)=Bx-variablesequalone.Hence,cirepresentsapointinP.Toseethat( 3 )isbindingonthispoint,notethatB)]TJ /F3 11.955 Tf 12.3 0 Td[((B)-250(jj)-251(jj+2)+1+jBjjjj+2j=jj+jj)]TJ /F3 11.955 Tf 18.82 0 Td[(1.Theright-handsideof( 3 )thusreducestojj+jj)]TJ /F3 11.955 Tf 18.45 0 Td[(1,andbecauseallbutonex-variableontheleft-handsideof( 3 )equalsoneinpointci,theinequalityisbindingatthispoint.Fori=1,...,B)-289(jj)-290(jj,denecn+iexactlyasc1,exceptthatcomponentsx1,...,xiandyi+1allequalone,whilecomponenty1equalszero.Thefeasibilityofthesepointsisestablishedusingthesameargumentasabove,withtheobservationthatthetotalnumberofone-valuedx-variablesisequaltoi+jj+jj(B)]TJ -417.22 -23.9 Td[(jj)-311(jj)+jj+jj=B.Notingthatyi+1=1forthesepoints,theright-handsideof( 3 )isminfjj+jj,B)]TJ /F3 11.955 Tf 12.66 0 Td[((i+1)+1+ji+1jg.Notingthati+1=;andB)]TJ /F4 11.955 Tf 12.11 0 Td[(iB)]TJ /F3 11.955 Tf 12.11 0 Td[((B)-234(jj)-235(jj)=jj+jj,theright-handsideof( 3 )equalsjj+jj. 52

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Therefore,( 3 )isbindingoncn+ibecauseallxicomponentsequalonecorrespondingtoi2[.Foreachi=B)-301(jj)-301(jj+1,...,n0)]TJ /F3 11.955 Tf 12.9 0 Td[(1,denecn+isuchthatcomponentsx1,...,xi+1andyi+2equalone,alongwithafewothersspeciedbelow.Thusfar,thepointsclearlybelongtoP,notingthati+1n0B.Wenowshowhowtomodifythesepointssothattheybecomebindingon( 3 )whileremaininginP.Theright-handsideof( 3 )equalsminfjj+jj,B)]TJ /F3 11.955 Tf 12.56 0 Td[((i+2)+1+ji+2jg,butnotingthatthesecondtermisnomorethanB)]TJ /F3 11.955 Tf 12.78 0 Td[((B)-291(jj)-291(jj+3)+1=jj+jj)]TJ /F3 11.955 Tf 19.58 0 Td[(2,theright-handsideevaluatestoB)]TJ /F4 11.955 Tf 12.68 0 Td[(i)]TJ /F3 11.955 Tf 12.68 0 Td[(1+ji+2j.Thecontributiontotheleft-handsideof( 3 )fromthex-variablesthusfarequalsji+2j,andwethereforeseektosetB)]TJ /F4 11.955 Tf 12.17 0 Td[(i)]TJ /F3 11.955 Tf 12.18 0 Td[(1morex-variablesequaltoone.(Indoingso,thetotalnumberofx-componentssettoonebecomes(i+1)+(B)]TJ /F4 11.955 Tf 12.51 0 Td[(i)]TJ /F3 11.955 Tf 12.51 0 Td[(1)=B,and( 3a )remainssatised.)Wethereforesetthex-componentscorrespondingtotheB)]TJ /F4 11.955 Tf 12.42 0 Td[(i)]TJ /F3 11.955 Tf 12.42 0 Td[(1highest-indexedelementsof[equalto1.Notethatifthe(B)]TJ /F4 11.955 Tf 12.18 0 Td[(i)]TJ /F3 11.955 Tf 12.18 0 Td[(1)sthighestelementof[isatleastaslargeasi+3,thenthexi+2componentremainsequaltozero(andsosettingtheyi+2componentequaltoonesatises( 3b )( 3g )),andtheadditionalcomponentssettoonearealldistinctfromindicesf1,...,i+1g,thusensuringthat( 3 )becomesbindingonthispoint.Toestablishtheconditionthatthe(B)]TJ /F4 11.955 Tf 11.64 0 Td[(i)]TJ /F3 11.955 Tf 11.64 0 Td[(1)sthighestelementof[isatleastaslargeasi+3,choose1sothati=B)-261(jj)-261(jj+.Byassumption(b),the(jj+jj)thhighestelementof[isnotlessthanB)-267(jj)-268(jj+3if6=;.(If=;,theclaimholdsbecauseallelementsof[aregreaterthann0.)NotingthatB)]TJ /F4 11.955 Tf 12.37 0 Td[(i)]TJ /F3 11.955 Tf 12.36 0 Td[(1=jj+jj)]TJ /F3 11.955 Tf 18.75 0 Td[((1+),the(B)]TJ /F4 11.955 Tf 12.37 0 Td[(i)]TJ /F3 11.955 Tf 12.36 0 Td[(1)sthighestelementof[isatleastB)-222(jj)-222(jj+4+=i+4.LetCbethe(n+n0)]TJ /F3 11.955 Tf 11.6 0 Td[(2)(n+n0+1)matrixsuchthateachrowi=1,...,n+n0)]TJ /F3 11.955 Tf 11.6 0 Td[(2isdenedbyci+1)]TJ /F4 11.955 Tf 12.87 0 Td[(c1.LeteCbethematrixobtainedbyeliminatingcolumnsx1,y1,andyBjjjj+2fromC,andobservethateCisalower-triangularmatrixwithnonzero 53

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diagonals.Thus,therowsofeCandCarelinearlyindependentandc1,...,cn+n0)]TJ /F12 7.97 Tf 6.59 0 Td[(1areafnelyindependent,provingthat( 3 )isfacet-dening.Wenowprovethat( 3 )denesafacettoconv(P)onlyiftheassumptionsofthetheoremaresatised.Firstsupposethat6=;yet1B)-280(jj)-281(jj+2,sothatassumption(b)fails.SinceB)]TJ /F4 11.955 Tf 12.38 0 Td[(i+1jj+jjforalli1)]TJ /F3 11.955 Tf 12.39 0 Td[(1andjij=0foralli1,inequality( 3 )canberewrittenasjjXi=1xi+jjXi=1x i1)]TJ /F12 7.97 Tf 6.59 0 Td[(1Xi=1(jj+jj)yi+minfjj+jj,B)]TJ /F5 11.955 Tf 11.95 0 Td[(1+1gy1+n0+1Xi=1+1minfjj+jj,B)]TJ /F4 11.955 Tf 11.95 0 Td[(i+1+jijgyi. (3)If1B)-189(jj)-190(jj+1thecoefcientony1becomesjj+jj.However,changingthecoefcientony1tojj+jj)]TJ /F3 11.955 Tf 18.29 0 Td[(1resultsinastrengthenedvalidinequalitysincey1=1forcesx1=0,meaningtheleft-handsidecanbeatmostjj+jj)]TJ /F3 11.955 Tf 18.31 0 Td[(1.Thus,( 3 )isnotfacet-deninginthiscase.Supposeinsteadthat1=B)-222(jj)-223(jj+2.Inthiscase,( 3 )furthersimpliestojjXi=1xi+jjXi=1x i1)]TJ /F12 7.97 Tf 6.59 0 Td[(1Xi=1(jj+jj)yi+(jj+jj)]TJ /F3 11.955 Tf 17.94 0 Td[(1)y1+n0+1Xi=1+1(B)]TJ /F4 11.955 Tf 11.96 0 Td[(i+1+jij)yi. (3)Weshowthat( 3 )isimpliedbyacombinationofothervalidinequalities.Considergenerating( 3 )using=f2,...,jjgand=,anddeningi=fj2:j
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Adding( 3 )to x11)]TJ /F12 7.97 Tf 6.59 0 Td[(1Xi=1yi+n0+1Xi=1+1yi,(3)whichisvalidby( 3b )and( 3d ),produces( 3 );therefore,( 3 )isnotfacetdening.Nowsupposethatcondition(b)holdsbutcondition(a)fails.Ifjj+jj
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summarizetheupdatedmodel.Denetheupdatedpolytope P0=8>>>><>>>>:(x,y)2RnRn0+1( 3a )( 3d ),( 3f ),( 3g )( 3 ),8j=2,...,n0( 3 ),8(,)2)]TJ /F14 11.955 Tf 64.47 64.56 Td[(9>>>>=>>>>;,(3)where)]TJ /F1 11.955 Tf 10.09 0 Td[(isthesetofall(,)satisfyingtheassumptionsofTheorem 3.3 .Wenowprovethatthisisavalidandidealformulation. Theorem3.4. System( 3 )isavalidformulationofP,i.e.,P0\(ZnZn0+1)=P. Proof. Let(x,y)2P.Constraints( 3a )( 3d ),( 3f ),and( 3g )areobviouslysatised,andconstraints( 3 )and( 3 )mustalsobesatisedduetoTheorems 3.1 and 3.2 .Thus,(x,y)2P0andPP0\(ZnZn0+1).Let(x,y)2P0\(ZnZn0+1).Inordertoshowthat(x,y)2P,weneedonlyshowthatconstraints( 3e )aresatised,astheremainderoftheconstraintsof( 3 )arepartofthedenitionofP0.Constraints( 3e )areimpliedby( 3 )becausethey-variablesarenonnegative. Theorem3.5. System( 3 )isanidealformulationofP,i.e.,P0=conv(P). Proof. Letcbeanyn-vectoranddbeany(n0+1)-vectorsuchthatatleastonecomponentofcordisnonzero,anddene f(x,y)=nXj=1cjxj+n0+1Xi=1diyi.(3)WeshowthatFargmax(x,y)ff(x,y):(x,y)2PgiscontainedinoneofthefacesofP0,thusprovingtheresult.Withoutlossofgenerality,wemayassumethatc1=d1=0,using( 3b )and( 3c )toeliminatex1andy1fromtheobjectiveifnecessary.Let(x,y)beanyelementofFandsupposethatcj<0forsomej2f2,...,ng.Ifjn0,itfollowsthatxj=1onlyifyi=1forsomei>j(orelseabettersolutioncouldbeobtainedbychangingxjtozero);hence,( 3 )correspondingtojisbindingat(x,y).Ifjn0+1,anysolutionwithxj=1 56

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isdominatedbythefeasiblesolutionobtainedbychangingxjtozero;thus,(x,y)mustbeinthefaceofP0correspondingtoxj0.Therefore,ifcj<0forsomej2f2,...,ng,itfollowsthatFiscontainedinafaceofP0.Next,supposethatF\f(x,y)2P:yi=1g=;forsomei2f1,...,n0+1g.Inthiscase,eachpointinFisbindingononeofthefaces( 3f ).Thus,fortheremainderofthisproof,weassumethatcj0,8j=2,...,n,andthatcanddaresuchthatF\f(x,y)2P:yi=1g6=;foreachi=1,...,n0+1.Let=f1,...,Ngdenotethesetofindicesj2f2,...,ngsuchthatcj>0,with1<2<0thenF\f(x,y)2P:y1=1g=;,andifdi<0thenF\f(x,y)2P:yi=1g=;.Therefore,wenowrestrictouranalysistothecaseinwhich6=;.Denei=\f2,...,i)]TJ /F3 11.955 Tf 12.44 0 Td[(1gandi=\fi+1,...,ng,foralli=1,...,n0,anddenen0+1=\f2,...,n0gandn0+1=n0.Accordingly,if(x,y)2Fwithyk?=1,thenx=1,82k?,andasmanyx,2k?,asallowedby( 3a )shouldbesettoone,i.e., X2k?x=minfjk?j,B)]TJ /F4 11.955 Tf 11.96 0 Td[(k?+1g.(3)Let(x,y)2Fandletk?2f1,...,n0+1gdenotetheindexsuchthatyk?=1.If1NB)]TJ /F4 11.955 Tf 12.76 0 Td[(n0,thenweprovethisclaimintwocases:(i)12n0+1,inwhichweshowthat( 3d )isbindingonallpointsinF,and(ii)n0+1=;,inwhichweshowthat( 3g )isbindingonallpointsinF.Incase(i),ifk?1then( 3 )and( 3d )implythatx1+y1=1.Ifinsteadk?1)]TJ /F3 11.955 Tf 12.69 0 Td[(1,thenB)]TJ /F4 11.955 Tf 12.69 0 Td[(k?+1B)]TJ /F4 11.955 Tf 12.7 0 Td[(n0Njk?j.From( 3 ),x0=1forall02k?;thus,x1=1andx1+y1=1,provingthat( 3d )isbindingonF.Incase(ii),12n0+1.Notingthatk?n0+1,observethatB)]TJ /F4 11.955 Tf 9.99 0 Td[(k?+1B)]TJ /F4 11.955 Tf 10 0 Td[(n0Njk?j,anditfollowsfrom( 3 )thatx0=1forall02k?.Thisimpliesthatx11isbindingonallofF.Thus,wemaynowassumethatNB)]TJ /F4 11.955 Tf 10.84 0 Td[(n0+1.NowsupposethatNB.Weshowthisimpliesthat( 3a )isbindingoneveryelementofF.Let(x,y)2F,andletk?2f1,...,n0+1gbetheindexsuchthatyk?=1. 57

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Thisimpliesthatyi=0foralli6=k?,andx1==xk?)]TJ /F12 7.97 Tf 6.58 0 Td[(1=1.Withonly(k?)]TJ /F3 11.955 Tf 12.66 0 Td[(1)x-variablesxedtoonethusfar,thecardinalityconstraint( 3a )permitsasmanyasB)]TJ /F4 11.955 Tf 12.59 0 Td[(k?+1morex-variablestotakevalueone.Notingthatjijjj)-276(jf2,...,igj=N)]TJ /F4 11.955 Tf 12.78 0 Td[(i+1B)]TJ /F4 11.955 Tf 12.78 0 Td[(i+1fori=1,...,n0,andn0+1=n0,itfollowsfrom( 3 )thatP2k?x=B)]TJ /F4 11.955 Tf 12.29 0 Td[(k?+1;thusPni=1xi=Band( 3a )mustbebinding.WehenceforthassumethatNB)]TJ /F3 11.955 Tf 11.95 0 Td[(1.Next,supposethat1minfn0+1,B)]TJ /F4 11.955 Tf 12.79 0 Td[(N+2g.Let(x,y)2F,andletk?2f1,...,n0+1gdenotetheindexsuchthatyk?=1.Ifk?1,then( 3d )(correspondingto1n0)and( 3 )implythatx1+y1=1.Ifinsteadk?1)]TJ /F3 11.955 Tf 11.02 0 Td[(1,observethaty1=0andB)]TJ /F4 11.955 Tf 11.97 0 Td[(k?+1B)]TJ /F5 11.955 Tf 11.97 0 Td[(1+2Njk?j,andfrom( 3 ),x1=1and( 3d )isbindingonallofF.Thusfar,wehaveeliminatedallcasesexceptthoseforwhich(a)B)]TJ /F4 11.955 Tf 12.77 0 Td[(n0+1NB)]TJ /F3 11.955 Tf 12.51 0 Td[(1and(b)either1n0+1or1B)]TJ /F4 11.955 Tf 12.51 0 Td[(N+3.Using=n0+1and=n0+1,conditions(a)and(b)aboveidenticallymatchtheassumptionsofTheorem 3.3 ,thereforeindicatingthat(n0+1,n0+1)2)]TJ /F1 11.955 Tf 10.09 0 Td[(andthecorrespondingconstraint( 3 )denesafacetofP0.Thisinequalityisgivenby NXi=1xin0+1Xi=1minfN,B)]TJ /F4 11.955 Tf 11.95 0 Td[(i+1+jijgyi.(3)WenowshowthatFiscontainedinthefaceofP0associatedwith( 3 ).Let(x,y)2Fandletk?2f1,...,n0gbetheindexsuchthatyk?=1.Weprovethisclaimintwocases:(i)k?B)]TJ /F4 11.955 Tf 12.03 0 Td[(N+1and(ii)k?B)]TJ /F4 11.955 Tf 12.02 0 Td[(N+2.Incase(i),observethatk?B)]TJ /F4 11.955 Tf 12.03 0 Td[(N+1n0andeither1n0+1k?+1or1B)]TJ /F4 11.955 Tf 12.13 0 Td[(N+3k?+2.Ineithercase,wendthatk?=;,and( 3 )reducesto X2k?xN.(3)Moreover,B)]TJ /F4 11.955 Tf 11.97 0 Td[(k?+1Njk?jcombinedwith( 3 )impliesthattheleft-handsideof( 3 )isequaltoN;thus,( 3 )isbindingonF.Incase(ii),B)]TJ /F4 11.955 Tf 12.08 0 Td[(k?+2N,andafter 58

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cancelingjk?jfromeachside,( 3 )becomes X2k?xB)]TJ /F4 11.955 Tf 11.95 0 Td[(k?+1.(3)Weprove( 3 )issatisedasanequalityinsubcases:(ii.a)1n0+1and(ii.b)1B)]TJ /F4 11.955 Tf 12.64 0 Td[(N+3.Incase(ii.a),jk?j=NB)]TJ /F4 11.955 Tf 12.63 0 Td[(k?+2;thus,(x,y)2Fcombinedwith( 3 )impliesthattheleft-handsideof( 3 )equalsB)]TJ /F4 11.955 Tf 12.59 0 Td[(k?+1.Incase(ii.b),ifk?=n0+1,wendthatjk?jj\fB)]TJ /F4 11.955 Tf 12.51 0 Td[(N+3,...,ngj)-269(jfB)]TJ /F4 11.955 Tf 12.51 0 Td[(N+3,...,n0gj=N)]TJ /F3 11.955 Tf 12.28 0 Td[([n0)]TJ /F3 11.955 Tf 12.28 0 Td[((B)]TJ /F4 11.955 Tf 12.29 0 Td[(N+3)+1]=B)]TJ /F4 11.955 Tf 12.28 0 Td[(n0+2=B)]TJ /F4 11.955 Tf 12.29 0 Td[(k?+3.Ifinsteadk?n0,wendthatjk?jj\fB)]TJ /F4 11.955 Tf 10 0 Td[(N+3,...,ngj)-59(jfB)]TJ /F4 11.955 Tf 10 0 Td[(N+3,...,k?gj=N)]TJ /F3 11.955 Tf 10 0 Td[([k?)]TJ /F3 11.955 Tf 10.01 0 Td[((B)]TJ /F4 11.955 Tf 10 0 Td[(N+3)+1]=B)]TJ /F4 11.955 Tf 10 0 Td[(k?+2.Ineithercase,jk?jB)]TJ /F4 11.955 Tf 12.44 0 Td[(k?+2,provingthatP2k?x=B)]TJ /F4 11.955 Tf 12.44 0 Td[(k?+1and( 3 )isbindingonF. Remark3.4. Eachinequalityin( 3 )isfacet-deningforconv(P),andthus,Theorem 3.5 implies( 3 )isaminimalformulationofconv(P).Proofsthattheinequalitiesin( 3 )carriedoverfrom( 3 )denefacetsarestraightforwardandthereforeomitted.2Becausethereareanexponentialnumberof-and-setsthatinducefacetsoftheform( 3 ),enumeratingtheseinequalitiesisimpractical.Inthefollowingparagraphs,weshowhowtogenerate,inpolynomialtime,amost-violatedinequality( 3 )given(x,y)2PnP0.Letk2fB)]TJ /F4 11.955 Tf 12.26 0 Td[(n0+1,...,B)]TJ /F3 11.955 Tf 12.27 0 Td[(1ganddene)]TJ /F7 7.97 Tf 6.77 -1.8 Td[(k=f(,)2)-342(:jj+jj=kg.Wedescribeaprocessforidentifying(?,?)thatmaximizestheviolationof( 3 )at(x,y)overallsets(,)2)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(k;thus,amost-violated( 3 )canbefoundbyrepeatingthisprocessforeachk=B)]TJ /F4 11.955 Tf 11.95 0 Td[(n0+1,...,B)]TJ /F3 11.955 Tf 11.95 0 Td[(1.Werstdenethequantity =n0+1Xj=1minfk,B)]TJ /F4 11.955 Tf 11.95 0 Td[(j+1gyj,(3) 59

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whichrepresentsalowerboundontheright-handsideof( 3 )evaluatedat(x,y).ThisisalowerboundbecauseB)]TJ /F4 11.955 Tf 12.09 0 Td[(j+1B)]TJ /F4 11.955 Tf 12.09 0 Td[(j+1+jjjforallj=1,...,n0+1.Notethatthelowerboundistightwhen=;.Wenowdescribehowinclusionofdifferentindicesin[causestheright-handsideof( 3 )toincrease.Forj=2,...,n,wedenejastheincreasetotheright-handsideof( 3 )duetoincludingjin[.Sinceinclusionofjin[ispreventedbycondition(b)ofTheorem 3.3 foreachj=2,...,minfn0,B)]TJ /F4 11.955 Tf 11.95 0 Td[(k+2g,denej=1fortheseindices.Forj=B)]TJ /F4 11.955 Tf 9.99 0 Td[(k+3,...,n0andi=j+1,...,n0+1,itfollowsthatifB)]TJ /F4 11.955 Tf 9.99 0 Td[(k+3,...,i)]TJ /F3 11.955 Tf 9.98 0 Td[(1g;thus,B)]TJ /F4 11.955 Tf 12.03 0 Td[(i+1+jijk)]TJ /F3 11.955 Tf 12.02 0 Td[(2forallsuchi,i.e.,theminimuminthei-thtermof( 3 )isgivenbyB)]TJ /F4 11.955 Tf 11.97 0 Td[(i+1+jij.Notingthatforthesevaluesofi,theminimumtermin( 3 )isgivenbyB)]TJ /F4 11.955 Tf 11.32 0 Td[(i+1,includingjin[increasesthecoefcientonyiin( 3 )byexactlyoneforeachi=j+1,...,n0+1;accordingly,denej=Pn0+1i=j+1yi.Forj=n0+1,...,nincludingjin[changesonlytheleft-handsideof( 3 ),sodenej=0.Assuch,inequality( 3 )evaluatedat(x,y)reducesto Xj2[(xj)]TJ /F5 11.955 Tf 11.95 0 Td[(j).(3)From( 3 ),itisapparentthattheeffectontheviolationofincludingeachalternativein[isindependentofalltheotheralternatives;hence,( 3 )canbemaximizedbychoosingthekalternativesinf2,...,ngwiththegreatest(xj)]TJ /F5 11.955 Tf 12.4 0 Td[(j)-values.Thiscanbeaccomplishedusingapolynomial-timesortingalgorithm. 3.3.2Obtainingconv(P)throughSSRLTIntheprevioussections,wederivedaminimalrepresentationofP0=conv(P).Alternatively,conv(P)couldhavebeenobtainedfromPusingtheReformulation-LinearizationTechnique(RLT)ofSheraliandAdams[ 56 57 ]orsimilarmethods(e.g.,Balasetal.[ 58 ]andLovaszandSchrijver[ 59 ])thatwouldliftPintohigherdimensioninordertoconstructatighterrepresentationofconv(P)andthenprojectouttheaddedvariables. 60

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WenowprovearesultthatallowstheuseofaspecializedRLTtoachievethesameresult. Lemma3.2. Supposey2f0,1gn0+1,anddeneQ(y)asthesetofallxsuchthat(x,y)2P.ThenallextremepointsofQ(y)areinteger-valued. Proof. Forxedy2f0,1gn0+1,( 3 )reducestoalinearsystemoftheform 266664eTI)]TJ /F4 11.955 Tf 9.29 0 Td[(I377775x266664BUL377775,(3)whereeisthen-vectorconsistingofallones,andUandLaren-vectorscontainingintegralupperandlowerbounds,respectively,onthex-variables.Sinceeachelementofeisequaltoone,eTistotallyunimodular.AppendingIand)]TJ /F4 11.955 Tf 9.3 0 Td[(ItoeTpreservestotalunimodularity,resultingintheconstraintmatrixof( 3 ).SinceB,U,andLareintegral,thedesiredresultholds. FromLemma 3.2 itfollowsthatconv(P)=conv[P\(RnZn0+1)].Thus,convexifyingPviaRLTonlyrequiresustoensurethatthey-variablesarebinary-valuedatallextremepoints.ThisconvexicationstepcanbeperformedusingtheSpecialStructuresRLT(SSRLT)ofSheralietal.[ 55 ],byexploitingthestructureofconstraint( 3b ).Wenowdescribehowinequalities( 3 )and( 3 )canbederivedthroughSSRLT.Followingthetechniquesdescribedin[ 55 ],we:(a)multiplytheconstraintsof( 3 )byykforeachk=1,...,n0+1andxj0by1)]TJ /F14 11.955 Tf 12.17 8.96 Td[(Pn0+1k=1ykforallj=1,...,n0+1(wherethelattersetofoperationsyieldequalityconstraints);(b)applytheidentitiesy2k=ykforallk=1,...,n0+1,ykxk=0forallk=1,...,n0,yjyk=0forallj6=k,andykxj=ykforj
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andj=minfn0+1,k+1g,...,n.Thisprocedureresultsinthefollowingconstraints: nXj=k+1zkj(B)]TJ /F4 11.955 Tf 11.95 0 Td[(k+1)yk,8k=1,...,n0+1, (3a)xj=j)]TJ /F12 7.97 Tf 6.59 0 Td[(1Xk=1zkj+n0+1Xk=j+1yk,8j=2,...,n0, (3b)xj=n0+1Xk=1zkj,8j=n0+1,...,n, (3c)0zkjyk,8k=1,...,n0+1,8j=minfn0+1,k+1g,...,n. (3d)Constraints( 3a )areobtainedbymultiplying( 3a )byyk,( 3b )and( 3c )arederivedbymultiplying1)]TJ /F14 11.955 Tf 12.79 8.97 Td[(Pn0+1k=1ykbyxj0forallj=2,...,n,and( 3d )resultfromthemultiplicationofykwith( 3d ),( 3e ),and( 3g ).TheremainingSSRLTinequalitiesareimpliedbyconstraintsin( 3 )or( 3 )andarethereforeomitted.DenePZasthesetofall(x,y,z)satisfying( 3 )suchthat(x,y)2P.Theconvergenceresultof[ 55 ]impliesthatconv(P)=f(x,y):(x,y,z)2PZg.Thatis,projectingthez-variablesoutofPZproducesalloftheinequalitiesin( 3 ).Inparticular,sincezkj0thez-variablescanberemovedfrom( 3b )iftheequalityischangedtoaninequality,directlyresultingin( 3 ).Toobtaininequalities( 3 ),letandsatisfytheassumptionsofTheorem 3.3 .Aggregatetheequalities( 3b )and( 3c )correspondingtoj2[toobtain( 3a ),whichproduces( 3b )uponreorderingsummations: Xi2I()xi+Xi2I()x i=Xj2 j)]TJ /F12 7.97 Tf 6.58 0 Td[(1Xk=1zkj+n0+1Xk=j+1yk!+Xj2 n0+1Xk=1zkj! (3a)=n0+1Xk=1 Xj2:j>kzkj+Xj2zkj+jkjyk! (3b)n0+1Xk=1(minfjfj2:j>kgj+jj,B)]TJ /F4 11.955 Tf 11.95 0 Td[(k+1g+jkj)yk. (3c) 62

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ThequantityPj2:j>kzkj+Pj2zkjmustbeboundedabovebyboth(B)]TJ /F4 11.955 Tf 12.08 0 Td[(k+1)ykfrom( 3a ),and(jfj2:j>kgj+jj)ykfrom( 3d ),leadingto( 3c ).However,fork=1,...,B)-103(jj)-104(jj+1,jfj2:j>kgj+jj=jj+jjB)]TJ /F4 11.955 Tf 10.54 0 Td[(k+1(withequalitywhenk=B)-252(jj)-252(jj+1)because1B)-252(jj)-252(jj+3or=;.ForeachkB)-304(jj)-305(jj+1increasingkbyone(a)decreasesB)]TJ /F4 11.955 Tf 12.94 0 Td[(k+1byexactlyoneand(b)decreasesjfj2:j>kgj+jjbyatmostone,meaningthatB)]TJ /F4 11.955 Tf 11.96 0 Td[(k+1jfj2:j>kgj+jjforallsuchk;thus,( 3c )becomes: Xi2I()xi+Xi2I()x in0+1Xk=1(minfjj+jj,B)]TJ /F4 11.955 Tf 11.95 0 Td[(k+1g+jkj)yk.(3)Wenowarguethat( 3 )isequivalentto( 3 )if(,)2)]TJ /F1 11.955 Tf 6.78 0 Td[(.Fromconditionof(b)ofTheorem 3.3 ,kfB)-224(jj)-224(jj+3,...,k)]TJ /F3 11.955 Tf 11.98 0 Td[(1gforallkB)-224(jj)-224(jj+4;therefore,B)]TJ /F4 11.955 Tf 12.17 0 Td[(k+1+jkjjj+jj)]TJ /F3 11.955 Tf 18.35 0 Td[(2forallsuchk.Moreover,forallkB)-239(jj)-240(jj+3,condition(b)ofTheorem 3.3 necessitatesthatk=;,provingthat( 3 )isequivalentto( 3 ). 3.3.3GeneralizedStepInequalitiesInpreviousstudiesofthemultiple-opponentproblem(e.g.,[ 32 ]andChapter 2 ofthisdissertation),avariable!representsthevaluep!iassociatedwiththealternativeithatmaximizesp!ioverallalternativesthathavenotbeeninterdicted.InChapter 2 ,stepinequalitiesaredevelopedandshowntobefacetsfor Y!=f(^!,^x):9(x,,y)feasibletomodel( 3 )with^and^x=xg,(3)thepolyhedronlinking!directlywithx.TheseinequalitiesareshowninChapter 2 tobegeneralizationsofthestepinequalitiesof[ 32 ],whichassumeeachopponentknowsthetruep-values.Wenowshowhowtheseinequalitiesmaybeobtainedbyappropriatelyprojectingouty-variables.Infact,thisanalysisactuallyleadstoafurther-generalizedstepinequality.Tothisend,werstsummarizethestepinequalitiesofChapter 2 63

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Forthepurposesofthisanalysis,weagaindropthesuperscriptsfromalldataandvariablesandassumethatforthescenarioofinterest,alternativesareorderedasin( 3 ).Weusethefollowingnotation,consistentwithChapter 2 : H=fh1,...,hmgf1,...,n0+1g,whereh1<i(`i)]TJ /F5 11.955 Tf 11.95 0 Td[(i)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xhi).(3)isvalidaslongasi2HimpliesGiHandpj>mforallj=hm+1,...,n0+1,andfacet-deningwhen12H.Wenowshowhowtoobtainageneralizedversionof( 3 )byappropriatelyprojectingouty-variablesfrom( 3 ).Forthispurpose,wedenethesetH+=fj>hm:pj
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where0i=i)]TJ /F5 11.955 Tf 12.02 0 Td[(mforalli2I(H)andp0j=pj)]TJ /F5 11.955 Tf 12.02 0 Td[(mforallj=1,...,n0.Forj2H,jjand`j0=j0.Sincei2Himpliesj2Hforalljiforalli2I(H)suchthat`i>iimplythat m+Xj2H0`jyj+Xi2I(H)0`iyhi)]TJ /F14 11.955 Tf 24.74 11.36 Td[(Xi2I(H):`i>i(`i)]TJ /F5 11.955 Tf 11.95 0 Td[(i)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xhi)+Xj2H+p0jyj.(3)Fromthedenitionof`j,thevaluesof`jandk(j)correspondingtoj2Handj2Hformanon-increasingsequenceandthersttwosummationscanbecombinedas m+Xi2I(H):`i=ihiXj=1(i)]TJ /F5 11.955 Tf 11.95 0 Td[(`i+1)yhi)]TJ /F14 11.955 Tf 24.74 11.36 Td[(Xi2I(H):`i>i(`i)]TJ /F5 11.955 Tf 11.96 0 Td[(i)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(xhi)+Xj2H+p0jyj.(3)Substituting( 3 )forallyhisuchthati2I(H)and`i=i,andobservingthatPhij=1(i)]TJ /F5 11.955 Tf -455.05 -23.91 Td[(`i+1)=0`1produces `1)]TJ /F14 11.955 Tf 24.74 11.36 Td[(Xi2I(H):`i=i(i)]TJ /F5 11.955 Tf 11.96 0 Td[(`i+1)xhi)]TJ /F14 11.955 Tf 24.74 11.36 Td[(Xi2I(H):`i>i(`i)]TJ /F5 11.955 Tf 11.96 0 Td[(i)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(xhi)+Xj2H+p0jyj.(3)IfH+=;,( 3 )isidentically( 3 ).SinceGiHforalli2H,theresultsfromChapter 2 implythat( 3 )isvalid;further,if12H,theinequalityisfacet-deningforY!.ThisgeneralizestothecasewhereH+6=;aswell:Thegeneralizedstepinequality( 3 )isfacet-deningifandonlyifGiHforalli2Hand12H.Proofofthisresultfollowscloselytotheproofofthecorrespondingresultfor( 3 )inChapter 2 ,andisthereforeomitted.OtherclassesoffacetsforY!canbeobtainedviaalternativeprojectionsofthey-variables.Consider,asanexample,theinstancewithn=9,B=n0=6,and 65

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[p1,...,p7]=[1,0.75,0.6,0.42,0.26,0.19,0.06],sothat =y1+0.75y2+0.6y3+0.42y4+0.26y5+0.19y6+0.06y7.(3)Thetermsin( 3 )canbegroupedinanumberofwaysthatleadtodifferentprojections.Considerforexample,rearranging( 3 )as =0.06(y1+y2+y3+y4+y5+y6+y7) (3a)+0.1(3y1+3y2+3y3+3y4+2y5+y6) (3b)+0.03y6 (3c)+0.06(y1+y2+y3+y4)+0.18(y1+y2+y3)+0.15(y1+y2)+0.25y1. (3d)Replacingthey-termsin( 3a ),( 3b ),( 3c ),and( 3d )respectivelywith( 3b ),( 3 )correspondingto=;and=f7,8,9g,( 3f ),and( 3 )resultsinthefollowinginequality, 1)]TJ /F3 11.955 Tf 11.95 0 Td[(0.25x1)]TJ /F3 11.955 Tf 11.95 0 Td[(0.15x2)]TJ /F3 11.955 Tf 11.95 0 Td[(0.18x3)]TJ /F3 11.955 Tf 11.96 0 Td[(0.1(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x7))]TJ /F3 11.955 Tf 11.95 0 Td[(0.1(1)]TJ /F4 11.955 Tf 11.95 0 Td[(x8))]TJ /F3 11.955 Tf 11.96 0 Td[(0.1(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x9),(3)whichisfacet-deningforY!,yetnotoftheform( 3 ).(Theproofofthisresultisstraightforwardandthereforeomitted.)Alternatively,( 3 )maybefactoredas =0.06(4y1+4y2+4y3+3y4+2y5+2y6+y7) (3a)+0.07(2y1+2y2+2y3+2y4+2y5+y6) (3b)+0.1(y1+y2+y3+y4)+0.12(y1+y2+y3)+0.15(y1+y2)+0.25y1. (3c)Projectingoutthey-variablesusing( 3 )with=f5gand=f7,8,9g,( 3 )with=;and=f7,8g,and( 3 )respectivelyon( 3a ),( 3b ),and( 3c ) 66

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produces 1)]TJ /F3 11.955 Tf 9.51 0 Td[(0.25x1)]TJ /F3 11.955 Tf 9.51 0 Td[(0.15x2)]TJ /F3 11.955 Tf 9.52 0 Td[(0.12x3)]TJ /F3 11.955 Tf 9.52 0 Td[(0.1x4)]TJ /F3 11.955 Tf 9.51 0 Td[(0.06(1)]TJ /F4 11.955 Tf 9.51 0 Td[(x5))]TJ /F3 11.955 Tf 9.52 0 Td[(0.13(1)]TJ /F4 11.955 Tf 9.52 0 Td[(x7))]TJ /F3 11.955 Tf 9.52 0 Td[(0.13(1)]TJ /F4 11.955 Tf 9.51 0 Td[(x3))]TJ /F3 11.955 Tf 9.51 0 Td[(0.06(1)]TJ /F4 11.955 Tf 9.52 0 Td[(x9),(3)whichisanotherfacet-deninginequalityforY!. 3.4ExtensionstotheMultiple-OpponentProblemTheprevioussectionfocusesondevelopingafullpolyhedralrepresentationoftheone-opponentproblem.Notingthatthemultiple-opponentproblemisstronglyNP-hard,theremustbeadditionalfacet-deninginequalitiesthatinvolvevariablesrepresentingactionsinmultiplescenarios.Moreover,notingthepolynomialequivalenceofseparationandoptimization(see,e.g.,[ 53 ]),someofthesefacetsmustbedifculttoobtain.Thatis,onewouldexpectsomeclassesoffacetstocorrespondtosolutionsforanNP-completeproblem.Wenowdescribeaninstanceofthemultiple-opponentproblem(whichincludessuchaclassoffacets)havingalternativesf1,...,ngandscenarios=f1,...,Mg,wherenM.Tothisend,werstpartitionthealternativessubsetfM+1,...,n)]TJ /F3 11.955 Tf 13.12 0 Td[(4gintoH2=fM+1,...,h0g,H3=fh0+1,...,h00g,andH4=fh00+1,...,n)]TJ /F3 11.955 Tf 12.58 0 Td[(4g.DeneB=7,andfor!=1,...,Mandj=1,...,8,letk!jdenotethej-thpreferredalternativeinscenario!(asinSection 3.1 ).Foreachscenario,supposen0=B.Forthisinstance,weareinterestedinthepolytopeconv[P],denedin( 3 ).Weassumethefollowingforeach!=1,...,M:k!1=!;k!j2Hjforj=2,3,4;andk!j=j+n)]TJ /F3 11.955 Tf 11.96 0 Td[(8forj=5,6,7,8.WenowestablishthatPhasdimensionn+6M. Lemma3.3. ThesetPhasdimensionn+6M. Proof. Each(x,y)2Psatises8Xj=1y!j=1,and (3)y!1=1)]TJ /F4 11.955 Tf 11.95 0 Td[(xk!1, (3) 67

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foreach!=1,...,M,analogousto( 3b )and( 3c ).Equalities( 3 )arealllinearlyindependentofeachotherbecauseeachy!1appearsexactlyonce.Equalities( 3 )mustbelinearlyindependentofeachother,andof( 3 )becausey!2appearsonlyonceforeach!=1,...,M.Thus,thedimensionofPisatmostthenumberofvariableslessthenumberoflinearlyindependentequalities,n+8M)]TJ /F3 11.955 Tf 11.95 0 Td[(2M=n+6M.WeshowthatthedimensionofPisatleastn+6Mbyprovidingn+6M+1afnelyindependentpointsc0,...,cn+6MinPoftheform(x1,...,xn,y11,...,y18,y21,...,yM8).Denec0suchthatcomponentsy!1,8!=1,...,Mareequaltooneandallothersareequaltozero.Fori=1,...,M,deneciexactlyasc0,butchangingthexicomponentfromzerotooneandswappingthevaluesofyi1andyi2.Fori=M+1,...,n,deneciexactlyasc0exceptforaoneinthexi-component.Fori=1,...,Mandj=1,...,6,denecn+6(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+jsuchthat:componentsxiandxki2,...,xkij+1areequaltoone;componentsyij+2andyi01,8i06=iareequaltoone;andallothercomponentsareequaltozero.LetCbethe(n+6M)(n+8M)matrixsuchthatforeachi=1,...,n+6M,rowiofCisgivenbyci)]TJ /F4 11.955 Tf 11.96 0 Td[(c0.Let~CbethesquaresubmatrixofCconsistingofcolumnsx1,...,xnandy!3,...,y!8forall!=1,...,M.Because~Cisalower-triangularmatrixwithnonzerodiagonals,therowsof~C,andthusofCaswell,mustbelinearlyindependent,provingthatc0,...,cn+6Mareafnelyindependent. Withk!1andk!5,...,k!8xed,theentireprobleminstancecanbedescribedbytheorderedtriplets(k!2,k!3,k!4)2H2H3H4foreach!=1,...,M.Notingthatthemultiple-opponentproblemisNP-hard(see[ 51 ])andhavingacompleteandpolynomiallyseparablerepresentationoftheone-opponentconvexhull,wewouldanticipatethatsomeofthefacetsofconv[P]containy-variablesfromseveraldifferentscenarios;indeed,differentcombinationsofthealternative(k!2,k!3,k!4)leaddirectlytofacet-deninginequalities. 68

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AmatchingisacollectionofscenariosAsatisfyingthefollowingproperty:Foranyi,i02A,i6=i0,itfollowsthatki26=ki02,ki36=ki03,andki46=ki04.ForanymatchingA,wemaystatethefollowinginequality: xn)]TJ /F12 7.97 Tf 6.59 0 Td[(3+Xi2Ayi51.(3)Validityof( 3 )isintuitive:Ifxn)]TJ /F12 7.97 Tf 6.59 0 Td[(3=1,thenone-scenarioconstraintsanalogousto( 3d )preventtheassignmentofanyyi5toone.Furthermore,itisnotpossibletohaveyi5=yi05=1fori,i02A,i6=i0sinceeachofki1,...,ki4mustbedistinctfromeachofki01,...,ki04andB=7.AmatchingAismaximaliftheredoesnotexistA0,AA0suchthatA0isamatching.Wenowshowthatwitheachmaximalmatchingofisassociatedwithauniquefacetofconv[P]. Theorem3.6. SupposeAisamaximalmatchingof.Then( 3 )isfacet-deningforconv[P]. Proof. Asarguedabove,( 3 )isvalidforconv[P],andthus,weneedonlyestablishthatthedimensionofthefaceofconv[P]denedbythisinequalityisn+6M)]TJ /F3 11.955 Tf -415.37 -23.91 Td[(1(followingfromLemma 3.3 ).Weconstructn+6Mafnelyindependentpointsc1,...,cn+6M2Poftheform(x1,...,xn,y11,...,y18,y21,...,yM8)thatarebindingon( 3 ).Fori=1,...,M,denecisuchthatcomponentsxi,xn)]TJ /F12 7.97 Tf 6.59 0 Td[(3,andyi2equaloneaswellasyi01foralli06=i(andallothercomponentsequalzero).Fori=M+1,...,ndenecisuchthatcomponentsxi,xn)]TJ /F12 7.97 Tf 6.58 0 Td[(3,andyi1,8i=1,...,M,areequaltooneandallothersareequaltozero.(Notethatincn)]TJ /F12 7.97 Tf 6.58 0 Td[(3,onlyonex-variable,xn)]TJ /F12 7.97 Tf 6.59 0 Td[(3,issettoone.)Fori=1,...,Msuchthati2A,denecn+6(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+jsuchthat: forj=1,2,componentsxi,xki2,...,xkij+1,xn)]TJ /F12 7.97 Tf 6.59 0 Td[(3,andyij+2equalone(andallothersequalzero); forj=3,componentsxi,xki2,...,xki4,andyi5equalone(andallothersequalzero);and, 69

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forj=4,5,6,componentsxi,xki2,...,xkij+1,andyij+2equalone(andallothersequaltozero).Amongthesesixpoints,onlytheonecorrespondingtoj=3hasxn)]TJ /F12 7.97 Tf 6.59 0 Td[(3=0andinthispointyi5=1,indicatingthatallarebindingon( 3 ).Fori=1,...,Msuchthati=2A,theremustexisti02Asuchthatki2=ki02,ki3=ki03,orki4=ki04.Forthisi,denecn+6(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+jexactlyasifiwereanelementofA,butforj=3,xi0,xki02,...,xki04,andyi05shouldalsoequalone.(Note:thisispossiblewithintheB=7restrictionbecausescenariosiandi0shareincommoneithertheirsecond,third,orfourthpreference.)LetCdenotethe(n+6M)(n+8M)matrixsuchthatrowiisgivenbyciforalli=1,...,n+6M.WeprovetheonlysolutiontoTC=0isi=0,8i=1,...,n+6M.Columnsyij+2fori2nAandj=1,...,6areidentitycolumns,implyingthatn+6(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+j=0.Columnsyij+2fori2Aandj=1,...,6nowcontainonlyone-variablenotalreadyshowntoequalzero,implyingthatn+6(i)]TJ /F12 7.97 Tf 6.59 0 Td[(1)+j=0.Havingalreadyshownthati=0forallin+1,columnsxifori=1,...,n)]TJ /F3 11.955 Tf 12.09 0 Td[(4andi=n)]TJ /F3 11.955 Tf 12.09 0 Td[(2,n)]TJ /F3 11.955 Tf 12.09 0 Td[(1,nnowimplythati=0.Fromcolumnxn)]TJ /F12 7.97 Tf 6.58 0 Td[(3,wenowndthatn)]TJ /F12 7.97 Tf 6.59 0 Td[(3=0aswell;thus,=0istheonlysolutionandc1,...,cN+6Mmustbelinearlyindependent(andthereforeafnelyindependent).Thatis,thefacedenedby( 3 )hasdimensionn+6M)]TJ /F3 11.955 Tf 11.95 0 Td[(1,asrequired. GivenamatchingA,itiseasytondamaximalmatchingA0byaugmentingthescenariosinAusingasimplegreedyalgorithm.Thedecisionproblemofndingathree-dimensionalmatching(3DM)ofcardinalitygreaterthanorequaltoanarbitraryconstantkisNP-complete(byatrivialreductionfrom3DMasdenedin[ 60 ]).Usingthesimpleconversionalgorithmdescribedabove,3DMisequivalenttotheproblemofndingamaximalmatchingofcardinalityatleastk,andtherefore,thefacetsdenedby( 3 )correspondtosolutionsofNP-completeproblems. 70

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3.5ComputationalResultsWenowpresentabriefcomputationalstudy,whichisdesignedtoillustratetheeffectivenessofourproposedinequalitiesinimprovingthelowerboundsyieldedbythelinearprogrammingrelaxationstomultiple-opponentC-BiPSNIPinstances.TorandomlygenerateC-BiPSNIPinstances,wesetjj=25andselectn2f50,100,150gandB2f0.5n,0.8ng.ForeachcombinationofnandB,wegenerateveinstancesof( 3 )byrandomizingthep-andp-values.For!=1,...,jjandj=1,...,n,wedrawindependentrealizationsofp!jfromacontinuousuniformdistributionon(0.25,0.75)withprobability0.6andsetp!j=0otherwise.Forall!andjsuchthatp!j>0,weindependentlygeneratep!juniformlyfromtheset(p!j)]TJ /F3 11.955 Tf 11.95 0 Td[(0.25,p!j+0.25).Inourimplementation,constraints( 3a )( 3d ),( 3f )( 3g ),and( 3 )areaddedaprioriforeachscenarioandconstraints( 3 )areaddedasneeded.Foreachrandominstance,weminimizeP!2Pn0+1j=1p!jy!j;i.e.,weassumeequalweightsf!,forall!2,inmodel( 3 ).Wesolveeachinstanceusing,inturn,threeimplementations:NoCuts,whichsimplysolvestheformulationusingthedefaultCPLEXsettings(anddoesnotmakeuseofthevalidinequalities( 3 ));AllCuts,inwhichallviolatedinequalities( 3 )areaddedattherootnodeinacut-and-branchfashion;andSignicantCuts,inwhichcutsareonlyaddedtotherootnodeiftheviolationexceeds1,i.e.,iftheleft-handsideof( 3 )lesstheright-handsideisatleast1.Table 3-2 showstheobjectivevalueofthebestintegersolution(columnIP)obtainedbyanyofthethreealgorithms,andthecorrespondingoptimalitygap(columnG(%)),computedastheabsoluteoptimalitygapdividedbytheobjectivelowerbound.Alsoshownaretheobjectivevaluesfortherootnodecontinuousrelaxationafteraddingcutsasprescribedineachalgorithm(columnLP).Foreachcut-addingalgorithm,wederivegapreduction(columnR(%))tomeasuretheeffectof( 3 )ontighteningthemultiple-opponentformulation.Thiscolumniscomputedas100(LPc)]TJ /F1 11.955 Tf 12.33 0 Td[(LP)=(IP)]TJ /F1 11.955 Tf 12.34 0 Td[(LP),whereLPcistherootnodecontinuousrelaxationvaluewithcutsandLPisthe 71

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Table3-2. Comparisonofcontinuousrelaxationsunderdifferentcut-addingstrategies. NoCutsAllCutsSignicantCuts nBIPG(%)LPLPR(%)CutsLPR(%)Cuts 502530.350.0023.4223.420.00023.420.000504026.880.006.8811.8024.6022811.1121.171301005031.201.5924.3424.340.02124.340.0001008028.4017.327.1112.1323.5928911.8022.011841507531.736.8024.4224.420.02124.420.00015012029.1928.867.1712.0822.3031711.8321.15210 continuousrelaxationvaluewithoutcuts.Foreachcut-addingalgorithm,wealsoindicatethenumberofcutsaddedattheroot-node.Table 3-2 showsthatvaryingnseemstohavelittleeffectonthegapreduction,butitalsosuggeststhatinequalities( 3 )aremoreimportanttothemultiple-scenarioformulationwhenBislargerelativeton.WhenB=0.5n,cutsprovidenoimprovementtotheLPrelaxationinmostinstancesandmodestimprovementinothers.However,whenB=0.8n,adding( 3 )consistentlyreducesthegapby20to25%.Interestingly,addingonlysignicantlyviolatedcutsprovidesimprovementtotherootnodelower-boundnearlyequivalenttothatobtainedbyaddingallviolatedcuts. 72

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CHAPTER4GEOGRAPHICALINTERDICTIONOFAMAXIMUMFLOWNETWORKWeconsideranetworkthatresidesinEuclideanspaceandadefenderwhosegoalistomaximizeowfromasourcenodetoasinknodeinthenetwork.Theproblemweexaminetakestheperspectiveofaninterdictor,whoseekstominimizethedefender'smaximumowbylocatingattacksintheregiononwhichthenetworkislocated.Attacklocationsarenotrestrictedtonodeorarclocations,andservetodiminisharccapacitiesinaccordancewiththedistancefromthearctotheattack.TheproblemweconsiderisknowntobeNP-hardasitgeneralizesthemaximumowinterdictionproblemstudiedbyWood[ 8 ].Wedeveloptwoapproachestosolvingthisproblembasedonsolvingasequenceoflower-boundingintegerprograms,andcomparetheefcacyofourapproachesonasetofrandomlygeneratedtestinstances. 4.1BackgroundWeconsidertheinterdictionofacapacitatednetworkthatexistsinEuclideanspace.Nodesinthisnetworkexistatapointinspaceand(directed)arcsconnectnodepairsinastraightline.Anopponentwishestomaximizeowfromasourcenodetoasinknodeacrossthenetwork,whileaninterdictorseekstominimizetheopponent'smaximumowbychoosingmultiplelocationstoattack.Inthisproblem,attacksaremadeatpointsinspace.Damageisinictedoneacharcbyreducingitscapacityasafunctionofthedistancefromthemidpointofthearctoanattack.WerefertothisproblemastheEuclideanmaximumownetworkinterdictionproblem(E-MFNIP).Inthischapter,weprovidemathematicalprogramming-basedapproachesforsolvingE-MFNIP.Themotivationforstudyingthistypeofproblemisduetotheprevalenceofnetworksincomplexsystemsandthevulnerabilityofthosesystemstoattack.Forinstance,networksthatrepresentreal-worldtransportation,logistics,andpowergridsystemshavewell-denedgeographicalcharacteristics.Thesesystemsaresubjecttodisruptions(e.g.,anearthquake)thatmaydamagemultiplecomponentsinthesamegeographical 73

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region.Mostnetworkinterdictionresearchneglectsanygeographicalcharacteristics,focusinginsteadonidentifyingnetworkcomponentsthataremostcriticaltosustainingfunctionality.Incontrast,ourmethodologyexplicitlyaccountsforthesimultaneousfailureofnetworkcomponentscorrelatedbythephysicallocationofanattack.Asaresult,thismodelprovidesvaluableinsightsintoidentifyingphysicallocationsofvulnerability.Overthelasthalf-decade,severalstudieshaveexaminednetworkresilienceundergeographicallycorrelatedfailures.Neumayeretal.[ 61 ]useanintegerprogrammingmodeltoidentifyasingleverticallinecutinabipartitenetworkthatmaximizesthetotalcapacityofallintersectedarcs.Thisworkisextendedin[ 62 ],whichconsidersageneralownetwork,twotypesofcuts(circularandlinesegment),andseveralmeasuresofnetworkperformance(includingtheworst-casemaximumowbetweenapairofnodes).Inarelatedwork,NeumayerandModiano[ 63 ]examinetheresilienceofanetworktoasingleprobabilisticgeographicalfailure.Agarwaletal.[ 64 65 ]considertheproblemoflocatingmultiplecircle-shapedattacksaroundtelecommunicationsnetworkinwhichtheprobabilityofacomponentfailuredependsonitsdistancefromtheepicenterofanattack.Themajorcontributionofthesepapersisagreedyalgorithmthatapproximatesthenetwork'sresilience(denedasitsabilitytotransmitaxedpatternoftrafc).Bernsteinetal.[ 66 ]modelgeographicallycorrelatedfailuresandensuingcascadingfailuresinapowergrid,anduseamethodbasedon[ 65 ]toidentifyattacklocationstowhichthegridisvulnerable.Survivabledesignofnetworkssubjecttogeographicalfailures[ 67 ]hasalsoreceivedsomeattentioninthecontextofunderseacablenetworks.OurproblemismostsimilartothatofNeumayeretal.[ 62 ],whoalsocharacterizeworst-casedisruptionsusingthemaximumowmetric.Tothebestofourknowledge,however,ourresearchistherstattempttomodeltheworst-caseeffectsofmultiplegeographicaldisruptionsinamaximumownetwork.Moreover,ourproblemismoregeneralthanthemaximumowmodelof[ 62 ]inthatarccapacitiesmaybedened 74

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generallyasa(possiblycontinuous)functionofdistancebetweenarcandattack;in[ 62 ],anattackeitherhasnoeffectonanarcorremovesitaltogether.Theremainderofthischapterisorganizedasfollows.InSection 4.2 ,wedevelopageneralmodelformaximumowinterdictionthatyieldsaMIPformulationforE-MFNIPwhencapacityfunctionsarepiecewise-convex.InSection 4.3 ,weextendthismodeltodevelopaMIPmodelthatprovidesalowerboundforE-MFNIPundermoregeneralcapacityfunctions.Wethendescribehowthelowerboundcanbedynamicallyimprovedwithinanimplicitenumerationalgorithm.Anticipatingthedifcultyofsolvingthismodel,Section 4.4 describesaproblemthatrestrictstheinterdictortoselectitsattacksfromanitelistofcandidatelocations.ThismodelcanbeemployedtoyieldbothalowerandupperboundfortheoptimalE-MFNIPobjectivevalue,givingrisetoadiscretize-and-renemethodforsolvingE-MFNIP.WesummarizecomputationalresultsfromsolvingE-MFNIPonrandomlygeneratedtestnetworksinSection 4.5 4.2MaximumFlowInterdictionModelsInthissection,wedevelopaMIPmodelthatmaybeusedinsolvingE-MFNIP.Webeginbyrecappingagenericsingle-stagemodelformaximumowinterdictioninSection 4.2.1 .InSection 4.2.2 ,wespecializethismodeltoobtainamodelforE-MFNIPunderpiecewise-convexcapacityfunctions.Section 4.3 describeshowthismodelmaybeusedinsolvingE-MFNIPinstanceswithmoregeneralcapacityfunctions. 4.2.1GenericSingle-StageFormulationLetG(N,A)beamaximumownetworkhavingnodesetN=f1,...,nganddirectedarcsetANN,wherenode1isthesourcenodeandnodenisthesinknode.Letcijdenotethecapacityofarc(i,j)2A,anddenevariablesxijastheamountofowonarc(i,j)2A.Also,letarc(n,1)beanuncapacitatedreturnarcfromnodentonode1,andletA=A[f(n,1)g.Themaximumow(denoted)fromnode1tonode 75

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ncanbeobtainedbysolvingthemodel =maxxn1, (4a)s.t.Xj2FS(i)xij)]TJ /F14 11.955 Tf 17.92 11.35 Td[(Xj2RS(i)xji=0,8i2N,:ui (4b)0xijcij,8(i,j)2A,:vij (4c)whereFS(i)=fj2N:(i,j)2AgandRS(i)=fj2N:(j,i)2Ag.Becausetheobjectivemaximizesxn1,itisnotnecessarytorequirexn10.NotethataggregatingConstraints( 4b )correspondingtonodesi=2,...,nimpliesConstraint( 4b )correspondingtonode1,andsoweomitthisconstraintinoursubsequentanalysis.Model( 4 )isalinearprogram(LP)andthus,couldequallybeobtainedbysolvingtheLPdualof( 4 ).IntroducingdualvariablesuassociatedwithConstraints( 4b )(withu10)andvassociatedwithupper-boundingConstraints( 4c ),theconstraintsare: ui)]TJ /F4 11.955 Tf 11.96 0 Td[(uj+vij0,8(i,j)2A, (4a)un=1, (4b)vij0,8(i,j)2A. (4c)LettingUbethefeasibilitysetthatencompassestheconstraintsof( 4 ),thedualmaximumowformulationis=minX(i,j)2Acijvij,s.t.(u,v)2U. (4)Inourproblem,theinterdictorchoosesanattackzfromsomefeasiblesetZ,andarccapacitiesarecomputedaccordingtoafunctionij:Z!Rforall(i,j)2A.Aftertheleaderselectsaninterdiction^z2Z,thefollowermaximizesowontheresultingnetwork,solving( 4 )withcijreplacedbyij(^z).Theinterdictor'sproblemisthentochooseaninterdictionthatminimizesthefollower'smaximumow,whichcanbestated 76

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as:?=minX(i,j)2Aij(z)vij,s.t.(u,v)2U,z2Z. (4)Whenijisdenednontrivially(i.e.,notconstant)forsome(i,j)2A,thismodelisnonlinearintheobjective.However,asshownin[ 8 ],forany^z2Zthereexistsasolutionto( 4 )inwhichvij2f0,1gforall(i,j)2A.Imposingtheserestrictions,wecanlinearizetheobjectivefunctionbydeningvariableswijforeach(i,j)2Atorepresenttheproductij(z)vij,andaddingtheinequalities wijij(z))]TJ /F4 11.955 Tf 11.95 0 Td[(Mij(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vij),8(i,j)2A,and (4a)wij0,8(i,j)2A, (4b)whereMijisanupperboundonij(z)overallz2Z.TheresultingformulationisaMIPthatislinearlyconstrainedwhenijisalinearfunction:?=minX(i,j)2Awij, (4)s.t.Constraints( 4a )and( 4b ),(u,v)2U,z2Z,vij2f0,1g,8(i,j)2A. Remark4.1. Model( 4 )generalizesseveralversionsofmaximumowinterdiction.Inthebasicmaximumownetworkinterdictionproblem(MFNIP),theleaderhasabudgetofKandcanremoveanyarc(i,j)2Aatacostofbij.Removinganarc(i,j)effectivelyreducesitscapacityfromcij(aconstant)tozero.SettingZ=fz2f0,1gjAj:P(i,j)2AbijzijKg,ij(z)=cij(1)]TJ /F4 11.955 Tf 13.07 0 Td[(zij),andMij=cijin( 4 )resultsinamodelequivalenttotheMFNIPmodelgivenin[ 8 ].(Hence,theoptimizationproblemgivenby( 4 )isNP-hard.)Whenbij=1,8(i,j)2A,intheabovedenitionofZ,thecardinalityconstrainedmaximumownetworkinterdictionproblem(C-MFNIP),orKmost-vitalarcsproblem,results.2 77

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4.2.2SpecicationtoE-MFNIPTypicallyz2ZconsistsofjAjdecisions,eachrepresentingtheamountofinterdictionimpartedonanarc.ThenoveltyofourresearchliesinourdenitionofthesetZ:WeassumethatGhasphysicalstructure,residinginq-dimensionalEuclideanspace,andthatatmostKinterdictionsaremadeatpointsonandaroundG.Inwhatfollows,weusethenotation[s],denedforanypositiveintegers,torefertothesetf1,...,sg.Thus,anelementz2Zisgivenasz=[z1j...jzK],wherezk2Rq,8k2[K],isthelocationofthek-thinterdiction.Forsimplicity,weassumethattheconstraintsetZisgivenas Z=fz2RqK:LzkU,k2[K]g,(4)whereLandUareq-vectors.Weassumethateachnodei2Nislocatedatapointp(i)2Rq,andhenceeacharccenter(i,j)2Aislocatedatthepointp(i,j)=0.5(p(i)+p(j)).Weassumethatthedamageinictedbyanattackonanarcisafunctionofthedistancebetweenthearcandtheattacklocation.Forsimplicity,wecomputedistanceusingtheManhattannorm,i.e.,kpk1Pqk=1jpkj.For^z2Rq,denedij(^z)=k^z)]TJ /F29 11.955 Tf 11.95 0 Td[(p(i,j)k1asthedistancebetween^zandarc(i,j).Inourmodel,thecapacityofarc(i,j)isdeterminedbythedistanceoftheclosestattacktop(i,j).Thatis,capacityfunctionsareoftheform ij(z)=fijmink2[K]dij(zk),(4)wherefij:R+!R+isafunctionthatmapsdistancetocapacity.WhenK=1,thisassumptionisintuitive.WhenK>1,thisassumptionisconservative(fromtheinterdictor'sperspective)becauseonlyoneoftheKinterdictionsinuencesthecapacityofeacharc.E-MFNIPistheproblemthatresultswhenZandaredenedasin( 4 )and( 4 ),respectively.Wedene?EastheoptimalobjectivefunctionvaluetoE-MFNIP. Remark4.2. WefocusprimarilyonscenariosinwhichtheoriginalproblemmayhaveasetS1of(multiple)sourcenodesandasetS2ofsinknodes.Thisproblemcanbe 78

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convertedintoaproblemhavingonlyonesourceandonesinkasfollows:(i)Createsourcenode1andsinknoden;(ii)Createarcs(1,j),8j2S1,andarcs(i,n),8i2S2;and(iii)Denef1j(d)=Mforallj2S1andfin(d)=Mforalli2S2,whereMisaverylargeconstant.2Wenowspecialize( 4 )tomodelE-MFNIPasaMIPhavingaconvexcontinuousrelaxationunderacertainclassoff-functions.Inthissection,weassumethatfijisgivenas fij(d)=mint2[T]gtij(d),(4)wheregtij:R+!R+isconvexandnondecreasingforeacht2[T].Denehtij=gtijdijandnotethathtijisaconvexfunctionbecausegtijanddijareconvexandgtijisnondecreasing.Moreover,observethatijcanberepresentedas ij(z)=mint2[T]gtijmink2[K]dij(zk) (4a)=mint2[T],k2[K]gtijdij(zk) (4b)=mint2[T],k2[K]htij(zk), (4c)where( 4b )followsbecausegtijisnondecreasing.Denetkijtoequal1iftheminimumin( 4b )isachievedbyt2[T]andk2[K]andtkij=0otherwise.Usingthe-variables,ijcanbeexpressedas ij(z)=minXt2[T],k2[K]htij(zk)tkij, (4a)s.t.Xt2[T],k2[K]tkij=1andtkij2f0,1g8t2[T],k2[K]. (4b)ThisexpressioncouldbesubstitutedforijinModel( 4 )toobtainavalidformulation.Furthermore,observethatdecreasingij(z)relaxes( 4a ),whichistheonlyplaceij(z)appearsinModel( 4 ).Thus,theminimizationoperatorcanbedroppedfrom( 4 ),becauseanoptimalsolutionwillalwaysexistinwhichij(z)takesitssmallestvalue 79

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allowedby( 4 ).TheresultingMINLPminimizesP(i,j)2Awijsubjectto(u,v)2U,z=[z1jjzr]2Z,v2f0,1gjAj,and( 4b )aswellas wijXt2[T],k2[K]htij(zk)tkij)]TJ /F4 11.955 Tf 11.96 0 Td[(Mij(1)]TJ /F4 11.955 Tf 11.96 0 Td[(vij),8(i,j)2A, (4a)Xt2[T],k2[K]tkij=1,8(i,j)2A,and (4b)tkij2f0,1g,8(i,j)2A,t2[T],k2[K]. (4c)Tolinearize( 4a ),lettkijrepresenttheproducthtij(zk)tkijandintroduceinequalitiessimilarto( 4 )toobtainthefollowingmodel,validforanyMtijsuchthatMtijhtij(z),8LzU,(i,j)2A,t2[T]. ?h=minX(i,j)2Awij (4a)s.t.(u,v)2U,wijXt2[T],k2[K]tkij)]TJ /F4 11.955 Tf 11.95 0 Td[(Mij(1)]TJ /F4 11.955 Tf 11.95 0 Td[(vij),8(i,j)2A, (4b)wij0,8(i,j)2A, (4c)Xt2[T],k2[K]tkij=1,8(i,j)2A, (4d)tkijhtij(zk))]TJ /F3 11.955 Tf 14.57 2.65 Td[(Mtij(1)]TJ /F5 11.955 Tf 11.96 0 Td[(tkij),8(i,j)2A,t2[T],k2[K], (4e)tkij0,8(i,j)2A,t2[T],k2[K], (4f)vij2f0,1g,8(i,j)2A, (4g)tkij2f0,1g,8(i,j)2A,t2[T],k2[K], (4h)z2Z.Asbefore,Model( 4 )isaMINLP,butthistimeitscontinuousrelaxationisconvex(becauseallh-functionsareconvex).Thesetractablecontinuousrelaxationscanbeusedtosolve( 4 )via,e.g.,branchandboundsolongasallf-functionscanbe 80

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expressedas( 4 ).Inthefollowingsection,wediscusstechniquesforapproximatingE-MFNIPwhenfcannotbeexpressedas( 4 ),basedonusingModel( 4 )withappropriatelydenedg-functions. 4.3SolvingoveraGeneralCapacityFunctionInmanyapplications,thef-functionscannotberepresentedintheformof( 4 ).Intheremainderofthissection,weoutlineaprocedureforusingModel( 4 )tolower-boundtheoptimalE-MFNIPobjectivefunctionvaluebydeningappropriatepiecewise-linearg-functions.Weassumethetrue-functionstaketheformof( 4 ),withfbeingnondecreasingandconcave.Theseassumptionsintuitivelymatchwhatonewouldexpectfromthebehaviorofarealisticinterdiction:Capacityincreasesastheinterdictionmovesfartherawayfromanarc(nondecreasing),andasmallchangeininterdictionlocationismorelikelytohaveapronouncedimpactonnearbyarcsthanonfarawayarcs(concave).Iffisalsopiecewise-linear,Model( 4 )becomesanexactmodelusinglinearg-functions.Iffisnotpiecewise-linear,g-functionscanbeselectedtoensurethat( 4 )providesalowerboundontheoptimalE-MFNIPobjective.Onemightreasonablyconjecturethat,undertheseassumptionsonf,optimalattacklocationsalwayscoincidewiththemidpointofanarc.Thisconjectureisfalse,asweillustratethroughthefollowingexample.Considerthetwo-dimensionalnetworkillustratedinFigure 4-1 inwhichallodd-numberednodesaresourcesandalleven-numberednodesaresinks.Theonlypossibleminimumsource-sinkcutisgivenbythearcs(1,2),(3,4),and(5,6),wherep(1,2)=(0,0),p(3,4)=(0.5,0.5),andp(5,6)=(1,0).If(forK=1)f12(d)=f34(d)=f56(d)=f(d)d0.75(astrictlyconcave,nondecreasingfunction),thenattackingeither(0,0),(0.5,0.5),or(1,0)reducesthecapacityofthiscutto2f(1)=2.Anattackat(0.5,0),whichdoesnotcoincidewiththemidpointofanarc,resultsinanevensmallerminimumcutcapacityof3f(0.5)=1.78. 81

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Figure4-1. Examplenetworkwithonlyonepossiblecutset. Wenowdescribeatechniqueforapproximatingij-functionsbydeningpiecewise-lineargij-functions.Supposebreakpointsa0,...,aTaregivensuchthat0=a0
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that(^z,^u,^v,^w,^,^)isafeasiblesolutionto( 4 )havingobjectivevalue?E.Foreach(i,j)2A,lettijandkijbedenedsuchthat(tij,kij)2argmint2[T],k2[K]gtijdij(^zk).For(i,j)2A,t2[T],andk2[K],dene^tkij=8><>:1ift=tijandk=kij0otherwise,andobservethat( 4d )and( 4h )aresatisedbythischoiceof^.Constraints( 4e )and( 4f )reducetotijkijijhtijij(^zk)=gtijdij(^zk)andtotkij0whent6=tijork6=kij.Accordingly,dene^as^tkij=8><>:mint2[T],k2[K]gtijdij(^zk)ift=tijandk=kij0otherwise,andnotethatConstraints( 4e )and( 4f )aresatised.Next,observethat( 4b )holdsbecause Xt2[T],k2[K]^tkij=^tijkijij (4a)=mint2[T],k2[K]gtijdij(^zk) (4b)fijdij(^zk) (4c)^wij+Mij(1)]TJ /F3 11.955 Tf 12.25 0 Td[(^vij), (4d)where( 4c )followsdueto( 4 ),and( 4d )isimpliedbyConstraint( 4a )ofModel( 4 ).Theremainingconstraintsof( 4 )arealsoconstraintsof( 4 )andhence,(^z,^u,^v,^w,^,^)isfeasibletoModel( 4 )withobjectivevalueP(i,j)2A^wij=?E. Remark4.3. ObservethatfneednotbeconcaveinordertoapplytheapproximationMIPinthissection.Weillustratetheuseofconvexg-functionstolower-bounda(nonconcave)stepfunctionfinFigure 4-3 .Suchalower-boundingschemeprovides 83

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Figure4-2. fijlowerboundedbyg1ijandg2ijoverz2Z(left)andrenedapproximationafterpartitioningZ(right). Figure4-3. Examplestepfunctionfij(left)andlower-boundingconvexgij-functions(right). awaytomodelcapacityfunctionssuchasthoseusedin[ 62 ]inwhicharcsareeithercompletelyremoved(ifanattackiswithinrdistanceunits)oraltogetherunharmed.2 Remark4.4. Itispossibletorenetheg-functionsdynamicallywithinthesolutionof( 4 )toobtainabetterapproximationforE-MFNIP.ThisalgorithmisbasedonpartitioningasearchspaceZintoregionsZandZ,reningtheg-functionstomoretightlyapproximatefovertheseregions,andthenbranchingtodeterminewhethereachattackbelongstoZorZ.WhetheraparticularattackisconstrainedtolieinZorZ 84

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determineshowtheg-functionscanbetightened.Assuch,wereplacethefunctionsgtijinModel( 4 )withgtkij.Figure 4-2 demonstratesthepartitioningofasetZintoZandZ,resultingintighterapproximationsoneitherside.Thegenericschemewepresentisbasedonthefollowingthreeconcepts.One,solving( 4 )oversomeregion^ZZyieldsalowerboundontheoptimalobjectivevalueto( 4 )inwhichz2^Z.Anupperboundcanalsobeobtainedbasedonany^z2Zbyevaluating(^z),themaximumowafterattack^z.Two,notethatZcanbedividedintotwosetsZandZbychoosingsomehyperplaneH,lettingHandHbethetwohalfspacesinducedbyH,andsettingZ=Z\HandZ=Z\H.Three,avalidbranchingschemegivenZ=Z[ZcreatesK+1subproblems,whereinthek-thsubproblem,z1,...,zk)]TJ /F12 7.97 Tf 6.58 0 Td[(1areconstrainedtocomefromZandzk,...,zKmustcomefromZ.ObservethatallinterdictionsolutionsthatarenotfeasibletoanyoftheseK+1subproblemaresymmetric(viareindexingattacklocationsz1,...,zK)toasolutionthatisfeasibletooneofthesubproblems.Usingtheseconcepts,wemayperformamodiedbranch-and-boundsearchofZ.Webeginasifsolving( 4 )overz2Zandaninitialsetofg-functionsviastandardbranch-and-bound.Inthisalgorithm,asolution^z2Zwithobjectivevalue^mayberetainedasanincumbentsolutiononlyif(i)^isthesmallestobjectivevalueamongknownintegralsolutionsand(ii)(^z))]TJ /F3 11.955 Tf 13.26 0 Td[(^islessthansometolerance.If^zisrejectedasanincumbentsolutionduetoreason(ii),theproblemisdividedintoK+1subproblemsasdescribedabove.Inthiscase,thisparticularsolutionto( 4 )issuchthatforsomet?2[T],k?2[K],and(i?,j?)2A,wehavethatvi?j?=1,t?k?i?j?=1,andgt?k?i?j?di?j?(^zk?)
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willrequirethattheobjectivevalueat^zinthemodiedModel( 4 )matchesthetruemaximumowafterattack^z.Figures 4-4 and 4-5 illustratethebranch-and-reneprocedurediscussedinthepreviousparagraph.Inthisexample,q=K=2andZ=[0,4][0,4].Arc(i,j)hascapacityfunctionfij(d)=10(1)]TJ /F3 11.955 Tf 12.03 0 Td[(0.5d),wheredisthedistancefrom(2,1)themidpointofarc(i,j)tothenearestattack.Model( 4 )hasbeendenedusingT=1,sothatasinglelinearg-functionlower-boundsfij,i.e.,g11ij(d)=g12ij(d)=gij(d)=1.94d.Thesolution^z1=(0.9,2.6),^z2=(2.2,1.8)hasbeenidentiedasapotentialsolutionto( 4 ).Note,however,thatgij(d)
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Figure4-4. Networkforpartition-and-reneexample. Figure4-5. Renementofg-functionsforpartition-and-reneexample. describedinSection 4.4.1 ,isarestrictionof( 4 )andthereforeallfeasiblesolutionstothisproblemyieldupperboundson?E.InSection 4.4.2 ,weproposeamodicationofthismodelthatprovidesalowerboundon?Ebyrelaxingthecapacityfunctions.Section 4.4.3 describesamethodologybywhichthislowerboundcanberenediterativelytoobtaintighterrelaxations. 87

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4.4.1DiscretizedE-MFNIPModelDeneR=fz2Rq:LzUgandconsideramodelthatselectsKattacksfromamonganitesetofcandidatelocationsPR.SupposeinterdictionsareplacedatlocationsSP,wherejSj=K.From( 4 ),theresultingcapacityofarc(i,j)2Aisgivenbyfij[minp2Sdij(p)]=minp2Sfij[dij(p)],becausefij:R+!R+isnondecreasing.GivenP,valuescpijfij[dij(p)]canbecomputedaprioriforeachp2Pandtreatedasconstants;thus,theobjectiveistoselectSPthatminimizesthemaximumowoveragraphinwhichthecapacityofarc(i,j)isminp2Sfcpijg.WerefertothisdiscretizedproblemasDE-MFNIP.ToformulateDE-MFNIPasaMIP,letvariablesypequaloneiflocationp2Pisattackedandzerootherwise.They-variablesareconstrainedtocomefromthesetY=fy2f0,1gjPj:Pp2Pyp=Kg.Foreach(i,j)2A,denecijmaxp2Pfcpijgandobservethatthecapacityofarc(i,j)isminp2Pfcij)]TJ /F3 11.955 Tf 12.02 0 Td[((cij)]TJ /F4 11.955 Tf 12.02 0 Td[(cpij)ypg.Given^y2Y,thefollowersolves( 4 )withConstraints( 4c )replacedbyxijcij)]TJ /F3 11.955 Tf 12.03 0 Td[((cij)]TJ /F4 11.955 Tf 12.03 0 Td[(cpij)^yp,8p2P.DeneDE(^y)asthemaximumowresultingfrom^y2Y.ThevalueofDE(^y)canbeobtainedalternativelybysolvingthefollowingdualmodel,wherepijisthedualvariableassociatedwiththenewcapacityconstraints: DE(^y)=minX(i,j)2AXp2P[cij)]TJ /F3 11.955 Tf 11.95 0 Td[((cij)]TJ /F4 11.955 Tf 11.96 0 Td[(cpij)^yp]pij, (4a)s.t.(u,v)2U, (4b)Xp2Ppij=vij,8(i,j)2A. (4c)Model( 4 )isalinearprogram;thus,if( 4 )hasanoptimalsolution,itmusthaveonethatiscomplementaryslackwithanoptimalsolutiontoitsdual.Hence,addingtheconstraintspij^yp,8(i,j)2A,p2P,doesnotchangetheoptimalobjectivevalue.Thisrelationshipalsopermitsustosetpijyp=pij,becausepijypandyp2f0,1g.We 88

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nowfree^yasavariable(y)tomodelDE-MFNIPasthefollowingMIP: ?DE(P)=minX(i,j)2AXp2Pcpijpij, (4a)s.t.Xp2Ppij=vij,8(i,j)2A, (4b)(u,v)2U, (4c)0pijyp,8(i,j)2A,p2P, (4d)y2Y. (4e)Integerrestrictionsneednotbeplacedonvariablesu,v,andinModel( 4 ),asproveninthefollowingtheorem. Theorem4.2. ThereexistsanoptimalsolutiontoModel( 4 )suchthatallu-,v-,and-variablesarebinary-valued. Proof. Let^ybeanyelementofY.Considertheproblemthatresultsfrom( 4 )wheny=^yisxed.Denes1ands2asvectorsofslackvariablesonconstraints)]TJ /F4 11.955 Tf 9.3 0 Td[(ui+uj)]TJ /F4 11.955 Tf 11.23 0 Td[(vij0andpij1,respectively,andobservethatConstraints( 4b )( 4d )reducetoasystemoftheform 266664C)]TJ /F4 11.955 Tf 9.3 0 Td[(I0I00IB0000I0I3777750BBBBBBBBBB@uvs1s21CCCCCCCCCCA=0BBBB@0011CCCCA,(4)plusnonnegativityconstraintsonallvariablesexceptu.DeneeMasthecoefcientmatrixin( 4 ).ObservethatCisthetransposeofthenode-arcincidencematrixforthearcsetA,andisthereforetotallyunimodular(TU).NotethateachcolumnofBismadeupofallzerosexceptforthecoefcientcorrespondingtopij,correspondingtoaparticular(i,j)2Aandp2P,whichequals)]TJ /F3 11.955 Tf 9.3 0 Td[(1.AppendingCwithasequenceof 89

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rowsorcolumns,eachcontainingallzerosexceptforasingle1or)]TJ /F3 11.955 Tf 9.3 0 Td[(1,preservestotalunimodularity.Therefore,thematrices C)]TJ /F4 11.955 Tf 9.3 0 Td[(I,264C)]TJ /F4 11.955 Tf 9.3 0 Td[(I0I375,264C)]TJ /F4 11.955 Tf 9.3 0 Td[(I00IB375,and266664C)]TJ /F4 11.955 Tf 9.3 0 Td[(I00IB00I377775,(4)areallTUsubmatricesofeM,andeMisitselfTU.Becauseallofright-handsidevaluesin( 4 )areeither0or1,variablesu,v,andmustallbebinary-valuedinanyextremepointsolutionof( 4 )inwhichyisxedtoequal^y.Existenceofanoptimalextremepointsolutionisguaranteedbecausethisproblemisalinearprogram. Remark4.5. ProblemDE-MFNIPremainsNP-hardaswebrieyillustratehere.ConsideraDE-MFNIPinstancedenedoveranetworkG(N,A)inwhichtheelementsofAareorderedasA=f(i1,j1),...,(ijAj,jjAj)gwherei`2Nandj`2Ndenetheoriginanddestinationnodesofthe`-tharc.Notethatthenodesrepresentedbyi`,`2f1,...,jAjg,andj`,`2f1,...,jAjg,neednotbedistinctfromeachother.SupposePisgivenasP=fp1,...,pjAjgandcp`ijisgivenas cp`ij=8><>:0ifi=i`andj=j`cijotherwise,(4)for`2f1,...,jAjg.Inthiscase,( 4 )modelstheproblemofndingKarcsthat,whenremoved,resultinthesmallestmaximumow.ThisproblemisexactlyC-MFNIP,whichisNP-hard[ 8 ],implyingthatDE-MFNIPisNP-hardingeneral.DE-MFNIPturnsouttobeageneralizationofanotherNP-hardproblemaswell.Supposethatvij2f0,1gisxedforall(i,j)2Asuchthatarcsf(i,j)2A:vij=1gformsacutsetinG(disconnectingowfromnode1tonoden).Then,DE-MFNIPreducestothep-medianproblem[ 68 ]wherePisthesetofpotentialfacilitylocationsandf(i,j)2A:vij=1gisthesetofcustomers.2 90

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Remark4.6. Preliminarycomputationalexperienceindicatesthatthecontinuousrelaxationof( 4 )canbeweak.Wenowproposeaclassofvalidinequalitiesthatcouldbeusedtotightenthisrelaxation.WesaythatarcsubseteAAisa(directed)pathifthereexistsasequenceofnodesfi`gr`=1f2,...,n)]TJ /F3 11.955 Tf 13.28 0 Td[(1gsuchthateA=f(1,i1),(ir,n)g[hSr)]TJ /F12 7.97 Tf 6.59 0 Td[(1`=1f(i`,i`+1)gi.ForapatheA,theinequalityweproposeprovidesalowerboundontheportionoftheobjective( 4a )sumarisingfromarcsineA.Tothisend,consideraxed^y2Yanddene^P=fp2P:^yp=1g.Let?eA(^y)denotetheoptimalobjectivevaluetoModel( 4 )whenAisreplacedbyeAandyisxedto^y,i.e., ?eA(^y)=minX(i,j)2eAXp2^Pcpijpij, (4a)s.t.Xp2^Ppij=vij,8(i,j)2eA, (4b)0pij1,8(i,j)2eA,p2^P, (4c)(u,v)2U.BecauseeAA,thefeasibleregionof( 4 )containsthesetofsolutionsfeasibleto( 4 )wheny=^y.Hence,if:Y!Risanyfunctionsuchthat?eA(y)(y)forally2Y,theinequalityP(i,j)2eAPp2Pcpijpij(y)mustbevalidfor( 4 ).BecauseeAisapath,theconstraintset(u,v)2Uin( 4 )reduces(viaaggregatingconstraintsui)]TJ /F4 11.955 Tf 12.29 0 Td[(uj+vij0)toP(i,j)2eAvij=1.DuetoConstraint( 4b ),(u,v)2Umaybereplacedin( 4 )by X(i,j)2eAXp2^Ppij=1.(4)Afterdoingthis,notethatwhenever( 4c )and( 4 )aresatised,thereexistvij,(i,j)2eA,suchthat( 4b )issatised.Therefore,( 4b )maybedroppedfromModel( 4 ).Consequently,solutionofModel( 4 )istrivial,andtheoptimalobjective 91

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valueisgivenby ?eA(^y)=minp2^Pp,(4)wherepmin(i,j)2eAcpij,8p2P.Thus,usingthefunction(y)=minp2P:yp=1pproducesthe(nonlinear)validinequality X(i,j)2eAXp2Pcpijpijminp2P:yp=1p.(4)Wenowlinearize( 4 )byintroducingauxiliarybinaryvariables.Tothisend,wesorttheelementsofpaccordingtop:Letfp<`>gjPj`=1denoteuniqueelementsofPsuchthatp<1>p.Denevariables`,`=1,...,jPj,toequal1ifp<`>isselectedastheminimumin( 4 )and0otherwise,i.e., `=8><>:1ifyp<1>==yp<`)]TJ /F12 7.97 Tf 6.59 0 Td[(1>=0andyp<`>=1,0otherwise.(4)Duetothecardinalityrestrictionimposedony2Y,itfollowsthatjPj)]TJ /F7 7.97 Tf 8.94 0 Td[(K+2=jPj)]TJ /F7 7.97 Tf 8.94 0 Td[(K+3==jPj=0andjPj)]TJ /F7 7.97 Tf 8.94 0 Td[(K+1=1)]TJ /F14 11.955 Tf 12.02 8.97 Td[(PjPj)]TJ /F7 7.97 Tf 8.94 0 Td[(K`=1`.Moreover,therelationshipbetweenandyiscapturedbythelinearconstraints `yp<`>,8`=1,...,jPj)]TJ /F4 11.955 Tf 17.93 0 Td[(K, (4a)`1)]TJ /F4 11.955 Tf 11.95 0 Td[(yp<`0>,8`=2,...,jPj)]TJ /F4 11.955 Tf 17.94 0 Td[(K,`0=1,...,`,and (4b)`yp<`>)]TJ /F14 11.955 Tf 11.95 11.36 Td[(X`0<`yp<`0>,8`=1,...,jPj)]TJ /F4 11.955 Tf 17.94 0 Td[(K. (4c)Usingthe-variables,( 4 )canberepresentedas X(i,j)2eAXp2PcpijpijjPj)]TJ /F7 7.97 Tf 8.94 0 Td[(K+1+jPj)]TJ /F7 7.97 Tf 8.94 0 Td[(KX`=1(p<`>)]TJ /F5 11.955 Tf 11.95 0 Td[(p)`.(4)Model( 4 )cannowbetightenedbyadding( 4 )alongwith( 4 )andbinaryrestrictionsonthe-variables.However,thisimprovementtothemodelcomesattheexpenseofaddingO(jPj)binaryvariablesandO(jPj2)constraints.Becauseofthe 92

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Figure4-6. Functionfij(d)lower-boundedbyrelaxedfunctionfij[maxf0,d)]TJ /F4 11.955 Tf 11.95 0 Td[(ag]. addedcomplexity,wesuggestusingaweakerversionof( 4 )thatincludesonly
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When(p)=0,8p2P,DE(P,)isidenticaltoModel( 4 );thus,?DE(P)=?DE(P,0). Theorem4.3. ForPR,?DE(P)?E. Proof. Clear,duetothefactthatPR. DE-MFNIPisequivalenttotheversionofE-MFNIPthatwouldresultfromrestrictingeachzktocomefromPinsteadoffromR.When(p)>0,wehavethatcpij(0)cpij((p)).Thisraisesthefollowingquestion:Howlargemust(p)bebeforewecanguaranteethat?E?DE(P,)?Toaddressthisquestion,forp2Rqanda0,deneB(p,a)astheballaroundpwithradiusa,i.e.,B(p,a)=fp2Rq:kp)]TJ /F3 11.955 Tf 12.17 0 Td[(pk1ag.For^PPand:^P!R+,deneB(^P,)=[p2^PB(p,(p)). Theorem4.4. SupposePisanitesubsetofRand(p)isgivenforeachp2P.IfRB(P,),then?E?DE(P,). Proof. Let(^z,^u,^v)denoteanoptimalsolutionto( 4 )inwhich^visbinary-valued.Fork2[K],let^pk2Pbeapointsuchthat^pk)]TJ /F3 11.955 Tf 11.67 0 Td[(^zk1isminimized,andnotethat^pk)]TJ /F3 11.955 Tf 11.67 0 Td[(^zk1(^pk)byhypothesis.Dene^A=f(i,j)2A:^vij=1g.Then ?E=X(i,j)2^Aij(^z)=X(i,j)2^Afij[mink2[K]dij(^zk)] (4a)=X(i,j)2^Amink2[K]fij[dij(^zk)] (4b)X(i,j)2^Amink2[K]fij[maxf0,dij(^pk))]TJ /F5 11.955 Tf 11.96 0 Td[((^pk)g] (4c)=X(i,j)2^Amink2[K]c^pkij((^pk)), (4d)where( 4b )followsbecausefijisnondecreasingand( 4c )followsasaresultofthetriangleinequalitydij(^zk)+^pk)]TJ /F3 11.955 Tf 11.66 0 Td[(^zk1dij(^pk).NowconsiderthesolutiontoDE(P,)inwhichu=^u,v=^v,andallothervariablesareequaltozeroexceptfor y^pk=1,8k2[K], 94

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yp=1,arbitrarilyforK)]TJ /F4 11.955 Tf 11.65 0 Td[(radditionalp2P,whereristhenumberofuniquepointsinf^p1,...,^pKg. ^pkij=1correspondingtoakthatminimizesdij(^zk)overk2[K],8(i,j)2^A.ThissolutionisfeasibletoDE(P,)withobjectivevalueequalto( 4d ),andtherefore?E?DE(P,). CombiningtheresultsofTheorems 4.3 and 4.4 ,wehavethat?DE(P,)?E?DE(P)foranyandPsuchthatRB(P,).Undermodestassumptionsonthef-functions,Pandmaybechosentoguaranteethat?E)]TJ /F5 11.955 Tf 12.73 0 Td[(?DE(P,)isarbitrarilysmall,asproveninthefollowingtheorem. Theorem4.5. LetPR,:P!R+,anda>0begivensuchthat(p)a,8p2P.Foreach(i,j)2A,supposefijisLipschitzcontinuousanddeneij0asthecorrespondingLipschitzconstant.Then?DE(P,)?E)]TJ /F4 11.955 Tf 12.25 0 Td[(a?,where?isaconstantthatdependsonlyonthe-valuesandthenetwork'sstructure. Proof. Byassumption,wehavethat fij(d2))]TJ /F4 11.955 Tf 11.96 0 Td[(fij(d1) d2)]TJ /F4 11.955 Tf 11.96 0 Td[(d1ij,80d1
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Asshownin[ 8 ],thereexistsanoptimalsolutionto( 4 )inwhichallu-andv-variablestakebinaryvalues.Moreover,givenanyu2f0,1gn,vij=1maybeoptimalonlyifui=0anduj=0.(Otherwisevij=0isfeasibleandhasasmallerobjectivevaluebecauseij(z)0,8z2Z.)Thus,itisvalidtoreplaceUin( 4 )with UU\8>>>>>>><>>>>>>>:(u,v)vijuj,8(i,j)2Avij1)]TJ /F4 11.955 Tf 11.96 0 Td[(ui,8(i,j)2Au2f0,1gnv2f0,1gjAj9>>>>>>>=>>>>>>>;.(4)Dene(p)=a,8p2Pandobservethat?DE(P,)?DE(P,)?E(a),wherethesecondinequalityfollowsbecauseDE-MFNIPisarestrictionofE-MFNIP.Therefore,itfollowsthat ?DE(P,)?E(a) (4a)=min(u,v)2U,z2ZX(i,j)2Amink2[K]fijmaxf0,dij(zk))]TJ /F4 11.955 Tf 11.95 0 Td[(agvij (4b)min(u,v)2U,z2ZX(i,j)2Amink2[K]maxffij(0),fij(dij(zk))]TJ /F4 11.955 Tf 11.95 0 Td[(aijgvij (4c)min(u,v)2U,z2ZX(i,j)2Amink2[K]maxffij(0)+aij,fij(dij(zk)gvij (4d)+min(u,v)2U,z2ZX(i,j)2A()]TJ /F4 11.955 Tf 9.3 0 Td[(aij)vij (4e)?E)]TJ /F4 11.955 Tf 11.96 0 Td[(a?, (4f)where(i)( 4c )followsfrom( 4 )becausefijisnondecreasing,and(ii)boundednessofUguaranteesanoptimalsolution(withobjectivevalue?)totheminimizationproblemin( 4e ). Theorem 4.5 guaranteesthat,withalargeenoughP-setandsmallenough-values,thelowerboundprovidedby?DE(P,)isarbitrarilycloseto?E.However,obtainingalowerboundinthisfashionisoftenimpracticalbecausethedifculty 96

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assoicatedwithsolvingDE-MFNIPgrowsquicklyasPincreases.InSection 4.4.3 ,weproposeanalternativetechniquefordevelopingtightlowerboundsbasedoniterativelybuildingthesetP.WenowproveconditionsunderwhichmodicationofthesetPguaranteesanincreasein?DE(P,). Theorem4.6. LetP1andP2bedisjointsubsetsofRwithanitenumberofelements,andletp?2Rn(P1[P2).Dene:P1[fp?g[P2!R+andsupposeRB(P1[fp?g,).IfB(P2,)=B(p?,(p?)),then?DE(P1[fp?g,)?DE(P1[P2,)?E. Proof. BecauseB(p?,(p?))B(P2,)andRB(P1[fp?g,),itfollowsthatRB(P1[P2,).Therefore,?DE(P1[P2,)?EfollowsfromTheorem 4.4 .Wenowshowthat?DE(P1[fp?g,)?DE(P1[P2,).Tothisend,weshowthatcp?ij((p?))cpij((p))foreveryp2P2.Ifdij(p)(p),thisistriviallytruebecausep(i,j)2B(p,(p))B(p?,(p?))andthereforecp?ij((p?))=cpij((p))=0.Supposeinsteadthatdij(p)>(p).Dene^pij=((p)=dij(p))p+(1)]TJ /F5 11.955 Tf 12.58 0 Td[((p)=dij(p))p(i,j)andobservethatp(i,j))]TJ /F3 11.955 Tf 12.17 0 Td[(^pij1=dij(p))]TJ /F5 11.955 Tf 12.77 0 Td[((p).Moreover,^pij2B(p?,(p?))becauseB(P2,)B(p?,(p?)),andtherefore dij(p?)^pij)]TJ /F29 11.955 Tf 11.96 0 Td[(p?1+p(i,j))]TJ /F3 11.955 Tf 12.17 0 Td[(^pij1 (4a)(p?)+dij(p))]TJ /F5 11.955 Tf 11.95 0 Td[((p), (4b)followsfromthetriangleinequality.Hence,maxf0,dij(p?))]TJ /F5 11.955 Tf 9.38 0 Td[((p?)gmaxf0,dij(p))]TJ /F5 11.955 Tf 9.38 0 Td[((p)g,andbecausefijisnondecreasing,itfollowsthatcp?ij((p?))cpij((p)).Now,let(^y,^u,^v,^)denoteanoptimalsolutiontoDE(P1[P2,)inwhichallvariablesareintegral.(Theorem 4.2 guaranteestheexistenceofsuchasolution.)Wenowconstructasolution(~u,~v,~y,~)toDE(P1[fp?g)asfollows.Dene~u=^uand~v=^vandobservethat(~u,~v)2U.Dene~yas: ~yp=8><>:^ypifp2P1maxp2P2f^ypgifp=p?.(4) 97

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IfPp2P1[fp?g~yp<>:^pijifp2P1Pp2P2^pijifp=p?.(4)Notethat~p?ij2f0,1gbecausePp2P2^pij^vij1,and( 4d )issatisedforp2P1because^and^yarefeasiblefor( 4 ).Whenp=p?,thisconstraintisalsosatisedbecause^pij^ypforallp2P2impliesthateithermaxp2P2f^ypg=1orPp2P2^pij=0.Next,weprovethat( 4b )issatisedeachoftwocases:~p?ij=0(case1)and~p?ij=1(case2).Incase1,wehavethat^pij=0,8p2P2.Inthiscase,wendthatPp2P1^pij=^vij=~vij.Therefore,Pp2P1[fp?g~pij=~vijand( 4b )issatised.Incase2,wehavethat^pij=1forsomep2P2,whichimpliesthatPp2P1^pij=0and^vij=~vij=1.Thus,Pp2P1[fp?g~pij=~vijand( 4b )issatised.Wehavenowshownthat(~u,~v,~y,~)satisesalloftheconstraintsofDE(P1[fp?g,).Therefore,?DE(P1[fp?g,)canbeupper-boundedbytheobjectivevaluecorrespondingto(~y,~u,~v,~): ?DE(P1[fp?g,)X(i,j)2A0@Xp2P1cpij((p))~pij+cp?ij((p?))~p?ij1A (4a)X(i,j)2A0@Xp2P1cpij((p))^pij+Xp2P2cpij((p))^pij1A (4b)=?DE(P1[P2,), (4c)where( 4b )followsbecause:(i)^pij=1foratmostonep2P2foreach(i,j)2A;and(ii)cp?ij((p?))cpij((p))forallp2P2and(i,j)2A.Combiningtheaboveresults,wehavethat?DE(P1[fp?g,)?DE(P1[P2,)?E,completingtheproof. 98

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AnimplicationofTheorem 4.6 isthatthecontentsofPcanbemodiedinsuchawaythatguaranteesanincreasein?DE(P,)withoutlosingthepropertythat?DE(P,)?E.ThisresultgivesrisetoamethodologyforsolvingE-MFNIP,whichwedescribeinthefollowingsection. 4.4.3Discretize-and-ReneSolutionMethodologyWenowdescribehow( 4 )canbeusedtosolveinstancesofE-MFNIP.Themethodsdescribedinthissectionarecorrectforgeneralq,butourfocusinthissectionisonproblemsinwhichq=2(e.g.,aswouldbethecaseinpowergrid,telecommunicationsnetwork,andtransportationsettings).First,specifyaninitialsetofpointsP0Randvalues0(p)2R+foreachp2PsuchthatRB(P0,0).Aninitiallowerboundfor?EcanbeobtainedbysolvingDE(P0,0).Solvingthisproblemrevealsaninitialsetofattacklocationsthatareoptimalfortherelaxedproblem.Next,setP0ismodied(andrenamedP1)toconsiderahigherdensityofcandidatelocationssurroundinglocationsthatwereoptimalforDE(P0,0).Corresponding1-valuesareassignedtothenewelementsofP1inaccordancewiththeassumptionsofTheorem 4.6 sothat?DE(P0,0)?DE(P1,1).Inthisfashion,thelowerboundisiterativelyimproveduntilitiswithinsomeacceptabletolerancegapofaknownupperbound.Upperboundsfor?Eareobtainedintwoways:(i)bysolvingDE(P,0)foranyP(seeTheorem 4.3 ),and(ii)byevaluatingtheobjectivevalueofanyfeasiblesolutionto( 4 ).RecallfromSection 4.1 that([z1,...,zK])denotesthemaximumowresultingfromattacksatlocationsz1,...,zK.Wenowprovideaformalalgorithmicdescriptionofourdiscretize-and-renesolutionalgorithm. Step1. SelectP0Rand0(p),8p2P0,suchthatRB(P0,0).SetUB=1anditerationcounters=0.Let">0beagiventoleranceparameter. Step2. SolveDE(Ps,s)andobtainanoptimalsolution(^y,^u,^v,^).Dene^P=fp2Ps:^yp=1g,andsupposetheelementsof^Pare^p1,...,^pK.Compute?DE(Ps,s),?DE(Ps),ands([^p1j...j^pr]). 99

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Step3. Ifminfs,?DE(Ps)g0isthesmallestvaluesuchthatRB(P0,r).Note,however,thatthelowerboundprovidedby( 4 )isgenerallytighterwhenissmaller.InordertominimizethevalueofrneededtoensurethatRB(P0,r),oneshouldselecttheelementsofP0tobesomehowevenlyspaced.Viewedfromanotherperspective,forxedr>0,oneshouldconstructP0usingthefewestpossiblepointssothatRB(P0,r).Forq=2andR=fz:LzUg,selectingP0=fL)]TJ /F4 11.955 Tf 10.81 0 Td[(r
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Figure4-7. Testnetwork. bycomparingthequalityofE-MFNIPsolutionsonatestnetworktosolutionswhichcouldbeobtainedbysolvingtheMFNIPmodelof[ 8 ].Figure 4-7 displaysanetworkwithsixsourcenodes,sixsinknodes,16intermediatenodes,and67undirectededgesrepresentedusing267=134directedarcs.Thenetworkissetinsideaten-by-tentwo-dimensionalsquare.Eachedge(i,j)isassignedanominalcapacitycij2f1,...,10gandthecapacityofedge(i,j)iscomputedascij(1)]TJ /F4 11.955 Tf 12.25 0 Td[(bd)wheredisthedistancetothenearestattackand0
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Figure4-8. Comparisonofobjectivevaluesasbisvaried. EachE-MFNIPinstancewassolvedtowithin2%ofoptimality.Figure 4-8 plotsthetheresultingE-MFNIPobjectivevaluesagainstb.Asanticipated,thegapbetweenE-MFNIPandMFNIPincreasesasbincreases.Whenb=0.9,theE-MFNIPsolutionresultsinamaximumowvaluethatis29%lessthantheowresultingfromtheMFNIPsolution.Figure 4-9 displaystheoptimalE-MFNIPsolutions(representingattacklocationswithastar)forb=0.01,0.1,0.3,...,0.9.TheMFNIPsolutionisnotpicturedbecauseitisalmostidenticaltotheE-MFNIPsolutionforb=0.01.Asbincreases,theoptimalattacklocationsmovefromaclosely-packedformationtoamorespreadoutformation.Thisbehaviorhasanintuitiveexplanation:Whenbislarge,attackshaveasignicanteffectonalargerrangeofdistancesmeaningthatapairofattacksmayhaveadiminishedeffectduetoredundancyiflocatedtooclosetogether.WenowcomparethediscretizedmethodsofSection 4.4 withthelower-boundingmodelofSection 4.3 .AllofthecomputationalrunsinthisresultsetwereexecutedonaDellPowerEdge2600machinewithtwoPentium43.2GHz/1Mcacheprocessors,usingCPLEX12.1tosolvealloftheintegerprogrammingmodels.Thediscretize-and-reneprocedure,codedinC++,usestheConcertTechnology2.9librarytosolveasequence 102

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Figure4-9. Plotofattacklocationsascapacityparameterbisvaried. ofintegerprogramswiththesameversionofCPLEX.Networkswereconstructedbyrandomlylocatingnodesinsideatenbytensquareandthenaddinganedgebetweeneachpairofnodeswithacertainprobability.Thecapacityofeachedge(i,j)iscomputedasintheexamplenetworkofFigure 4-7 usingb=0.5.Wenowdescribeinfurtherdetailthesix-stepprocessusedtogeneratethesenetworks.One,parametersn2Z+,S2Z+,and1arespeciedaprioritocontrolthenumberofnodes,numberofsourcesandsinks,andthearcdensity,respectively.Two,thetwo-dimensionalsquareR=fz2R2:0z110,0z210gisdividedintoSequallysizedbands,B=fz2R2:0z110,()]TJ /F3 11.955 Tf 11.88 0 Td[(1)(S=10)z2(S=10)g,2[S].Three,nodelocationsp(i),i2[n]arechosenuniformlyinRsuchthatatleasttwonodesarelocatedinBforeach2[S].Four,foreach2[S],theleft-mostnodeinB(i.e.,theonewiththesmallestz1-coordinate)isdesignatedasasourcenode,andtheright-mostnodeisdesignatedasasink.Five,foreach2[S],edgesareconstructedto 103

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Table4-1. Randomnetworkgenerationproles. ProlenSArcs A2861.7578B28101.7558C2861.40130D28101.40140E4362.00134F43102.00116G4361.60228H43101.60200 connectallofthenodeslocatedinBbyasinglepaththatpassesthrougheachofthenodesfromlefttoright.Six,foreachi,j2[n]suchthati
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4. Discretize-and-rene:WeusetheapproachdescribedinSection 4.4.3 ,whereP0and0arechosenaccordingtoRemark 4.7 usingr=1.Wereport OandO ,thebestknownupperandlowerbounds,respectively.Discretize-and-reneinstancesweresolvedtowithinonepercentofoptimalityorterminatedaftertwohours.Foreachinstance,wereportthesolutiontimeT(s)(ortheoptimalitygapforinstancesnotcompletewithintwohours).Wereport1,[L:T1]and[L:T2]failtoproduceastrictlypositivelowerboundwithintwohours.Thismaybeduetotheweakness 105

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Figure4-10. Upperandlowerboundsobtainedateachiterationofdiscretize-and-reneforFnetworkwithK=5attacks. Figure4-11. Solutiontimesforthreesubroutinesateachiterationofdiscretize-and-rene. ofModel( 4 )resultingfromtheinclusionofmultiplelevelsofbigMconstraints.Algorithms[L:r1.0],[L:r0.5],[U:r1.0],[U:r0.5],and[U:mp]allsingle-solveimplementationsofModel( 4 )consistentlysolvewithinsecondsandprovidemuchtighterbounds,eveninthecases([L:r1.0]and[U:r1.0])thatusethesmallestP-set.Algorithm[U:mp]producesthebestsolutionamongallalgorithmsin27of32instances,suggestingthatsolvingModel( 4 ),withPdenedasthesetofarcmidpoints,maybean 106

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effectiveheuristicmethodforidentifyingE-MFNIPsolutions.However,wecautionfutureresearchersagainstrelyingsolelyupon[U:mp]tosolveE-MFNIPbecause(i)[U:mp]providesnolowerboundandthereforenoguaranteeofoptimality,and(ii)ourexperiencesuggeststhatthequalityof[U:mp]solutionsisheavilydependentuponthetypeoff-functionsused.Forinstance,consideranynetworkinwhichthedistancebetweenanytwoarcmidpointsisatleastr>0andthecapacityfunctionsaregivenas fij(d)=8><>:0drcijd>r,(4)forall(i,j)2A.Thus,if(i,j)and(i0,j0)areuniquearcssuchthatr
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Table4-2. ComputationalresultsfornetworksAthroughD.[L:T1][L:T2][L:r1.0][L:r0.5][U:r1.0][U:r0.5][U:mp][D&R] Net.KT(s)O T(s)O T(s)O T(s)O T(s) OT(s) OT(s) OT(s)O O A1<6.497420.5<18.8120.6<28.0125.7<23.86123.623.8332262.51428%1.70<4.3017.28<17.8116.3<10.820010.710.850.000.00<0.6132.26<14.9411.5<4.467894.464.4970.000.00<0.0010.00114.029.18<0.0011420.000.00B1<20.64453.5<48.6455.9<62.2262.2161.321560.761.4344.3%6.51273%8.99<20.8231.3<47.7245.1139.9278339.640.050.000.00<12.7220.0<40.9335.7228.3533928.128.470.000.00<6.59312.2<35.4227.8118.6111118.618.7C1725.336.7%47.5162.4466.9<81.2477.1674.215773.674.33339%2.390.13122.9233.8160.6151.5245.7405445.445.850.000.00<11.2520.7150.9341.3231.41.13%31.131.470.000.00<5.25511.8<45.2133.8721.83.21%21.221.9D1246.039.4%97.12131.013143.61161.112158.55157.34846155.7157.33281%6.890.31267.41191.0<121.53120.813116.75.34%110.9116.850.000.00238.41362.9<104.1399.5492.210.2%84.192.770.000.00122.51143.9<91.4285.0173.611.8%66.974.7 108

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Table4-3. ComputationalresultsfornetworksEthroughH.[L:T1][L:T2][L:r1.0][L:r0.5][U:r1.0][U:r0.5][U:mp][D&R] Net.KT(s)O T(s)O T(s)O T(s)O T(s) OT(s) OT(s) OT(s)O O E178.92676624.3121.1823.3<29.0327.41226.630826.526.73449%0.480.00<5.4518.69119.2115.7211.7150411.611.750.000.00<0.7822.72114.6110.313.397383.393.4270.000.00<0.0010.14112.618.6310.80834720.8060.810F11315.123.2%38.7139.5747.2154.5653.9352.6203652.252.63726%0.881.43119.4624.1140.2438.5733.21.92%32.733.350.000.00<6.83212.0<33.2<28.4120.3667420.220.470.000.00<2.92157.25129.3324.5113.8487513.713.8G14716.4296%12.1238.31045.5<57.11256.64955.8382654.655.130.000.00212.0719.9140.7638.62133.84.89%32.133.750.000.0023.0847.14<30.3324.7917.711.4%15.817.670.000.0010.0032.39<26.8218.739.2613.5%8.619.77H14625.264.6%43.71068.81675.0385.91684.02784.1706083.083.830.000.00227.01538.8162.71257.32655.27.67%51.655.650.000.00413.6323.4151.8343.9638.45.76%36.738.870.000.0026.661614.9145.71138.42829.916.34%27.231.6 109

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CHAPTER5CONCLUSIONInthisdissertation,weperformin-depthanalysisofseveralinterdictionproblems.InChapter 2 ,weextendtheBiPSNIPmodelof[ 32 ]inanumberofways,beginningwithareformulationoftheproblemthathasfewerconstraintsandhasatightercontinuousrelaxation.Wedevelopaclassoffacet-deninginequalitiesandacorrespondingseparationprocedure.TheseinequalitiesgeneralizetheBiSNIPinequalitiesgivenin[ 32 ],andtheyappeartosignicantlyimprovecomputationalefciency.WesolveaninstancederivedfromtheU.S.roadnetworkanddiscusstheresults.Thedifcultyoftheprobleminstancesinvestigatedherelendevidencetothenecessityofdevelopingeffectiveheuristictechniquesforthisclassofproblems.Anotherinterestingfollow-oninvestigationmayseektondapproximationalgorithmsthatyieldhigh-qualitysolutionsusingreasonablecomputationaleffort.InChapter 3 ,weanalyzefurtheranalyzetheBiPSNIPmodelbycharacterizingtheconvexhullofinteger-feasiblesolutionstothesingle-scenariocardinality-constrainedproblem.Thispolyhedronhasanexponentialnumberoffacetsthatariseduetotheinclusionofaconstraintlimitingthenumberofinterdictions.Thecutsdeningthesingle-scenariopolyhedronarethenreconsideredfromtheperspectiveofsolvingamultiple-scenarioproblem.ConsistentwiththeNP-hardnessresultfrom[ 51 ],wearguethatthemultiple-scenariopolyhedronmaycontainfacetsthataredifculttoseparate;however,thesingle-scenariocutsarestillabletoimprovetheLPrelaxationsignicantlywhenBislargerelativeton.Ourexperienceisthatthemultiple-scenariopolyhedronvariesdrasticallydependingontherelativeorderingofp-valuesinthedifferentscenarios.Aninterestingexperimentcouldcompareinstancesinwhichthep-valuesfromdifferentscenarioswerecorrelated,andreportresultssimilartothosegiveninTable 3-2 .Moreover,althoughtheone-scenario 110

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cutsclearlytightentheLPrelaxationinmanyinstances,efcientimplementationprocedureshavenotbeenstudiedextensively.Weleavethesestudiesforfuturework.InChapter 4 ,wedevelopanewmodelformaximumowinterdictionwhichexplicitlymodelsthegeometricrelationshipbetweenanattackandnetworkcomponents.Theproblemwemodel,denotedE-MFNIP,generalizesthetraditionaldiscretemaximumowinterdictionproblem,whichisknowntobeNP-hard.Duetothestructureofthecapacityfunctionsweuse,existenceofacompactsingle-stageintegerlinearprogrammingformulationisunlikelysowedevelopalternatesolutionapproacheswhichfallintwoclasses:(i)methodsfromSection 4.3 thatusecontinuousvariablestoselectattackcoordinates,and(ii)methodsderivedfromsolvingadiscretizedversionofE-MFNIP,denotedDE-MFNIP.ComputationalresultsindicatethatthemethodsderivedfromDE-MFNIParemuchmoreeffectiveinobtaininghigh-qualitysolutionstoE-MFNIPthantheSection 4.3 methodsandaremuchfaster,too.Amongthediscretization-basedapproacheswedevelop,thecentralcontributionsofthischapterareDE-MFNIPmodelsthatgiveupperandlowerboundsforE-MFNIP.Weshowhowthesemodelscanbecombinedintoadiscretize-and-renealgorithmtoobtainoptimalsolutionsforE-MFNIP.Thisalgorithmsuccessfullyidentiesoptimalsolutions,butrequiresiterativesolutionofDE-MFNIPinstancesthatbecomeprogressivelymoredifculttosolve.Mostoftheeffortassociatedwithdiscretize-and-renecomesfromimprovingthelowerboundthatultimatelyprovidestheguaranteeofoptimality,suggestingthatheuristicapproachesbasedonsolvingDE-MFNIPmayalsobeeffective.BasedontheanalysisinChapter 4 ,manyofthemosteffectivemethodsforsolvingE-MFNIParebasedonsolvingDE-MFNIP.Thus,futureresearchdevelopingimprovedsolutiontechniquesforDE-MFNIPcouldbeofbenetinsolvingE-MFNIP.Weidentifysomepotentialdiscretization-basedE-MFNIPheuristics,butothertypesofheuristicsandapproximationscouldproveusefulaswell.Additionalinvestigationsmay 111

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seektondbestimplementationsofthealgorithmsweproposeaswellasextendourtechniquestodifferenttypesofdistanceandcapacityfunctions. 112

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REFERENCES [1] R.D.Wollmer,Removingarcsfromanetwork,OperationsResearch,vol.12,pp.934,1964. [2] A.McMastersandT.Mustin,Optimalinterdictionofasupplynetwork,NavalResearchLogisticsQuarterly,vol.17,pp.261,1970. [3] UnitedNations.(2012,June)Treatyonthenon-proliferationofnuclearweapons.[Online].Available: http://www.un.org/disarmament/WMD/Nuclear/NPT.shtml [4] S.H.Lubore,H.D.Ratliff,andG.T.Sicilia,Determiningthemostvitallinkinaownetwork,NavalResearchLogisticsQuarterly,vol.18,pp.497,1971. [5] P.M.Ghare,D.C.Montgomery,andW.C.Turner,Optimalinterdictionpolicyforaownetwork,NavalResearchLogisticsQuarterly,vol.18,pp.37,1971. [6] H.D.Ratliff,G.T.Sicilia,andS.H.Lubore,Findingthenmostvitallinksinownetworks,ManagementScience,vol.21,pp.531,1975. [7] H.W.CorleyandH.Chang,Findingthenmostvitalnodesinaownetwork,ManagementScience,vol.21,pp.362,1974. [8] R.K.Wood,Deterministicnetworkinterdiction,MathematicalandComputerModeling,vol.17,pp.1,1993. [9] C.A.Phillips,Thenetworkinhibitionproblem,inProceedingsofthe25thAnnualACMSymposiumontheTheoryofComputing,1993,pp.776. [10] J.O.RoysetandR.K.Wood,Solvingthebi-objectivemaximum-ownetwork-interdictionproblem,INFORMSJournalonComputing,vol.19,pp.175,2007. [11] D.S.Altner,O.Ergun,andN.A.Uhan,Themaximumownetworkinterdictionproblem:Validinequalities,integralitygaps,andapproximability,OperationsResearchLetters,vol.38,pp.33,2010. [12] D.FulkersonandG.Harding,Maximizingtheminimumsource-sinkpathsubjecttoabudgetconstraint,MathematicalProgramming,vol.13,pp.116,1977. [13] B.Golden,Aprobleminnetworkinterdiction,NavalResearchLogisticsQuarterly,vol.25,pp.711,1978. [14] M.O.Ball,B.L.Golden,andR.V.Vohra,Findingthemostvitalarcsinanetwork,OperationsResearchLetters,vol.8,pp.73,1989. [15] H.W.CorleyandD.Y.Sha,Mostvitallinksandnodesinweightednetworks,OperationsResearchLetters,vol.1,pp.157,1982. 113

PAGE 114

[16] K.Malik,A.Mittal,andS.Gupta,Thekmostvitalarcsintheshortestpathproblem,OperationsResearchLetters,vol.8,pp.223,1989. [17] A.Bar-Noy,S.Khuller,andB.Schieber,Thecomplexityofndingmostvitalarcsandnodes,UniversityofMaryland,InstituteofAdvancedComputerStudies,CollegePark,MD,Tech.Rep.,1995. [18] L.Khachiyan,E.Boros,K.Borys,K.Elbassioni,V.Gurvich,G.Rudolf,andJ.Zhao,Onshortpathsinterdictionproblems:Totalandnode-wiselimitedinterdiction,TheoryofComputingSystems,vol.43,pp.204,2008. [19] E.IsraeliandR.K.Wood,Shortest-pathnetworkinterdiction,Networks,vol.40,pp.97,2002. [20] R.D.Wollmer,Algorithmsfortargetingstrikesinalines-of-communicationnetwork,OperationsResearch,vol.18,pp.497,1970. [21] C.LimandJ.C.Smith,Algorithmsfordiscreteandcontinuousmulticommodityownetworkinterdictionproblems,IIETransactions,vol.39,pp.15,2007. [22] J.C.Smith,C.Lim,andF.Sudargho,Survivablenetworkdesignunderoptimalandheuristicinterdictionscenarios,JournalofGlobalOptimization,vol.38,pp.181,2007. [23] R.L.Church,M.P.Scaparra,andR.S.Middleton,Identifyingcriticalinfrastructure:Themedianandcoveringfacilityinterdictionproblems,AnnalsoftheAssociationofAmericanGeographers,vol.94,pp.491,2004. [24] R.L.ChurchandM.P.Scaparra,Ther-interdictionmedianproblemwithfortication,GeographicalAnalysis,vol.39,pp.129,2007. [25] F.Liberatore,M.P.Scaparra,andM.S.Daskin,Analysisoffacilityprotectionstrategiesagainstanuncertainnumberofattacks:Thestochasticr-interdictionmedianproblemwithfortication,ComputersandOperationsResearch,vol.38,pp.357,2011. [26] M.P.ScaparraandR.L.Church,Abilevelmixed-integerprogramforcriticalinfrastructureprotectionplanning,ComputersandOperationsResearch,vol.35,pp.1905,2008. [27] ,Anexactsolutionapproachfortheinterdictionmedianproblemwithfortication,EuropeanJournalofOperationsResearch,vol.189,pp.76,2008. [28] K.J.Cormican,D.P.Morton,andR.K.Wood,Stochasticnetworkinterdiction,OperationsResearch,vol.46,pp.184,1998. [29] U.JanjarassukandJ.T.Linderoth,Reformulationandsamplingtosolveastochasticnetworkinterdictionproblem,Networks,vol.52,pp.120,2008. 114

PAGE 115

[30] O.Berman,D.Krass,andC.W.Xu,Locatingow-interceptingfacilities:newapproachesandresults,AnnalsofOperationsResearch,vol.60,pp.121,1995. [31] A.Gutfraind,A.Hagberg,andF.Pan,OptimalinterdictionofunreactiveMarkovianevaders,inProceedingsofthe6thInternationalConferenceonIntegrationofAIandORTechniquesinConstraintProgrammingforCombinatorialOptimizationProblems,Springer,Berlin,2009,pp.102. [32] D.P.Morton,F.Pan,andK.J.Saeger,Modelsfornuclearsmugglinginterdiction,IIETransactions,vol.39,pp.3,2007. [33] F.PanandD.P.Morton,Minimizingastochasticmaximum-reliabilitypath,Networks,vol.52,pp.111,2008. [34] M.D.Bailey,S.M.Shechter,andA.J.Schaefer,SPAR:stochasticprogrammingwithadversarialrecourse,OperationsResearchLetters,vol.34,pp.307,2006. [35] R.Hemmecke,R.Schultz,andD.L.Woodruff,Interdictingstochasticnetworkswithbinaryinterdictioneffort,inNetworkInterdictionandStochasticIntegerProgramming,D.L.Woodruff,Ed.Kluwer,Boston,MA,2002. [36] D.L.W.HaraldHeld,RaymondHemmecke,Adecompositionalgorithmappliedtoplanningtheinterdictionofstochasticnetworks,NavalResearchLogistics,vol.52,pp.321,2005. [37] H.BayrakandM.D.Bailey,Shortestpathnetworkinterdictionwithasymmetricinformation,Networks,vol.52,pp.133,2008. [38] G.G.Brown,M.Carlyle,J.Salmeron,andR.K.Wood,Defendingcriticalinfrastructure,Interfaces,vol.36,2006. [39] J.Salmeron,Deceptiontacticsfornetworkinterdiction:Amulti-objectiveapproach,OperationsResearchDepartment,NavalPostgraduateSchool,,Monterey,CA,Tech.Rep.,2011. [40] C.M.RoccoandJ.E.Ramirez-Marquez,Abi-objectiveapproachforshortest-pathnetworkinterdiction,ComputersandIndustrialEngineering,vol.59,pp.232,2010. [41] N.Assimakopoulos,Anetworkinterdictionmodelforhospitalinfectioncontrol,ComputersinBiologyandMedicine,vol.17,pp.413,1987. [42] Y.WangandS.-K.Au,Spatialdistributionofwatersupplyreliabilityandcriticallinksofwatersupplytocrucialwaterconsumersunderanearthquake,ReliabilityEngineeringandSystemSafety,vol.94,pp.534,2009. 115

PAGE 116

[43] G.Brown,M.Carlyle,D.Diehl,J.Kline,andK.Wood,Atwo-sidedoptimizationfortheaterballisticmissiledefense,OperationsResearch,vol.53,pp.745,2005. [44] J.Salmeron,K.Wood,andR.Baldick,Analysisofelectricgridsecurityunderterroristthreat,IEEETransactionsonPowerSystems,vol.19,pp.905,2004. [45] ,Worst-caseinterdictionanalysisoflarge-scaleelectricpowergrids,IEEETransactionsonPowerSystems,vol.24,pp.96,2009. [46] J.P.Watson,R.Murray,andW.E.Hart,Formulationandoptimizationofrobustsensorplacementproblemsfordrinkingwatercontaminationwarningsystems,JournalofInfrastructureSystems,vol.15,pp.330,2009. [47] G.G.Brown,W.M.Carlyle,R.Harney,E.Skroch,andR.K.Wood,Interdictinganuclear-weaponsproject,OperationsResearch,vol.57,pp.866,2009. [48] J.F.Benders,Partitioningproceduresforsolvingmixed-variablesprogrammingproblems,NumerischeMathematik,vol.4,pp.238,1962. [49] M.ScaparraandR.Church,Protectingsupplysystemstomitigatepotentialdisaster:Amodeltofortifycapacitatedfacilities,UniversityofKent,WorkingPaper209,2010. [50] P.Cappanera,Mariapaolascaparra,TransportationScience,vol.45,pp.64,2011. [51] F.Pan,Stochasticnetworkinterdiction:Modelsandmethods,Ph.D.dissertation,UniversityofTexasatAustin,2005. [52] A.J.MillerandL.A.Wolsey,Tightformulationsforsomesimplemixedintegerprogramsandconvexobjectiveintegerprograms,MathematicalProgramming,vol.98,pp.73,2003. [53] G.L.NemhauserandL.A.Wolsey,IntegerandCombinatorialOptimization,2nded.NewYork,NY:JohnWiley&Sons,1999. [54] L.A.Wolsey,IntegerProgramming.NewYork,NY:Wiley-Interscience,1998. [55] H.D.Sherali,W.P.Adams,andP.J.Driscoll,Exploitingspecialstructuresinconstructingahierarchyofrelaxationsfor0-1mixedintegerprograms,OperationsResearch,vol.46,pp.396,1998. [56] H.D.SheraliandW.P.Adams,Ahierarchyofrelaxationsbetweenthecontinuousandconvexhullrepresentationsforzero-oneprogrammingproblems,SIAMJournalonDiscreteMathematics,vol.3,no.3,pp.411,1990. [57] ,Ahierarchyofrelaxationsandconvexhullcharacterizationsformixed-integerzero-oneprogrammingproblems,DiscreteAppliedMathematics,vol.52,no.1,pp.83,1994. 116

PAGE 117

[58] E.Balas,S.Ceria,andG.Cornuejols,Alift-and-projectcuttingplanealgorithmformixed0-1programs,MathematicalProgramming,vol.58,pp.295,1993. [59] L.LovaszandA.Schrijver,Conesofmatricesandsetfunctionsand0-1optimization,SIAMJournalonOptimization,vol.1,pp.166,1991. [60] M.R.GareyandD.S.Johnson,Computersandintractability:aguidetothetheoryofNP-completeness.W.H.Freeman,1979. [61] S.Neumayer,G.Zussman,R.Cohen,andE.Modiano,Assessingtheimpactofgeographicallycorrelatednetworkfailures,inMILCOM,2008ProceedingsIEEE,2008,pp.1. [62] ,Assessingthevulnerabilityoftheberinfrastructuretodisasters,inINFOCOM,2009ProceedingsIEEE,2009,pp.1566. [63] S.NeumayerandE.Modiano,Networkreliabilitywithgeographicallycorrelatedfailures,inINFOCOM,2010ProceedingsIEEE,2010. [64] P.Agarwal,A.Efrat,S.Ganjugunte,D.Hay,S.Sankararaman,andG.Zussman,Networkvulnerabilitytosingle,multiple,andprobabilisticphysicalattacks,inMILCOM,2010ProceedingsIEEE,2010,pp.1824. [65] ,TheresilienceofWDMnetworkstoprobabilisticgeographicalfailures,inINFOCOM,2011ProceedingsIEEE,2011. [66] A.Bernstein,D.Bienstock,D.Hay,M.Uzunoglu,andG.Zussman,Powergridvulnerabilitytogeographicallycorrelatedfailuresanalysisandcontrolimplications,DepartmentofElectricalEngineering,ColumbiaUniversity,Tech.Rep.,2011. [67] W.Wu,B.Moran,J.Manton,andM.Zukerman,Topologydesignofunderseacablesconsideringsurvivabilityundermajordisasters,inWAINA,2009ProceedingsIEEE,2009. [68] P.MirchandaniandR.Francis,DiscreteLocationTheory.JohnWiley&Sons,1990. 117

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BIOGRAPHICALSKETCH KellySullivanwasbornin1984inLittleRock,Arkansas.HereceivedB.S.andM.S.degreesinIndustrialEngineeringfromtheUniversityofArkansasin2006and2008,respectively.HejoinedtheIndustrialandSystemsEngineeringdepartmentattheUniversityofFloridainAugust2008andwillcompletehisPh.D.inAugust2012.Hisresearchinterestsincludeintegerprogramming,robustoptimization,large-scaleoptimization,interdiction,andhomelandsecurityapplications. 118