Multilevel Discrete Formulations and Algorithms with Applications to New Production Introduction Games and Network Inter...

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Multilevel Discrete Formulations and Algorithms with Applications to New Production Introduction Games and Network Interdiction Problems
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english
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Hemmati, Mehdi
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Smith, Jonathan Cole
Committee Members:
Geunes, Joseph Patrick
Richard, Jean-Philippe P
Thai, My Tra

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Subjects / Keywords:
bilevel -- interdiction -- network -- optimization
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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Industrial and Systems Engineering thesis, Ph.D.
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Abstract:
In this dissertation, we study multilevel optimization andnetwork interdiction theory, and apply this theory across several applicationsettings.  The common theme of all ourproblems involves competitive settings in which the beneficiary of a systemseeks robust design, protection, or fortification actions to compete againstexternal factors (e.g., uncertainty or an intelligent player) that may affectthe system, either intentionally or unintentionally.  We propose discrete bilevel mathematicalprograms for several applications in finite stopping problems underuncertainty, network interdiction problems, a new product introduction game,and competitive facility location problems. We then provide the necessary theory that makes the problems amenable toexact solution techniques. In our first problem, we consider an extension to theoptimal stopping problem in which items offered to a customer are alsoassociated with profits from the seller's viewpoint.  For this problem, we seek an ordering thatinduces the customer to purchase an item that maximizes the seller'sprofit.  By incorporating uncertainty inthe items' values, profits, and customer stopping thresholds, we studycharacteristics of two optimization philosophies, the ``max-min profit'' andthe ``maximum expected profit,'' for the seller.  In particular, we analyze the computationaltractability of the resulting optimization models and devise an exact solutiontechnique for one special case of the problem. The second problem that we study is closely related to theclassical Set Covering Problem, which has applications in new productintroduction games and competitive facility location problems.  In our problem, each clause contains anordered partial set of the items.  Eachplayer incurs a cost for each item that is selected, and receives rewards basedon satisfied clauses.  In particular, aclause is satisfied only by the highest-ranked item (if any) that the playerselects, and the reward granted to the player from this clause depends on theitem that satisfies the clause.  We studya nonzero-sum two-player game in which each player aims to maximize its ownprofit through selecting the items and satisfying the clauses based on theprioritized set covering rule. Our third problem addresses another nonzero-sum two-playergame over a network in which an adversarial player aims to spread its influence(control) on the nodes over a number of time stages, while the other playeraims to protect the network against the spread of the adversary's influence.  In this problem, we consider the cascadingeffect of the initial influence (via a threshold-based diffusion model), whichresults in the gradual spread of the influence on uninfluenced nodes.  Accordingly, the budget-restricted attackeraims to maximize the damage that the network incurs due to the spread of theinfluence resulting from an initial attack. By taking the defender's perspective, we study protection patterns withthe aim of minimizing the maximum damage incurred by the network.  This problem has several applications inrecommending protection policies for social and computer networks, and inprescribing strategic locations to participate in immunization programs.The common approach to solve discrete bileveloptimization problems is to obtain an equivalent (possibly nonlinear)mixed-integer single-level optimization problem by applying the duality theoryto the (convex) second-level problem. This approach, in particular, cannot be applied to our second and thirdproblems due to the nonconvexity of the second-level problems.  Accordingly, we propose reformulations anddecomposition techniques to devise formulations that are amenable tospecially-tailored cutting-plane methods. Moreover, when possible, we investigate various separation problems tostrengthen our suggested cutting planes. To demonstrate the efficacy of our solution techniques, we conductcomputational studies by using CPLEX as the mixed-integer solver and, whenapplicable, we compare the efficiency of our methods with existing approachesin the literature.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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 Mehdi Hemmati.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Smith, Jonathan Cole.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-02-28

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MULTILEVELDISCRETEFORMULATIONSANDALGORITHMSWITHAPPLICATIONSTONEWPRODUCTIONINTRODUCTIONGAMESANDNETWORKINTERDICTIONPROBLEMSByMEHDIHEMMATIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013MehdiHemmati 2

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Idedicatethisworkto BehjatGhafouri,thekindestmother(RIP), HoushangHemmati,themostwonderfulfather, FirouzehArbab,themostsupportivestep-mother,and SaharandSinaHemmati,thebest-eversiblings. 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmysincerestgratitudetomywonderfuladvisor,Dr.J.ColeSmith,forhisinvaluablesupportduringmystudiesattheUniversityofFlorida.Wordscannotexplainmyappreciationforhispatience,kindness,andenthusiasminguidingmethroughmydoctoralresearchaswellashisextraordinaryhelpinmattersfarbeyondmyresearch.HisencouragementhastrulyhelpedmetosurvivesometoughdaysinmylifenotsolelyasaPhDstudent,butratherasaperson.Ineverfeltbeingframedinanordinaryadvisor-studentrelation,butratheritwasanamazingfriendshipbetweentwopersons,onewiseandmatureandtheotheroneinexperienced.IalsowouldliketothankDr.JosephP.Geunes,Dr.Jean-PhilippeP.Richard,andDr.MyT.Thaiforservingonmysupervisorycommitteeandprovidinginsightfulviewpointsandsuggestions.Inparticular,IamthankfulforhavingthechanceoftakingoperationsresearchrelatedcourseswithDr.Richard,whoisoneofthemostamazingteachersIcouldhaveinmylife.Iowemanyofmyrelatedknowledgetohim.StudyinghereattheUniversityofFloridagavemetheopportunitytoknowsomeofmymostwonderfulfriendsandenjoymytimesasaPhDstudent.Specialthankstomywonderfulbrothers,EhsanSalariandBehnamBehdani,forhelpingmefromtheveryrstdayofbeinginGainesville.Iowethemmanydaysofunbelievablesupport.IalsowouldliketoexpressmyappreciationforhavingthechancetoenjoymytimeswithCinthiaC.Perez,ClayKoshnick,MichealC.Prince,AndrewN.Romich,JohannaAmaya,KellyM.Sullivan,andJorgeSefairwithwhomIhavehadmanyunforgettablememories.Itrulyappreciatetheirhelpandsupport.Ialsowouldliketothankmyotherfriends,SiqianShen,Aye-nurArslan,DeonBurchet,ShantihSpanton,BitaTadayon,ChrysasVogiatzis,JoseWalteros,ZehraMelisTeksan,RuiweiJiang,DmytroKorenkevych,AlexeySorokin,andRezaSkandariforbeingwonderfulcolleagues.Finally,Iwouldliketothankmyfamily.TheirunconditionalloveandendlesssupporthavemademewhoIam.Iwouldbecertainlylostinmylifejourneywithoutthem. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2FINITEOPTIMALSTOPPINGPROBLEMS:THESELLER'SPERSPECTIVE 15 2.1IntroductionandLiteratureStudy ....................... 15 2.2Seller'sProblemwithanOptimalCustomer ................. 18 2.3Max-MinProblem ................................ 25 2.4MaximizationofExpectedProt ........................ 33 3AMIXED-INTEGERBILEVELPROGRAMMINGAPPROACHFORACOMPETITIVEPRIORITIZEDSETCOVERINGPROBLEM .................... 41 3.1IntroductionandLiteratureStudy ....................... 41 3.2ProblemFormulation .............................. 45 3.3ExactSolutionMethod ............................. 47 3.3.1Cutting-PlaneAlgorithm ........................ 49 3.3.2FollowerandSeparationSubproblem ................. 52 3.3.2.1Thesameactionrestriction ................ 54 3.3.2.2Thesingleproductrestriction ............... 56 3.3.2.3Thesameactionwithonefewerproductrestriction ... 57 3.3.2.4Thesameactionwithonemoreproductrestriction ... 59 3.4ComputationalResults ............................. 60 3.4.1ImplementationDetailsandInstanceGeneration .......... 60 3.4.2Results ................................. 63 4ACUTTING-PLANEALGORITHMFORSOLVINGAWEIGHTEDINFLUENCEINTERDICTIONPROBLEM ............................. 66 4.1IntroductionandLiteratureStudy ....................... 66 4.2ProblemFormulation .............................. 71 4.3ExactSolutionMethod ............................. 74 4.3.1ReformulationandObjectiveBounds ................. 74 4.3.2Cutting-PlaneAlgorithm ........................ 78 4.3.3SpreadNetworkInequalities ...................... 81 4.3.3.1Spread-network-basedcuttingplanes ........... 81 5

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4.3.3.2Spreadnetworkmodicationstrategy ........... 85 4.4Attacker'sProblemSolutionApproach .................... 90 4.4.1Reformulation1:ExponentialSetModel ............... 90 4.4.1.1Model ............................. 91 4.4.1.2Benders'decomposition ................... 92 4.4.2Reformulation2:CompactModel ................... 96 4.5ComputationalResults ............................. 99 4.5.1ImplementationDetails ......................... 99 4.5.2Resultsfortheattacker'sproblem ................... 102 4.5.3Resultsforthedefender'sproblem .................. 105 5CONCLUSIONSANDFUTURERESEARCH ................... 108 APPENDIX AAPPENDIXONREPRESENTATIONOFTHRESHOLDVALUES ........ 112 BPROOFOFTHEOREM3.1 ............................. 115 REFERENCES ....................................... 117 BIOGRAPHICALSKETCH ................................ 123 6

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LISTOFTABLES Table page 2-1Seller'sproblemexample .............................. 25 3-1Productintroduction:parametersusedtogeneratetestinstances ....... 62 3-2Facilitylocation:parametersusedtogeneratetestinstances .......... 63 3-3ComparisonofCPAimplementations ....................... 63 3-4ComparisonofaugmentedCPAimplementations ................ 64 3-5PerformancecomparisonforthebestCPA,ACPA2,HYB,andMITS ...... 65 4-1Sizecomparisonofattacker'sproblemformulations. ............... 99 4-2Parametersusedtogeneratetestinstances ................... 100 4-3Computationalresultsfortherstscenariooftheattacker'sproblemonADDnetworks ....................................... 103 4-4Computationalresultsforthesecondscenariooftheattacker'sproblemonADDnetworks .................................... 104 4-5Computationalresultsfortheattacker'sproblemonSFnetworks ........ 105 4-6ComputationalresultsofCPAimplementations .................. 106 7

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LISTOFFIGURES Figure page 2-1Sequenceofitemsinanoptimalstoppingproblem. ................ 17 4-1AninstancewithQ=3andT=2,intheabsenceofprotectednodes. ..... 67 4-2AninstancewithQ=3andT=2,withnodes6and9protectedbythedefender. ....................................... 67 4-3TwopossiblespreadnetworksforFigure 4-2 ................... 82 4-4SpreadnetworkmodicationusingTheorem 4.4 ................. 87 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMULTILEVELDISCRETEFORMULATIONSANDALGORITHMSWITHAPPLICATIONSTONEWPRODUCTIONINTRODUCTIONGAMESANDNETWORKINTERDICTIONPROBLEMSByMehdiHemmatiAugust2013Chair:J.ColeSmithMajor:IndustrialandSystemsEngineering Westudymultileveloptimizationandnetworkinterdictiontheory,andapplythistheoryacrossseveralapplications.Thecommonthemeoftheseproblemsinvolvescompetitivesettingsinwhichthebeneciaryofasystemseeksrobustdesign,protection,orforticationactionstocompeteagainstexternalfactors(e.g.,uncertaintyoranintelligentplayer)thatmayaffectthesystem. First,weconsideranextensiontotheoptimalstoppingprobleminwhichoffereditemsarealsoassociatedwithprotsfromthesellersviewpoint.Weseekanorderingthatinducesthecustomertopurchasetheitemthatmaximizesthesellersprot.Byincorporatinguncertaintyintheproblemparameters,westudytwooptimizationphilosophies,themax-minprotandthemaximumexpectedprot,andinvestigatethecomputationaltractabilityoftheresultingoptimizationmodels. Next,weconsideraStackelberggamethatarisesinnewproductintroductioninwhichtworms,aleaderandafollower,competewiththeaimofmaximizingtheirprotbyintroducingtheirsetofproducts.Knowingthatthesuccessofnewproductsofferedbytheleaderalsodependsonthefollowersresponse,wesolveamixed-integerbilevelprogram(MIBMP)tondthesetofproductstobeintroducedbytheleaderwiththeaimofmaximizingitsprot. 9

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Third,westudyacompetitionscenariooveranetworkinwhichanadversarialplayeraimstospreaditsinuenceonthenodesoveranumberoftimestages,whiletheotherplayeraimstoprotectthenetworkagainstthespreadoftheadversarysinuence.Inthisgame,thenetworkincursdamageforeachinuencednode,andthedefendersaimistominimizethemaximumdamageincurredbythenetwork.WesuggestanMIBLPforthisinterdictionproblem,andweproposeacutting-planealgorithmwithseveralvalidinequalities. Ourapproachtosolvetheseproblemsemploysreformulationsanddecompositiontechniquestodeviseformulationsthatareamenabletospecially-tailoredcutting-planemethods.Moreover,weinvestigatevariousseparationproblemstostrengthenthesecuttingplanes.WealsoconductcomputationalstudiesbyusingCPLEXasthemixed-integersolverand,whenapplicable,wecomparetheefciencyofourmethodswithexistingapproachesintheliterature. 10

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CHAPTER1INTRODUCTION Mathematicalprogramshavebeentraditionallyintroducedforapplicationsthatinvolveasingledecisionmaker,whoaimstochooseabestdecisionamong(possiblyasignicantlylarge)numberoffeasiblesolutions.Althoughmanyclassicaloptimizationproblemswithasingledecisionmakerareconsideredtheoreticallyintractable,signicantlylargeinstancesoftheseoptimizationproblemscannowbesolvedduetotheadventofimprovedcomputationalpowerandstate-of-the-artcommercialsoftwarepackages,whichemployvitaladvancesintherelatedtheory.Accordingly,problemshavinghundredsofthousandsvariablesandconstraintscannowbesolvedwithinreasonablecomputationallimits. Theworldofoptimizationproblems,however,alsoencompassesapplicationsthatinvolvetwoormoredecisionmakers.Forexample,applicationsthataddresscompetitionbetweenactivebusinessagentsthataimtomaximizetheirmarketshareinvolvetwodecisionmakers.Here,eachagent'sdecisionsmustbeoptimalwithrespecttonotonlyitsownconstraints,butalsowithrespecttotheotheragent'sdecisions(thatmayintentionallyorunintentionallyrestricttherstagent'savailabledecisions).Theseproblemsarecalledmultileveloptimizationproblems,andconsideredtobehighlyintractableevenonsmallinstances.Thesituationisaggravatedinthepresenceofdiscretevariables.Inthiscase,multileveloptimizationproblemsoftencannotbedirectlysolvedusingavailablecommercialsoftwarepackagesunlesstheproblemissomehowconvertedviareformulationintoasingle-levelproblem. Closelyrelatedtomultilevelproblemsareoptimizationproblemshavinguncertainparameters,particularlywhenaconservativedecisionmakerseeksoptimalpoliciestomitigatetheeffectsoftheworstpossibleoutcomes.Oneframeworktostudytheseproblemsistoperceiveanexternalagentthataimstoinducearealizationofprobabilisticparameters,sothatthedecisionmakerfacestheworstpossibleoutcomes 11

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dependingonhisorherdecisions.Werefertotheseproblemsasinterdictionproblems.Theinterdictionmodelingparadigmisapowerfulmechanismtostudyconservativedecisionmakinginseveralapplicationssuchashomelandsecurityanddisasterplanning.Stochasticprogrammingandrobustoptimizationaretwootherwidely-knownmodelingparadigmsthatallowincorporationofuncertaintyinoptimizationproblems.Whiletheformeraims,forexample,tomitigatetheundesiredeffectsofuncertaintyoveralong-termhorizon,thelatterseekspoliciesthatareoptimalwithrespecttoworst-casescenarios.Infact,robustoptimizationandinterdictiontheoryemployaverysimilarpointofviewinmodelingconservativedecisionmakingscenarios. Inthisdissertation,weprimarilyfocusonmultileveloptimization,interdictiontheory,andrelatedapplications.Inparticular,westudyapplicationsarisinginniteoptimalstoppingunderuncertainty,networkinterdiction,productintroduction,andcompetitivefacilitylocation.Weproposevariousbilevelprogramshavingdiscretevariablesinbothlevels,andwestudytheoreticalcharacteristicsofmathematicalformulationsaswellasefcientsolutiontechniques.NotethatChapters 2 4 areeachindividuallyself-contained.Accordingly,thenotationrequiredforeachchapterareintroducedatthebeginningofthechapter.Thisenablesthereadertostudychaptersindependently. InChapter 2 ,westudyanextensiontotheniteoptimalstoppingproblem.Inthisproblem,acustomeraimingtobuyoneitemreceivesaseriesofproductoffersfromaselleronebyone.Thecustomerisawareofthe(nite)numberofproductoffersandtheminimumandmaximumpossiblevaluesofeachitem,andmustpurchaseexactlyoneitem.Whenanitemispresentedtothecustomer,(s)heobservesitsvalue,anddetermineswhethertopurchasetheitemortopermanentlydismisstheitem.Thecustomer'sobjectiveistomaximizethevalueofthepurchaseditem.Inourstudy,weconsidertheproblemfromtheviewpointoftheseller,whowishestomaximizeprotassociatedwiththesolditem.Hence,thesellerseeksanoptimalsequenceofitemstosell,giventhatthecustomeractsaccordingtosomenear-optimaldecision-makingrules. 12

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Ourstudytakestheperspectivethatthecustomermaynotactoptimallyduetoimperfectdecision-makingstrategiesand/ortotheseller'suncertaintyintheitems'valuestothecustomer.Weinvestigatedifferentoptimizationphilosophiesforthesellerbyconsideringmax-minandmax-expectationobjectiveswhencustomerbehaviorisnotcompletelypredictable,anddiscusstheproblemtractabilityinthesecases. InChapter 3 ,weexamineamixed-integerbilevelprogramming(MIBLP)problemforacompetitivesetcoveringproblem.Theclassofproblemsweconsiderisapplicabletoseveralelds,includingnon-cooperativeproductintroductionandfacilitylocationgames.Intheprioritizedsetcoveringproblem,thereexistsasetofitemsandclauses.Itemsmaycorrespondtopotentialfacilitylocationsorproductsthatcanbeintroducedtoamarket.Theclausesmaybeassociatedwithcustomersormarketsegments,eachofwhomprioritizestheset-coveritemsaccordingtotheirinterestintheseitems.Weconsideratwo-playerStackelberggame,inwhichtheleaderselectsasetofitems,andthenthefollowerselectsanothersetofitemswithknowledgeoftheleader'saction.Everyselecteditemincursacosttotheplayers.Eachclauseissatisedbytheselecteditemhavingthehighestpriority,resultinginarewardfortheplayerthatintroducedthehighest-priorityselecteditem.Inthisproblem,eachplayeraimstooptimizeitsownobjective,incontrasttoapriorproductintroductionstudyinwhichthefollowerattemptstominimizetheleader'sprot.WedevelopanMIBLPmodelforthisprobleminwhichbinarydecisionvariablesappearinbothstagesofthemodel.Asthemaincontributionofthischapter,wethenproposeseveralvariationsofanexactcutting-planealgorithmtosolvethisproblem,andexaminetheefcacyofthemethodsonrandomlygeneratedtestinstances. InChapter 4 ,weconsiderabileveldefender-attackergamethattakesplaceonanetwork,inwhichtheattackerseekstotakecontrolover(orinuence)asmanynodesaspossible.Thedefenderactsrstinthisgamebyprotectingasubsetofnodesthatcannotbeinuencedbytheattacker.Withfullknowledgeofthedefender's 13

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action,theattackercantheninuenceaninitialsubsetofunprotectednodes.Theinuencethenspreadsoveranitenumberoftimestages,whereanuninuencednodebecomesinuencedattimetifathresholdnumberofitsneighborsareinuencedattimet)]TJ /F4 11.955 Tf 10.69 0 Td[(1.Theattacker'sobjectiveistomaximizetheweightednumberofnodesthatareinuencedoverthetimehorizon,wheretheweightsdependbothonthenodeandonthetimeatwhichthatisinuenced.Thisdefender-attackergameisespeciallydifculttooptimize,becausetheattacker'sproblemitselfisNP-hard,whichprecludesastandardinner-dualizationapproachthatiscommoninmanyinterdictionstudies.Weprovidethreemodelsforsolvingtheattacker'sproblem,anddevelopatailoredcutting-planealgorithmforsolvingthedefender'sproblem.Wethendemonstratethecomputationalefcacyofourproposedalgorithmsonasetofrandomlygeneratedinstances. 14

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CHAPTER2FINITEOPTIMALSTOPPINGPROBLEMS:THESELLER'SPERSPECTIVE 2.1IntroductionandLiteratureStudy Webeginthischapterbydescribingthefollowingstoppingproblem,whichisaclassicalproblemthathasreceivedsubstantialattentionacrossseveraldisciplines.Acustomermustpurchaseoneitemoutofasetofitemsthatarepresentedoneatatimetothecustomer.Whenanitemispresentedtoacustomer,thecustomerevaluatesitsvalue,andmustdecidewhethertopurchasetheitem(thusendingthegame)orpermanentlydiscard(orreject)theitem.Thecustomer'sobjectiveistomaximizethevalueofthepurchaseditem.(Therewardisequaltothevalueofthepurchaseditem,andisindependentofallotheritems'values.)Thecustomerisawareatthebeginningofthegameofthenumberofitems(denotedbyn)thatwillbepresentedandtheprobabilitydistributionusedtogeneratetheitems'values.Observethatifoneitemremains,thecustomermustpurchaseit.Itisthusstraightforwardtoseethatthismodelcapturesthecaseinwhichthecustomerdoesnotneedtobuyanitem,whichwewouldallowbysimplylettingthenalitemhaveavalueequaltothecustomer'svalueofpurchasingnothingatall. Thisgameisaspecialcaseofthebroadclassofoptimalstoppingproblemsinwhichthecustomermustdeterminewhentostopthegame(e.g.,bypurchasinganitem).See[ 20 ]foranearlysummaryofoptimalstoppingproblemanalysis,and[ 32 ]foracomprehensivemodernstudyofthisclassofproblems.Infact,theoriginalsecretaryproblem(see,e.g.,[ 31 33 34 59 ])isgivenasabove,butwheretheobjectiveistopickthemost-valuableitemfromamongthesetofallitems.Thereisnorewardforpickinganyotheritemthanthebestone.Therearenumerousversionsofthesegames,includingvariationsinwhichnisunknownorinnite,inwhichrewardsaregivenforsolutionsotherthanthemost-valuableone,andinwhichanattempttopurchasetheitemmayfail[ 4 37 57 69 ]. 15

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Thestoppingproblemconsideredinthischaptersatisestheso-calledmemorylessproperty,inthesensethatprioractionsthatoccurredinthisgamedonotaffecttheremainderofthegame.Thecustomeronlyneedstoknowhowmanyitemsremainandthevalueofthenextitem(inadditiontotheboundaryconditionsofthegame)tomakethenextdecision;priorinformationregardingthevaluesofprevious(rejected)itemsisirrelevant.Thismemorylesspropertyenablesustoemploydynamicprogrammingtechniques(see,e.g.,[ 8 ])tosolvetheabovestoppingproblem.(WeprovidethetechnicaldetailsforthisalgorithminSection 2.2 .) Inthischapter,weintroduceanewproblemfromtheseller'sperspective.Inthisproblem,eachitemisalsoassociatedwithaprot(independentfromtheitem'svaluetothecustomer)thatthesellermakesifthecustomerpurchasestheitem.Thesellermustdetermineasequenceoftheitemstopresenttothecustomer,sothatthecustomer(actingrationally,i.e.,optimallyinhis/herownbestinterests)wouldchooseanitemthatresultsinamaximumpossibleprottotheseller.Thisproblemisnontrivial,becauseplacingthemost-protableitemstooearlyinthesequencemayresultinthecustomerrejectingthoseitems,inhopesofndinganitemwithmorevaluetothecustomerlaterinthesequence.Placingthemost-protableitemslateinthesequenceincurstheriskofhavingthecustomerterminatesearchbeforeencounteringtheseitems. Moreover,thisproblemisfurthercompoundedthefactthathumandecision-makerstendnottooptimallysolvestoppingproblemsinlaboratorysettings.Wereferthereadertoanexcellentintroductorytreatmentofthismaterialin[ 6 ],andtorecentresultsappearingin[ 7 43 61 ].Acommonthemeinthislineofliteratureisthatdecision-makerstendtoterminatetheirsearchtoosoon.Itispotentiallyveryriskyforthesellertoassumethatacustomerwillactrationally,inthesensethatthecustomerfollowsanexactoptimizationalgorithminselectinganitem. Wepresentanexampletoillustratethesituation.Supposethatn=6,andthatthecustomerbelievesthattheitems'valuesareuniformlydistributedintheinterval[0,100]. 16

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Items1,...,6arearrangedinnonincreasingorderofprottotheseller(sothatitem1ismostpreferred).ConsiderthesituationgiveninFigure 2-1 ,whichdepictsasequencethatthesellerhaschosentopresenttoacustomer.Thesellerhasevidentlygambledthatthecustomerwillrejectitem5,whichhasavalueof70tothecustomer.Ideally,thecustomerwouldthenpurchaseitem1,resultinginthemostprottotheseller. Indeed,anoptimalcustomerwillrejectitem5withsixitemsremaining,andpurchaseitem1withveitemsremaining.(InSection 2.2 ,wewillillustrateanoptimaldecision-makingframeworkforthecustomer.Inparticular,itturnsoutthatanoptimallybehavingcustomerwouldbuythesixth-to-lastitemifitsvalueexceeds77.5,andwouldbuythefth-to-lastitemifitsvalueexceeds74.2.)However,thesellerriskshavingaconservativedecision-maker(ratherthananoptimalcustomer)thatstopstheprocesstooearlyandpurchasesitem5,orstopstheprocesstoolateandpurchasesitem6. Figure2-1. Sequenceofitemsinanoptimalstoppingproblem. Thechallengeinthischapteristoanalyzetheseller'sproblemfromthreedifferentperspectives.WebegininSection 2.2 byassumingthatthecustomeractsinanoptimalmanner,breakingtiesinamannerthatisdisadvantageoustotheseller.Thatis,whenthecustomer'sdecisionisoptimaleithertorejectorpurchaseanitem,thecustomer 17

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takestheoppositeactiondesiredbytheseller.(Thispessimisticassumptioncaneasilybemodied.) Wethenaccommodatethestochasticnatureofhumandecision-makersbyaccountingforrandomnessinthecustomer'soptimaldecision-makingpolicies,andinthecustomer'sperceptionoftheitems'values.Forinstance,wecanmodeladecision-makerwhotendstostoptoosoonbyadjustingthetrueitemvaluestobehigherthantheyactuallyare.However,whendealingwithuncertainty,wemustspecifyanobjectivethatreectstheseller'soptimizationphilosophy.Arisk-aversesellermayattempttomaximizetheminimumprotthatcanbemade,givenaboundedlyrationalcustomer.Wediscussthisproblem,alongwithasimplemodelforboundingcustomerrationality,inSection 2.3 Althoughasellerwhoisplayingthisgameoncemaypreferanoutcomewithlimitedriskofsellingalow-protitem,asellerwhoplaysthisgamerepeatedlywouldmorelikelyprefertomaximizeexpectedprotinstead.Hence,inSection 2.4 ,wepresentaprobleminwhichthesellermaximizesexpectedprotgivenadiscreteprobabilitydistributionfunctionofthecustomer'sproblem-solvingparameters.Wedemonstratethatevenrestrictedversionsoftheexpected-valueproblemareNP-hard,andarethusquitedifculttosolveintheworstcase. 2.2Seller'sProblemwithanOptimalCustomer Webeginbyintroducingnotationforthisproblemandpresentingtheoptimaldynamicprogrammingstrategyforthecustomer.Foreachitemi=1,...,n,theitem'svaluetothecustomerisvi,andtheitem'sprottothesellerisbi.Forthesakeofsimplicity,weexaminethecaseinwhichthecustomerassumesthatitemvaluesaregeneratedfromtheuniformdistributionontheinterval[0,100].Thelowerandupperboundsgivenherearearbitrary,andthefollowingdiscussioneasilyaccommodatesanygeneric(nite)bounds.Also,thelogicbehindourproceduresdoesnotchangeifthe 18

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valuesareassumedtobenonuniformlygenerated,solongasconditionalexpectationsarenite. AsstatedinSection 2.1 ,weassumethatthecustomerdecisionsarenotaffectedbythevaluesofthepreviouslyseenitems,butonlybythenumberofremainingitems.Moreprecisely,thecustomer'sdynamicprogrammingalgorithmemploysbackwardrecursion,startingfromthesituationinwhichonlyoneitemremains.Inthiscase,thecustomermustpurchasetheitem,andthecustomer'sexpectedvaluewillbe50.Now,iftherearetwoitemsremainingintheset,thecustomerwillpurchasetheitemifitsexpectedvalueisatleast50,andwillrejecttheitemotherwise.Whentwoitemsremain,thereisa50%chancethatthecustomerrejectstheitem,becauseitsvaluedoesnotexceed50(leavingthecustomerwithanexpecteditemvalueof50fromthelastitem),anda50%chancethatthecustomeracceptstheitembecauseitsvaluebelongstotheinterval[50,100](yieldinganexpectedvalueof75fromtheseconditem).Thecustomer'soverallexpectedvaluewithtwoitemsleftisthus62.5. Ingeneral,whenexaminingtheithiteminthesequence,wedenetitobethecustomer'sexpectedvaluefrompurchasinganitem.Arational(oroptimal)customerappliesthefollowingrecursiveformulatocomputethesevalues: tn=50, (2a)ti=Pr(V>ti+1)ti+1+100 2+Pr(Vti+1)ti+18i=1,...,n)]TJ /F4 11.955 Tf 11.95 0 Td[(1, (2b) whereVistheuniformrandomvariablereectingthevalueofanyitem;hence,Pr(Vk)is1ifk100,0ifk0,andk=100otherwise,withPr(V>k)=1)]TJ /F4 11.955 Tf 12.42 0 Td[(Pr(Vk).(Notethatthisprocesscanbeadaptedfornonuniformdistributions,andfordistributionsthatdependonhowmanyitemsremain.)Hereafterwerefertot-valuesasthresholds,becausetheyreecthowanoptimalcustomermakesdecisions:Iftheithitem'svalueofferedtothecustomerexceedsti+1,thenthecustomerselectsit.Fornotational 19

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convenience(allowingustohandlethecasecorrespondingtothelastitem),wedenetn+1=. Withoutlossofgenerality,weassumeb1bn.Wediscusstheseller'sproblemintermsofassigningitemstoslotsinthesequencepresentedtothecustomer.Theseller'sgoalwillbetoinducethecustomertobuyitem1ifpossible,andfailingthat,tobuyitem2,andsoon.Conceptually,thesellerwillplaceitem1intheearliestslotisuchthatv1>ti+1,anditem1ispurchasedifthecustomerreachesslotiinthesearch.Accordingly,wedene pi=minfp2f1,...,ng:vi>tp+1g8i=1,...,n.(2) Foritemi=1,...,n,pirepresentstheearliestpossibleslotinanysequenceinwhichthecustomerwouldchooseitemi,assumingthatthesearchisstillactive.Ofcourse,ifp1=1,theproblemistrivial:Placeitem1intherstslot,withtheremainingslotsarbitrarilydetermined,andthecustomerselectsitem1.Otherwise,thesellerseekstoplaceless-protableitemsearlyinthesequencewhosevaluesaresmallenoughthatthecustomerrejectsthem,untilreachingitem1.Dene: Np=fi2f1,...,ng:vii,thenaniteminNpisplacedatslotp,forallp=1,...,pi)]TJ /F3 11.955 Tf 12.24 0 Td[(i,withnoitemappearingtwiceinslots1,...,pi. 20

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Proof. First,supposethattheconditionsofthispropositionholdtrue.Thecustomerwillpurchaseitemipositionedatpiifnoitemisselectedbeforepi.Byassumptionthecustomerrejectsanyitemhiplacedinaslotppi.Thenbyswitchingtheitemsinslotskandpi,thecustomerwouldstillpurchaseitemiinpi.Giventhatitemiisplacedinslotpi,wenextestablishthataschedulemustexistinwhicheachitemj2J=f1,...,)]TJ /F2 11.955 Tf 36.46 0 Td[(gisplacedinslotspi)]TJ /F4 11.955 Tf 12.53 0 Td[(,...,pi)]TJ /F4 11.955 Tf 12.53 0 Td[(1(inanyorder).BecauseitemsinJcannotbepurchasedbythecustomer,theycanbeplacedanywhereinthesequence.Supposethatanitemj2JisnotoneofthejJjitemsthatimmediatelyprecedesitemiinpositionpi,andswapjwithanitemk=2Jthatiscurrentlypositionedinsomeslotp2fpi)]TJ /F4 11.955 Tf 12.38 0 Td[(,...,pi)]TJ /F4 11.955 Tf 12.38 0 Td[(1g.Ifjhadfolloweditemi,thenthecustomerwouldrejectjinsteadofkinslotp.Otherwise,p2f1,...,pi)]TJ /F3 11.955 Tf 12.41 0 Td[(ig,anditemkismovedearlierinthesequence.Ifitemkwasrejectedinitsoriginalposition,itwouldalsoberejectedatanearlierposition.Thecustomerstillrejectsitemj(inanyposition),andbecausetheremainingitemsareunaffected,wouldstillpurchaseitemi.Afterexecutingallsuchswaps,andrearrangingitemsinJasnecessary,weobtainaschedulesatisfyingtheconditionsoftheproposition. UsingProposition 2.1 ,thesellercanemploythefollowinggreedyprinciple.Startingwithitem1,examinewhetherthereexistsasequenceinwhichitem1isplacedinp1andtheearlierpositionsp=1,...,p1)]TJ /F4 11.955 Tf 12.57 0 Td[(1containitemsbelongingtoNp.Ifso,anoptimalsolutioncanbeconstructedbasedonthissequence.Otherwise,thesellerattemptstosellitem2,andsoon,untilanitemcanbesold. 21

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Computationally,ourchallengeistosolvetheproblemofplacingitemsinslots1,...,pi)]TJ /F3 11.955 Tf 12 0 Td[(i(whenpi>i)asefcientlyaspossible.First,itisusefultodeneamaximumsafeslotforeachitemias: fi=8>><>>:0ifvit2maxfp2f1,...,ng:vii,andthatthereexistsan(optimal)sequencewithitemiplacedinslotpi,whichinducesthecustomertopurchaseitemi.Thenthereexistsasolutioninwhichitemsscheduledinpositions1,...,pi)]TJ /F3 11.955 Tf 11.95 0 Td[(iappearinthesameorderastheyappearinF. Proof. Considerasequencewithipositionedinpi,andasubsetofitemsinthesetfi+1,...,nginpositions1,...,pi)]TJ /F3 11.955 Tf 10.4 0 Td[(i(suchasolutionisknowntoexistbyProposition 2.1 ).Itemsintherstpi)]TJ /F3 11.955 Tf 12.21 0 Td[(islotshavingthesamef-valuecanclearlyberearrangedtomeetthesequenceofitemsinF.Now,consideritemsjandkwithfj>fk,butitemj(inslotp0)orderedbeforeitemk(inslotp00).Supposethatweswaptheseitems'slots.Itemkinslotp0wouldstillberejected,becauseitemkwasrejectedinthelaterslotofp00.Itemjwouldberejectedinslotp00becausekwasrejectedinthesameslot,andthefactthatfj>fkindicatesthatvj
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(ifpi>i)arescheduledaccordingtoLemma 1 .WeformallystatethisprocessinAlgorithm 1 Algorithm1Seller'sProblemwithanOptimalCustomer 1: Computepiandfi,8i=1,...,n.ObtainFbysortingtheitemsinnondecreasingorderoftheirvalues,excludingitem1,andbreakingtiesindecreasingorderofitemindex. 2: Initializej=1andlist=;;letjlistjdenotethenumberofelementsinlist. 3: whilejj+jlistjdo 13: Setj=j+1. 14: ifitemjbelongstolistthen 15: Letposjbethepositionofitemjinlist. 16: Deleteitemjfromlist. 17: ifthereexistsanitemk2Fwithfkposjthen 18: FindanitemkhavingthesmallestvalueoffkinF,subjecttofkposj. 19: Insertitemkintopositionposjoflist,anddeletekfromF. 20: endif 21: endif 22: endwhile 23: AnoptimalsequenceschedulesitemsasinProposition 2.1 ,withtheorderingoftherstminfpj)]TJ /F4 11.955 Tf 11.96 0 Td[(1,j)]TJ /F4 11.955 Tf 11.95 0 Td[(1gitemsgivenbylist. Therstwhile-loopinAlgorithm 1 establisheslist,whichaidsusinmaximizingthenumberofitemsthataretoplacedatthefrontofthesequence.Afterthisrststepiscomplete,thesecondwhile-loopterminateswhenj+jlistjisatleastaslargeaspj:Whenthishappens,schedulingacombinationofitemsonlist(inpositionsp
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tentativelyscheduledposition.WeseekareplacementinthelistinStep 18 .Ifsuchanitemexists,thesizeoflistremainsconstant,andotherwise,itshrinksbyone. EachvaluepiandficanbecomputedinO(logn)stepsbybinarysearch,foratotalofO(nlogn)operations.ThesortingoperationforcomputingFtakesO(nlogn)timeaswell.Theoperationsintherstwhile-loopareO(n)incomplexity.Inthesecondwhile-loop,wemustpotentiallydeleteanitemfromlist(whichcanbedoneinconstanttime),nditemkinStep 18 (whichrequiresO(logn)steps),andperhapsinsertanelementbackintolist(whichcanbedoneinconstanttime).However,ifapositionindexiskeptexplicitlyateachiteration,thealgorithmrequiresO(n)operationsateachupdate.Instead,inStep 15 ,wendthepositionofjinlistviabinarysearch.(NotethatitemsappearinlistinthesameorderthattheyappearedinF;furthermore,thetie-breakingcriteriapresentinoursortingofFensuresthatthesortingoperationyieldsauniquesequenceF.Thesefactspermitustoexecutebinarysearchonlist.)Step 15 thereforealsotakesO(logn)steps,andsothesecondwhile-looprequiresO(nlogn)steps.Finally,recoveringthesequenceattheendofthealgorithmisanO(n)operation,andsotheoverallcomplexityofthisalgorithmisO(nlogn). Toillustratethisprocess,considerthen=10examplewhosev-,t-,p-,andf-valuesaredepictedinTable 2.2 .Initially,wehaveF=f8,6,9,7,5,10,3,2,4gandlist=f9,7,10,3,2,4g, whereitems8and6donotbelongtolistbecausef8=f6=0,anditem5doesnotbelongtolistbecauseitwouldbethethirditem,andf5=2<3.Followingthecreationoflist,F=f8,6,5g.Wetrytondasequencesuchthatitem1canbesold,butthisisimpossiblebecause1+jlistj=7andp1=10.Thatis,evenifeveryiteminlistisorderedbefore1,thecustomerwouldpurchaseanyitem(otherthan1)intheseventhposition. 24

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Table2-1. Seller'sproblemexample 12345678910 vi30687860838583928481ti86.185.083.682.080.077.574.269.562.550.0pi10859313124fi9748202013 Havingruledoutsellingitem1,thesellerattemptstosellitem2.First,item2isremovedfromlist,andisnotreplaced:Item2sitsinthefthpositionoflist,andnoiteminFhasanf-valueofatleast5.Hence,item2couldonlybesoldifp22+jlistj=7(representingtheve-itemsequenceoflistbeingscheduled,followedbyitem1,followedbyitem2),butp2=8andtheitemcannotbesold. Turningourattentiontosellingitem3,weremoveitem3fromlist(againwithoutreplacement),andtestp33+jlistj=7.Thistime,therelationshipholds,anditem3canindeedbesoldtothecustomer.AccordingtoProposition 2.1 andLemma 1 ,oneoptimalsequencebeginswithsequence:97213, withtheremainingveitemsbeingscheduledarbitrarily. 2.3Max-MinProblem Thestoppingproblemwediscussinthischapteressentiallyencompassesthreedifferentparametersets:prots,thresholds,andcustomervalues.AvitalassumptionthatwemadeinSection 2.2 isthatallparametersareknownwithcertainty,andthatthecustomerwillemployanoptimalstrategytoselectanitem.However,inamorerealisticsetting,theremaybesomedegreeofuncertaintyabouttheparameters,and(especiallyinthecaseofahumandecision-maker)aboutthestrategythatthecustomeruses.Thesellershouldthereforeincorporateknowledgeofthisuncertaintyintothesequence.Weconsiderinthissectionaconservativeseller,whoseeksasequencethatmaximizes 25

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protinaworst-casescenario.(ThecaseofasellerwhowishestomaximizeexpectedprotinsteadgivenaparticulardatadistributionisexploredinSection 2.4 .) Notethatitisnotgenerallypossibletospecifyasingleworst-casescenario(i.e.,anoutcomeofrandomdatathatismostdamaging)fortheseller.However,givenanestablishedsequenceofitems,itispossibletodetermineaworst-casesetofdatathatresultsintheminimumprotfortheseller.Hence,inthismodelingstrategy,wedeneanuncertaintysetUofallpossiblecombinationsofthresholdandcustomervalues(withprotvaluesbeingdeterministic).Ingeneral,theonlyassumptionsthatwemakeregardingthestructureofUisthatitisnonempty,andthethresholdvalues tthatbelongtoUmustsatisfy t1 tn>t0n+1=.Notethatfromagame-theoreticperspective,onecanviewUasthesetthatdenesboundedlyrationalbehaviorfromthecustomer. Denote`(x,U)astheminimumprotpossiblegivenasequenceofitemsxoverallpossibledataoutcomesinU.Theproblemconsideredinthissectionseekstomaximize`(x,U)overallpermutationsofitemsx,whichiswhywerefertothisproblemasamax-minoptimizationproblem.Thismodelingphilosophyisexactlythatembodiedbytherobustoptimizationcommunity;wereferthereaderto[ 9 ]forathoroughmathematicalprogrammingdiscussionofrobustoptimization. Becauseidentifyingtheworst-casedatascenariocorrespondingtoanyitemsequencexisanoptimizationproblem,itisconvenienttoenvisionathirdpartyadver-sarywhoseekstheworstpossibledataoutcomeinU,giventheseller'ssequenceofitems.Hence,wenowexaminethisproblemasaStackelberggameinwhichthesellerarrangestheitemstobesoldinsomesequence,theadversarymanipulatesdata(withintheallowableuncertaintysetU),andthecustomerfollowsthepreviouslystatedoptimalstoppingstrategyforselectinganitem,butbasedontheparametersmanipulatedbytheadversary. 26

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AsimplegeneralizationofAlgorithm 1 isnotsufcienttosolvethemax-minproblemdescribedabove.(Indeed,Corollary 1 inSection 2.4 willdemonstratethatthisproblemisNP-hardingeneral.)However,weillustrateinthissectiononestrategyforsolvingamax-minproblemgivenaspecicclassofU-sets.Consideranysetinwhichthethresholdvaluesarexedattheiroptimalt-valuesascomputedbyEquations 2a and 2b ,8i=1,...,n,andwheretheitemvaluesvarerestrictedasfollowsforsomenonnegativeintegerK: v0i)]TJ /F4 11.955 Tf 11.95 0 Td[(yiviv0i+yi8i=1,...,n (2a)nXi=1yiK (2b)yi2f0,1g8i=1,...,n. (2c) where0.Here,v0iactsasanominalvalueforeachi=1,...,n.NotethatConstraints 2a statethatthetruevalueforitemiissomewhereintheinterval[v0i)]TJ /F4 11.955 Tf 11.74 0 Td[(,v0i+].Constraint 2b statesthatonlysomeKparametersvimaydeviatefromtheirnominalvalues,andConstraint 2c stateslogicalrestrictionsonthey-variablesthatcontrolwhichparametersdeviatefromtheirnominalvalues. OurapproachtosolvethisproblemfollowsthegeneralstrategyemployedinAlgorithm 1 :Thesellersearchesforasequencethatresultsinsomeitemi=1,...,nbeingchosenbythecustomer,stoppingwhenthehighest-protitemcanbesold.However,wemustnowtaketheadversary'sroleintoaccount,notingthatthecustomervaluesmaybemodiedfromtheirnominalvaluesbytheadversarytopreventanitemfrombeingsold,orinducea(low-prot)itemtobesold. Forconvenience,wedene: J1i=f1,...,i)]TJ /F4 11.955 Tf 11.96 0 Td[(1g8i=1,...,n, (2a)J2i=fi+1,...,ng8i=1,...,n. (2b) 27

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Also,givenasequenceofitems,deneposiasthepositionofitemiinthesequence. Wenextprovidethefollowingproposition,whichestablishestheformofoptimalsequencesgivenanuncertaintysetoftheform. Proposition2.2. Letibethe(smallest)indexoftheitemsoldtothecustomerinanoptimalsequencegivenanuncertaintysetoftheform.AnoptimalsequenceconsistsofitemsinsetsA1,...,A5(someofwhichmaybeempty)intheorder A1{A2{A3{A4{itemi{A5(2) suchthat: A1consistsofitemsj2J2isuchthatv0j+tposj+1,includingthelastelementofA2, A3consistsofitemsj2J2isuchthatv0j
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A4issimilartoA2,withoutthecaveatthattheadversarymusttakeactiontopreventtheirsale. A5consistsofallitemsinJ2inotcontainedinA1orA3. Consideranyoriginalsequenceinwhichitemicanbesold,whichdoesnotsatisfy 2 .Weshowthattherealsoexistsamodiedsequencesatisfying 2 suchthatitemiisstillsold. First,considerthecaseinwhichtheadversaryisforcedtouseKmodicationstopreventaniteminJ1ifrombeingsoldintheoriginalsequence,beforeitemiiseventuallysold.(Forsimplicity,wesaythattheadversaryhasattackedaniteminJ1iiftheadversarysetsvi=v0i)]TJ /F4 11.955 Tf 12.38 0 Td[(topreventitfrombeingsold.)Intheoriginalsequence,letp0bethepositionoftheearliestscheduleditemthatbelongstoJ1i,andletp00bethepositionoftheKthitemthatisattackedinJ1i,notingthatp00tposj+1;thelatterinequalityalsoindicatesthatiftheadversaryhadtoattacktheiteminslotp0intheoriginalsequence,itmuststilldosowhenthisitemisinthelaterslotposjinthemodiedsequence.Afterrepeatingthisprocedure,allitemsinA1willprecedethoseinA2. Furthermore,supposethatthelastiteminA2isnotattackedbytheadversary.ByswappingtheorderofanyitemjinA2thatisattackedwiththelastiteminA2,wehavethatitemjisstillattackedinitslaterslot.Hence,asequenceexistsinwhichthelastiteminA2isattacked. Now,saythatthelastiteminA2endsatslotq0intheoriginalsequence.IfnoitemsinJ2iarescheduledafterA2andbeforei,thenA3isempty.Else,supposethatthelastiteminJ2ischeduledbeforeitemiisinpositionq00.IfthereisnoiteminJ1ipositioned 29

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insomeslotqsuchthatq0
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J1ibesortedinnonincreasingorderoftheirnominalvalues,yieldingthesequence(1),...,(jJ1ij).Also,letJ2ibesortedinnonincreasingorderoftheirnominalvalues,yieldingthesequence(1),...,(jJ2ij).Then: 1. ItemsinA1areorderedinthefollowinggreedyfashion:Setj=1andq=1,anddetermineifv0(j)+psuchthatv0(1)>tq+1.Then,item(j)isplacedintheearliestpositionq>pos(j)]TJ /F9 7.97 Tf 6.59 0 Td[(1)suchthatv0(j)>tq+1,foreachj=2,...,K.Allotherpositionsp+1,...,pos(K)thathavenotbeenassignedanitem(ifany)areassignedunscheduleditemsfromJi1inanyarbitraryorder. 3. ItemsinA3areorderedaccordingtothesamegreedyalgorithmasforA1,exceptwestartatpositionq=jA1j+jA2j+1,ignorethoseitemsthathavebeenscheduledinA1,andtestwhetherv0(j)p0.Wecouldgenerateamodiedsequencebyswappingthepositionofitemsjandk.Byassumption,itemkwouldnotbepurchasedbythecustomer(evenifitsvalueweremodiedbytheadversary).Ifp00posi,then 31

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itemiispurchasedbeforeitemjwouldbeencounteredbythecustomerinthemodiedsequence.Byperformingallsuchswaps,werecoveramodiedsequenceinwhichclaim1holdstrue.Notethatclaim3alsoholdstruebythesameargument. Toshowthatclaim2holdstrue,weusesimilarmechanicsasintheproofthatclaims1and3holdtrue.Letp0betheminimumindexinA2,withitemj2J1iagainbeingpositionedinslotp0,suchthatitemjisattackedbytheadversaryinthecurrentsequence(e.g.,v0j>tp0+1),andwhereahigher-valueditemk2J1iispositionedinslotp00>p0.Consideramodiedsequenceobtainedbyswappingthepositionofitemsjandk.Itemkmustbeattackedbytheadversaryinthemodiedsequencebecausevk>vj.Notethatitemjmuststillbeattackedinthemodiedsequenceaswell,because(a)itwasattackedintheoriginalsequenceinpositionp0,(b)itemjismovedtoalaterslotp00,and(c)bythesequencerule 2 ,wehavethatp00
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fewerthanKattacks.Thus,thereisnoimplicitrequirementtoorderitemsinA4thatforcestheadversarytoattack.ThejusticationforthegreedyalgorithmusedtoarrangeitemsinA1isthesameasgiveninLemma 2 ObservethatitisdifculttopredictbeforesolvingaproblemwhetheranoptimalsequencewillbegeneratedaccordingtoLemma 2 or 3 ,andiftheformer,whichvalueofp(=jA1j)shouldbeused.Therefore,analgorithmusedtosolvethecaseinwhichouruncertaintysetisoftheformwillconsidereachpossibilityallowedbyLemmas 2 and 3 Tostatethisalgorithm,whentryingtosellitemi=1,...,nafterestablishingthatitems1,...,i)]TJ /F4 11.955 Tf 9.89 0 Td[(1cannotbesold,wesortitemsinJ1iandJ2iinnonincreasingorderoftheirnominalvalues,toyieldthesequences(1),...,(J1i)and(1),...,(jJ2ij),respectively.WerstattempttoordertheitemsaccordingtoLemma 3 .Ifitemicannotbesoldbythissequence,wesetanintegerparameterP=jA1jinthesequenceproducedbyLemma 3 ,andattempttocreateasequencegeneratedaccordingtoLemma 2 foreachp=0,...,P.(Notethattheremaynotbeasequencepossibleforsomevaluesofp:Ifpistoosmall,thentheremaynotbeenoughitemsinJ1itollallslotsbetweenslotp+1andpos(K),wherethelastiteminA2mustbescheduled.)Ifnosequencecanbefoundthatsellsitemi,thenweseti=i+1andrestarttheprocess.Thealgorithmendsassoonasasequenceisfoundthatsellsitemi. NotethatthisalgorithmcanbeperformedinO(n3)steps,givenbytheO(n2)complexityofsearchingthroughsequencesproducedbyLemmas 2 and 3 intryingtosellitemi,andtheO(n)numberofitemsithatmustbeexploredbythealgorithm.WeleaveforfutureresearchtheexplorationofamoresophisticatedalgorithmthatwouldattempttoreducetheeffortrequiredtosearchthesequencesgivenbyLemmas 2 and 3 2.4MaximizationofExpectedProt Inthissection,weexamineanalternativecharacterizationofuncertaintythatarisesintheseller'sproblem.Here,theprot,value,andcustomerthresholddataareall 33

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potentiallyuncertain.WemodelthisuncertaintybyconsideringasetQofscenarios,wherescenarioq2Qoccurswithprobabilityq,andinwhichallprot,value,andthresholddataassociatedwithscenarioqissuperscriptedbyq(e.g.,vqireectsthevalueofitemiinscenarioq,andsoon).Denoteq0astheprobabilityofrealizingscenarioq2Q,wherePq2Qq=1. UnlikeinSection 2.3 ,weinsteadexaminetheproblemofmaximizingexpectedprot(whichwecallproblemEXP),ratherthanmaximizingtheminimumprotthatcouldbeobtainedfromasequence.Moreformally,problemEXPseeksasequenceofallitems,suchthattheexpectedprot(givenbythesummationoverallq2Qoftheitem'sprotthatwouldbesoldinscenarioqmultipliedbyq)ismaximized.ThefollowingtheoremanditscorollarystatethatoncesomedegreeofuncertaintyisincorporatedtotheproblemstudiedinSection 2.2 (whetherforthemax-minormax-expectationcase),theproblemcanbecomesubstantiallydifcultingeneral. Theorem2.1. ProblemEXPisNP-hard,evenwhenallprotvaluesaredeterministicandthecustomerusesoptimalthresholdvalues. Proof. Thecorrespondingdecisionproblem,EXPD,isstatedasfollows:ForagivenparameterG,doesthereexistasequencewhoseexpectedprotisatleastG?WewillshowthatEXPDisNP-complete,whichimpliesthatEXPisNP-hard.Assumingthatthecustomersolvesthestoppingprobleminpolynomialtime,EXPDclearlybelongstoNP:Foranygivensequencewecaneasilydeterminetheitemthatischosenbythecustomerineachscenario,computetheexpectedprot,andchecktoseeiftheexpectedprotisatleastG. NextweshowthatEXPDisNP-completebytransformingtheclassical3SATproblemtoanequivalentinstanceofEXPD.First,wedene3SATasfollows[ 35 ]. ConsiderasetofclausesConasetU=fu1,...,ungofnbinaryvariables(whichcaneithertakevaluesoftrueorfalse).Eachclausecontainsthreeliteralvalues,eachofwhichisatrueorfalsevalueforoneparticular 34

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variable.DoesthereexistatruthassignmentofbinaryvaluestothevariablesinU,i.e.,anassignmentoftrueorfalsevaluesforeveryvariableinUsuchthateveryclausehasaliteralthatmatchessomevalueintheassignment? WedenoteuTi(uFi)asaliteralhavingatrue(false)valueforvariablei.Forinstance,ifaclauseconsistsofliteralsfuT1,uF3,uT6g,thenanytruthassignmentmustsatisfytheconditionthateitheru1=true,oru3=false,oru6=true. Considerany3SATinstancewithnvariablesandmclauses,denotedbyCj,8j=f1,...,mg.WedeneitemsTi,8i=f1,...,ng,correspondingtouTiliteralsandFi,8i=f1,...,ng,correspondingtouFiliteralsinourEXPDinstance.Eachofthese2nitemshasaprotof1.Wedeneonefurtheritem,denotedbyZ,whichhasaprotof0.Letthecustomer'sthresholdvaluesbeoptimalforher,ascomputedinEquations 2a and 2b .Also,wedeneparametert?tobeavaluesatisfyingtherelationshiptn+1t?
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note,foranysequencewedeneearlyitemsastherstnitemsinthesequence,andlateitemsasthenextnitems(butnotthelastiteminslot2n+1). First,supposethatthereexistsasolutiontothe3SATinstance.Toconstructasequence,weplaceitemFiinslotianditemTiinslotn+iifthe3SATvariableuiistrue,8i=1,...,n.Otherwise,ifuiisfalse,weplaceitemTiinslotianditemFiinslotn+i,8i=1,...,n.ItemZispositionedinslot2n+1.Notethatinanyscenario,anearlyitemwillnotbechosen,becausetheearlyitemvaluesequaleithert?or0,whicharelessthantn+2andwillnotbechosenbythecustomerwhentheybelongtotherstnpositionsofthesequence.Ifalateitemexistshavingavalueoft?inscenarioq,thecustomerwillbuyonesuchitemataprotof1tothesellerinscenarioq.Forscenarioq=1,...,n,notethateitherTqorFqisalateitem,andhasvaluet?inscenarioq.Forscenarioq=n+1,...,n+m,oneofthethreeitemsinclauseq)]TJ /F3 11.955 Tf 11.27 0 Td[(ncorrespondstoalateitemhavingvaluet?,duetotheassumptionthatthelateitemscorrespondtoa3SATtruthassignment.Ineveryscenario,aprotof1isobtained,andsotheexpectedprotisG=1.Hence,theEXPDinstancehasasolution. Next,supposethatthereexistsasolutiontothetransformedEXPDinstance.NotethatthisisequivalenttoenforcingtheconditionthatitemZisnotchosenbythecustomerinanyscenario.Wewillshowthatthelateitemsinsuchasequencecorrespondtoasolutiontothe3SATinstance. ObservethatitemZmustbeplacedinslot2n+1.Ifnot,someitemTi(orFi)mustbethelastiteminthesequence.IfitscomplementaryitemFi(Ti)isanearlyitem,thecustomerskipstheearlyiteminscenarioiandchoosesitemZ.Otherwise,itemFi(Ti)isalateitem,whichimpliesthereexistsapairofitemsTkandFkforsomekthatarebothscheduledasearlyitems.ThisresultsinitemZbeingchoseninscenariok.Hence,itemZmustbepositionedinslot2n+1inourEXPDsolutioninordertoavoidsellingitemZintherstnscenarios.Moreover,bythesamelogic,exactlyoneoftheitemsTiorFimustbealateitem,fori=1,...,n. 36

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Usingtheaboveobservations,wesetthevariableuitotrue(false),ifitemTi(Fi)isalateitem.BecauseZisnotchoseninanyscenarioq=n+1,...,n+m,alateitemcorrespondstoaliteralinclauseCjforeachj=1,...,m.Therefore,theproposed3SATsolutionisfeasibletothe3SATinstance. Finally,notethatthetransformationcreatesapolynomialnumberofitemsandscenarios.Itishenceapolynomialtransformationift?ispolynomiallyrepresentable(i.e.,ifwerequirethatthenumberofbitsrequiredtorepresentt?ispolynomiallyboundedbynandm).Forexpediency,wecouldassumethatthecustomerusesthresholdswithprecisionboundedbyapolynomialfunctionofn.Takingt?=(tn+1+tn+2)=2,wehavethatanencodingoft?canalsoperformedusingapolynomialnumberofbits.Moregenerally,evenifthecustomerusesexactthresholdswithunlimitedprecision,wedemonstrateintheAppendix A thatavaluefort?,encodedusingapolynomialnumberofbits,canbecomputed.Thiscompletestheproof. Indeed,aconsequenceofthistheoremisthatthegeneralmax-minproblem(foranyarbitraryU6=;withmonotonethresholdvalues)mustalsobeNP-hard. Corollary1. Themax-minproblemisNP-hardforgeneraluncertaintysetsU,evenwhenallprotvaluesaredeterministicandthecustomerusesoptimalthresholdvalues. Proof. WecanusethesametransformationgivenforTheorem 2.1 ,whereUconsistsofthediscretevaluesetsgivenabovewiththethresholdvaluesdeterminedbyEquations 2a and 2b .Wewouldtheninsistonaworst-caseprotof1,whichturnsouttobeidenticaltorequiringthattheexpectedprotequals1.Thus,themax-minproblemwithgeneraluncertaintysetsisalsoNP-hard.Observethatwedonotmakeanyclaimsregardingtheinclusionofadecisionvariantofthemax-minproblemintheclassNP,becauseitisnotclearthatsolvingtheadversary'sproblemisgenerallyachievableinpolynomialtimeforgeneralU. 37

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TheimplicationofTheorem 2.1 isthatnopolynomial-timesolutionexistsforproblemEXP(unlessP=NP).Wethusprovideamixed-integerprogramming(MIP)modelforthisproblem,whichcanbesolvedbystandardMIPtechniques[ 53 ].(Indeed,stochasticprogrammingmodelsliketheonewefaceinthissectionareoftensolvedbydecompositiontechniques[ 11 ],butthedevelopmentofthesealgorithmsisbeyondthescopeofthiscapter.) ToformulatethisMIP,weletN=f1,...,ngforconvenience,anddenethefollowingsetofdecisionvariables.Letxij,8i2N,j2N,beabinarydecisionvariablethatequals1ifitemiisinslotjofthesequenceand0otherwise.Also,letzqj,8j2N,q2Q,beabinarydecisionvariablethatequals1iftheiteminslotjischosenbythecustomerinscenarioq,and0otherwise.Wedeneanewparameteraqij,8i2N,j2N,q2Q,whichequals1ifitemicouldpossiblybechosenbythecustomerinscenarioqifplacedinslotj,i.e.,: aqij=8>><>>:1ifi2N,j2fpi,...,ng,q2Q,0otherwise.(2) 38

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WebeginbystatinganonlinearMIPthatmodelsproblemEXP: maxXq2QqXi2NbiXj2Nxijzqj (2a)s.t.Xj2Nxij=18i2N (2b)Xi2Nxij=18j2N (2c)zqjaqijxij)]TJ /F7 7.97 Tf 13.62 15.21 Td[(j)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xk=1zqk8i2N,j2N,q2Q (2d)zqjXi2Naqijxij8j2N,q2Q (2e)zqj1)]TJ /F7 7.97 Tf 13.62 15.21 Td[(j)]TJ /F9 7.97 Tf 6.59 0 Td[(1Xk=1zqk8j2N,q2Q (2f)xij2f0,1g8i2N,j2N (2g)zqj2f0,1g8j2N,q2Q. (2h) Theobjectivefunction 2a calculatestheexpectedprot:Foreachscenarioq2Q,ifzqj=1,thenthesellerreceivesaprotofbi,weightedbyprobabilityq,ifitemi2Nisplacedinslotj2N(i.e.,ifxij=1).Constraints 2b and 2c guaranteethateachitemisassignedtoexactlyoneslotandviceversa.Constraints 2d 2f enforcetheconditionthatzqjequals1ifandonlyifjistherstslotforwhichtheassigneditemisatisestheconditionvqi>tj+1inscenarioq.Toseethis,consideranitemipositionedinslotj(xij=1),andobservethatifvqitj+1,thenaqij=0.Thus,Constraints 2e implythatzqj=0inthiscase.Ifhowevervqi>tj+1,andthusaqij=1,thentherearetwocasestoconsider.Iftheredoesnotexistaslotbeforejwhosecorrespondingitemhasbeenchosen(e.g.,Pj)]TJ /F9 7.97 Tf 6.58 0 Td[(1k=1zqk=0),thenConstraints 2d forcezqj=1.Iftheredoesexistak2f1,...,j)]TJ /F4 11.955 Tf 12.65 0 Td[(1gsuchthatzqk=1,thenConstraint 2f forcezqj=0asdesired.ObservethereforethatConstraints 2h canbereplacedsimplywith 39

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zqj0,8j2N,q2Q,notingthatz-variablesmustbebinary-valuedgivenbinaryx-values. Althoughtheobjectivefunctionisnonlinear,itcanbeeasilyconvertedtoalinearfunctionduetothefactsthat(a)nonlinearityonlyarisesduetothenonlineartermsxijzqj,and(b)thex-variablesarerestrictedtobebinary-valued.Byintroducinganewsetofvariables qij,8i2N,j2N,q2Q,whicharedesignedtotakeonthevalueofxijzqj,weobtainthefollowinglinearMIP: maxXq2QqXi2NbiXj2N qij (2a)s.t.Constraints( 2b ){( 2h ), (2b) qijxij8i2N,j2N,q2Q, (2c) qijzqj8i2N,j2N,q2Q. (2d) ObservethatConstraints 2c and 2d force qijxijzqj;equalitycomesfromthefactthatoptimizationwillforcethe -variablestotakeontheirlargestpermissiblevalues. 40

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CHAPTER3AMIXED-INTEGERBILEVELPROGRAMMINGAPPROACHFORACOMPETITIVEPRIORITIZEDSETCOVERINGPROBLEM 3.1IntroductionandLiteratureStudy Weaddressinthischapteratwo-playerStackelberggameonaprioritizedsetcoveringproblem.Inthe(one-player)prioritizedsetcoveringproblem,thereexistsasetofitemsthatcanbeselectedinordertosatisfyasetofclauses.Specically,eachclausecontainsanorderedpartialsetoftheitems.Theplayerincurscostsforeachitemthatisselected,andreceivesrewardsbasedonsatisedclauses.Inparticular,aclauseissatisedonlybythehighest-rankeditem(ifany)thattheplayerselects,andtherewardgrantedtotheplayerfromthisclausedependsontheitemthatsatisestheclause. Inthetwo-playerproblemweconsider,theplayersactinaStackelbergleader-followerfashion,inwhichthefolloweractswithfullknowledgeoftheitemsselectedbytheleader.Theleaderisawareofthefollower'sobjective,andmakesitsdecisionsinanticipationthatthefollowerwilloptimizeitsresponsetotheleader'sdecision.Themaincontributionthatwemakeinthischapterisanexactsolutionmethodforabilevelprogrammingmodelrepresentingthisgame,wherethebilevelprograminvolvesbinaryvariablesrepresentingdecisionsmadebyeachplayer. Whilethefocusofthischapterisonthegeneraltwo-playerprioritizedsetcoveringproblem,webrieydiscusstwoscenariosinwhichthisparticularproblemarises. Anaturalsettingforthisproblemarisesinnewproductdevelopmentandintroduction,whichisoneofthemostimportantstrategicdecisionsforrmsinacompetitivemarket[ 5 30 46 48 ].Here,thesetcoveringitemsmayrepresentpotentialproductsthatcanbedevelopedbyacompany,andtheclausesmayrepresentcustomers(ormarketsegments)thatwishtopurchaseproducts.Eachcustomerhasaprioritizedlistofproductsthat(s)hewouldbuyifavailable.Aftertheleaderrmintroducesasetofproducts,thefollowerrm(whichmayactuallycompriseasetofcompetingrms),respondsbyintroducingitsownsetofproducts.Customersthenchooseaproductthathasthemostutilitytothem.Theintroductionofthefollower'sproductsmaythereforesubstantiallyreducetheprotsanticipatedbytheleader. 41

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Thisconcernisespeciallyrelevantinthepresenceofpredatoryrmsthatexplicitlyseektominimizetheleader'sprot.Smithetal.[ 68 ]studyatwo-stageproductintroductiongamesimilartotheonementionedabove,butinwhichthefollowerseekstominimizetheleader'sprot.Inthiscase,theleaderestablishesaproductintroductionstrategythatisrobusttoanypossibleactionstakenbythefollower.However,thepredatorymodelmaybefartooconservativeinsomepracticalsettings.Thecurrentstudy,bycontrast,focusesonthecaseinwhichthefollowersimplyactstomaximizeitsownprotsratherthantominimizetheleader'sprots.Itisworthnotingthatthealgorithmemployedin[ 68 ]isbasedontheprinciplethattheleader'sobjectivefunctionislimitedbytheworst-possiblefollower'sresponsetotheleader'sactions.Asaresult,thisalgorithmisnotvalidfortheproblemconsideredinthischapter,andtheapproachtakeninthepresentchaptermustbefundamentallydifferentfromtheonetakenin[ 68 ]. Anotherapplicationareainwhichthisproblemarisesisincompetitivefacilitylocation.Inthissetting,thereexistsasetofpotentialfacilitylocations(thesetcoveringitems)andgeographicallylocatedcustomers(setcoveringclauses).Customerswillgravitatetothemostconvenientlocatedfacility,andhenceacustomer'spreferencelistisgovernedbythedistancefromfacilitylocations.Theleader,inmakingaone-timedecisiononwheretodeployfacilities,maythusbeconcernedabouttheplansofacompetitorinattractingcustomersacrosstheregionunderconsideration. Onceagain,thepresenceofapredatoryfollowerhasbeenstudiedintheliterature,basedonmin-maxcutting-planeprinciples.Wereferthereaderto[ 22 23 45 58 60 ]forrecentresearchinthiseld.Toourknowledge,though,noresearchhasyetbeentailoredtothecaseinwhichthefollowerisinterestedinmaximizingitsownprot,ratherthanminimizingtheleader'sprot.Moreover,theapproachestakeninthesepapersonceagaincannotbedirectlyextendedtotheproblemunderinvestigationinthepresentchapter. Basedontheforegoingapplicationareas,weusethemoreintuitivetermsproductstodescribethesetcoveringitems,andcustomerstodescribethesetcoveringclausesthroughoutthischapter. Multilevelprogramsaremathematicalprogramsinwhichsomeofthedecisionvariablesareconstrainedtobeoptimalwithrespecttosomeothermathematicalprograms[ 47 ].Thesemodelsoftenariseinthecontextofl-level,nonzero-sumgames,wherethestrategyoftheplayerataspeciclevelisknownbytheupper-levelplayers.Theavailabilityofdifferentdecisionstothelower-levelplayersissignicantlyaffected 42

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bytheactionstakenbytheupper-levelplayers[ 3 ].Theseproblemsarereferredtogenerallyasmathematicalprogramswithequilibriumconstraints[ 17 65 ]. Inparticular,bilevelprogramsareofsubstantialimportanceduetothefactthattheyhavebeenwidelyappliedtoStackelberggamesettingsthatinvolvetwointeractingplayersindifferentlevels,pursuingdifferentobjectives,butusingasetofcommonresources.VicenteandCalamai[ 72 ]andDempe[ 26 ]conductliteraturesurveysonthebilevelprogramsandtheirapplications.AmorerecentoverviewofbilevelprogrammingisgivenbyColsonetal.[ 24 ]. Linearbilevelprograms(i.e.,bilevelproblemsinwhichboththeupper-andlower-levelproblemsarelinearprograms,givenxedvaluesoftheotherplayer'svariables)arebyfarthemoststudiedcasesamongmultilevelprograms.Suchproblemsexhibitpropertiesthatmakethemamenabletomethodsthatuseacombinationofbranch-and-boundandimplicitsearchoverextremepoints.Branch-and-boundcanbeappliedtoa0-1mixed-integerprogramconvertedfromthelinearbilevelprogrambysubstitutingthelower-leveloptimalityconstraintwithanequivalentKKTsystem[ 1 ].Implicitsearchmethodsexploitthefactthatanoptimalsolutionofalinearbilevelprogramwillalwaysexistatanextremepointoftheconstraintset[ 10 ]. CandlerandTownsley[ 18 ]suggestanimplicitsearchmethodinwhichasubsetofallpossibleoptimalbasesforthelower-levelproblemisexaminedinordertoreachanoptimalbasicfeasiblesolutiontotheupper-levelproblem.BialasandKarwan[ 10 ]reportseveralalgorithmstosolvelinearbilevelprograms,includingapproachesthatprovablyidentifylocaloptimalsolutions.Astudyofcharacterization,solutionapproaches,andrelatedmodelsforlinearbilevelprogramshavealsobeenconductedbyWenandHsu[ 74 ].Bystudyingamixed-integerprogrammingreformulationofthelinearbilevelproblem,Audetetal.[ 1 ]introducethreesetsofvalidinequalitieswithinabranch-and-cutframework.Theseinequalitiesareshowntobevalidforallbilevelfeasiblesolutions,whilecuttingoffthesolutionsthatviolatethecomplementarity 43

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constraints.InexactsolutionapproachesusingGeneticAlgorithmsandTabuSearchmethodshavebeenalsostudiedintheliterature;see[ 36 ]and[ 41 ].ThereaderisreferredtotheextensivestudiesbyBard[ 3 ]andDempe[ 25 ]forathoroughtreatmentoflinearbilevelprogrammingmethods. Discretebilevelprograms,however,entailadifferentsetofchallenges.MooreandBard[ 51 ]proposeaspeciallytailoredbranch-and-boundalgorithmformixed-integerbilevelprograms.Theydevelopnode-fathomingrulestoidentifythenodesinwhichtherelaxedproblemisinfeasible,orisnotbetterthantheincumbentsolution.Utilizingthoserules,theydevelopabranch-and-boundmethodthatcomputesanoptimalsolutionwithinanitenumberofsteps.AnotherexactsolutionmethodsuggestedbyThirwaniandArora[ 16 71 ]iterativelyeliminatesoptimalsolutionstobilevelprogrammingrelaxationsinacutting-planefashion,wheneverthesesolutionsarenotoptimaltothelower-levelproblem.DeNegreandRalphs[ 27 ]employasimilarapproachbysolvingthesamebilevelprogrammingrelaxation,butprescribedifferentcuttingplanescomputedfromtheconstraintsthatarebindingattherelaxation'soptimalpoints.Mitsos[ 49 ]derivesanotherfamilyofvalidinequalitiesthatcanbeaddedtothesamerelaxationsofthemixed-integerbilevelproblem,whichleadstoanitelyconvergentalgorithm. Morecomplexformsofbilevelprogramshavebeenalsostudied.Mitsosetal.[ 50 ]presentasolutionmethodtondaglobalsolutionfornonlinearbilevelprograms.Theconvergenceproofoftheiralgorithmisbasedonnecessaryconditionsrequiredforbothlower-andupper-levelproblems.Byconvertingthelower-levelproblemintoamulti-parametricprogrammingproblem,Faiscaetal.[ 29 ]proposeanothersolutionmethodtoreachaglobalsolutionofbilevelprogramswithquadraticobjectivefunctions.Ozaltinetal.[ 56 ]studyabilevelstochasticknapsackprobleminwhichuncertaintyispresentintheright-hand-sideparameters. Theremainderofthischapterisorganizedasfollows.InSection 3.2 ,westatetheformaldenitionofourproblem.Wethendevelopaclassofexactsolutionmethods 44

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fortheproblembasedoncutting-planegenerationforaso-calledhigh-pointprobleminSection 3.3 .Finally,wereportcomputationalresultsofourproposedalgorithminSection 3.4 3.2ProblemFormulation DenesetN,withn=jNj,astheproductsinthesetcoveringproblem,andletMbethesetofcustomers.Customeri2Mhasanorderedproductpreferencelist,Oi=(p1i,p2i,...,pk(i)i),thatrepresentstherelativeutilityofeachproducttocustomeri.Customeriwillpurchasethehighest-rankedproductamongallproductsinOithathavebeenselectedbyeitherplayer,orwillpurchasenoproductatallifnoiteminOiisavailable. Theleaderstartsthegamebyselectingitssetofproducts.Withtheknowledgeoftheleader'sdecision,thefollowerthenselectsitssetofproducts.Therevenueearnedbytheplayersfromcustomeriiscomputedasfollows.Supposethatcustomeripurchasesproductj.Ifoneplayerselectsproductj,itearnsarevenueofrij.Ifbothplayersselectproductj,therevenuewillbedividedbasedonacoefcientij2[0,1],suchthattheleaderearnsijrij,andthefollowerearns(1)]TJ /F8 11.955 Tf 12.93 0 Td[(ij)rij.Theleader's(follower's)costtoselectproductjisgivenbybj(cj).Furthermore,weimposeabudgetlimitBfortheleader,whichlimitsthetotalcostincurredinselectingtheleader'sproducts. Notethatthe-valuesdiscussedaboveareusefulinexpandingthescopeofproblemsthatcanbeconsideredunderthisframework.Ifforinstanceselectingsomeproductj2Nshouldblockthefollowerfromselectingthesameproduct,thensettingij=1foreachi2Mequivalentlymodelsthisrequirementbynullifyinganysharedrevenuesthatthefollowercouldobtainfromrepeatingtheleader'sselectionofproductj. Tomodelthisproblem,wedenedecisionvariablesxj=1iftheleaderselectsproductj2N,andxj=0otherwise.Similarly,denedecisionvariablesyj=1ifthefollowerselectsproductj2N,andyj=0otherwise.Letwijandzijbevariables 45

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representingtherevenueearnedfromthesaleofproductj2Ntocustomeri2Mbytheleaderandthefollower,respectively.Also,denesetsHij,8i2M,j2Oi,asthesetofproductsthathaveahigherrankthanproductjinthepreferencelistforcustomeri.Wehavethefollowingformulation. max)]TJ /F3 11.955 Tf 11.95 0 Td[(bTx+Xi2MXj2Oiwij (3a)s.t.bTxB (3b)wijrijxj8i2M,j2Oi (3c)wijrij(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)8i2M,j2Oi,k2Hij (3d)wijrij(1)]TJ /F3 11.955 Tf 11.96 0 Td[(yk)8i2M,j2Oi,k2Hij (3e)wijrij(1)]TJ /F4 11.955 Tf 11.96 0 Td[((1)]TJ /F8 11.955 Tf 11.96 0 Td[(ij)yj)8i2M,j2Oi (3f)xj2f0,1g8j2N (3g)yispartofanoptimalsolutionto( 3{2 ), (3h) whereProblem 3 isdenedasfollows,giventheleader'sdecisionvariablevalues,x: (ProblemFx):x=max)]TJ /F3 11.955 Tf 11.95 0 Td[(cTy+Xi2MXj2Oizij (3a)s.t.zijrijyj8i2M,j2Oi (3b)zijrij(1)]TJ /F3 11.955 Tf 11.96 0 Td[(yk)8i2M,j2Oi,k2Hij (3c)zijrij(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xk)8i2M,j2Oi,k2Hij (3d)zijrij(1)]TJ /F8 11.955 Tf 11.96 0 Td[(ijxj)8i2M,j2Oi (3e)yj2f0,1g8j2N. (3f) 46

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Theobjectivefunction 3a representstheleader'sprot.TheproductselectionbudgetisenforcedbyConstraint 3b .Wenextguaranteethatwij=0ifproductjhasnotbeenofferedbytheleader(Constraints 3c ),iftheleaderselectsatleastoneproductk2Hij(Constraints 3d ),orifthefollowerwillselectaproductk2Hij(Constraints 3e ).Finally,Constraints 3f statethatifproductjhasbeenselectedbybothplayers,thentheleadercannotearnmorethanijrijfromsellingproductjtocustomeri.Binarinessofx-variablesisenforcedbyConstraints 3g ThebilevelnatureofthegameisenforcedbyConstraint 3h .NotethatConstraint 3h permitstheleadertoselectanyvectorythatisanoptimalfollower'sresponsegiventheleader'sactionx.Assuch,thismodelisoptimisticinthatitassumesthatthefollowerbreakstiesamongalternativeoptimalsolutionsinfavoroftheleader.Forthefollowerproblem,theobjectivefunction 3a andtheConstraints 3b 3e aredenedanalogouslytotheleaderproblem. Intherestofthischapter,wedeneX=fx2f0,1gn:bTxBgaspossibledecisionstheleadercantake.Weconcludethissectionbyestablishingthefollowerproblem'scomplexity(seeAppendix B fortheproof). Theorem3.1. ProblemFxisNP-hardinstrongsense. AnimmediateresultofTheorem 3.1 isthatProblem 3 isalsoNP-hard. 3.3ExactSolutionMethod Inthissection,wedescribeanexactsolutionmethodtailoredforthebilevelproblemstatedintheprevioussection.NotethatrepresentingConstraint 3h vialinearinequalitiesisdifcult,becauseitimposestheoptimalityofy-variablestoamixed-integerprogram,whichprohibitsusfromsubstitutingConstraint 3h withanequivalentKKTsystem.Therefore,wesuggestareformulationtoProblem 3 thatisamenabletosolutionviaacutting-planealgorithm. Givenx2X,letybeanyfollower'sdecisionvector.(Notethattheoptimalw-andz-variablesareeasilycomputablegivenvaluesforxandy;hence,werefertoa 47

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leader/followersolutionpairbyjust(x,y)whereconvenient.)Wecall(x,y)abilevelfeasiblesolutiontoProblem 3 ify(alongwithsomez)solvesproblemFx.LettingbethesetthatcontainsallbilevelfeasiblesolutionsofProblem 3 ,wecanstatethefollowingreformulationofProblem 3 max)]TJ /F3 11.955 Tf 11.95 0 Td[(bTx+Xi2MXj2Oiwij (3a)s.t.bTxB (3b)w2W(x,y) (3c)z2Z(x,y) (3d)x,y2f0,1gn (3e)(x,y)2, (3f) whereW(x,y)isthepolyhedralsetthatisdenedbyConstraints 3c 3f ,andZ(x,y)isthepolyhedralsetdenedbyConstraints 3b 3e Wedenethehigh-pointproblem(HPP)astherelaxationofProblem 3 obtainedbyomittingConstraints 3f .UsingtheconceptintroducedbyMooreandBard[ 51 ],wecanequivalentlydeneHPPastheproblemobtainedbycombiningallconstraintsinProblem 3 and 3 ,anddiscardingConstraint 3h .ThemotivationfordeningHPPistocopewiththedifcultyofobtainingtheexplicitdenitionoftheset.OurapproachstartsbysolvingHPP,andverieswhetherthecomputedoptimalsolutionofHPPbelongsto.Ifso,thissolutionmustbeoptimalto 3 ,becauseHPPisarelaxationof 3 .Otherwise,wecanaugmentHPPwithacuttingplaneandre-solveHPPinaniterativefashion.InSection 3.3.1 ,weformallystatethiscutting-planealgorithm,anddescribeauxiliaryseparationroutinesinSection 3.3.2 forgeneratingcuttingplaneswithinthisscheme. 48

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3.3.1Cutting-PlaneAlgorithm Let(x,y)be(partof)anoptimalHPPsolution.IfyisoptimaltoFx,then(x,y)2,andhence(x,y)mustbeoptimaltoProblem 3 .Otherwise,weneedtoidentifyacuttingplane,i.e.,avalidinequalitythatisviolatedbythecurrentbilevelinfeasiblepoint(x,y).Webeginbystatingalowerboundonx,foranyx2X. Lemma4. Letebeavectorofnones.Thenex,8x2X. Proof. Becausexjej,8j2N,andx2X,FxisarelaxationofFe,whichimpliesthatex. Thenextpropositionstatesoneclassofcuttingplanes. Proposition3.1. Let(x,y)beabilevelinfeasiblesolution,anddeneM=x)]TJ /F8 11.955 Tf 12.38 0 Td[(e.ThefollowinginequalityisvalidtoProblem 3 ,andcutsoff(x,y). )]TJ /F3 11.955 Tf 11.95 0 Td[(cTy+Xi2MXj2Oizijx)]TJ /F3 11.955 Tf 11.96 0 Td[(MXj2N)]TJ /F4 11.955 Tf 5.48 -9.69 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)xj+xj(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xj)(3) Proof. Notethattheleft-hand-sideof 3 representsthefollower'sobjectivefunctionvalue.Toseethat 3 isvalid,supposerstthatx=x,andthattheright-hand-sideof 3 reducestox.Inthiscase, 3 simplystatesthatthefollower'sobjectivefunctionvaluemustbeatleastx,whichisvalidbyourdenitionof.Alsobecause(x,y)isbilevelinfeasible,)]TJ /F3 11.955 Tf 9.3 0 Td[(cTy+Pi2MPj2Oizij
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Step2(LowerBound). SolveFx,andobtainanoptimalsolutiony.Letbetheoptimalobjectivefunctionvalueto 3 withx=xandy=y.IfLB,setLB=,andlet(x,y)betheincumbentsolution.ProceedtoStep3. Step3(Termination/CutRoutine) IfLB=UB,terminatewiththeincumbentsolutionbeingoptimal.Otherwise,addacuttingplane 3 toC,andreturntoStep1. Theorem3.2. AlgorithmCPAidentiesanoptimalsolutiontoProblem 3 inanitenumberofiterations. Proof. LetbetheoptimalobjectivevaluetoProblem 3 .Because 3 isvalid,aftereachexecutionofStep1inCPA;also,because(x,y)isfeasibleto 3 ,aftereachexecutionofStep2inCPA.NowsupposebycontradictionthatCPAdoesnotterminatenitely.BecauseXisaniteset,CPAwouldhavetoencountersomesolution^x2XmultipletimesinStep1and2ofthealgorithm.AtthersttimeCPAencounters^x,inequality 3 isaddedtoCwithrespectto^x.However,theproofofProposition 3.1 impliesthatifx=^x,thefollowervectorymustbebilevelfeasibleinallsubsequentiterationsofthealgorithm.Therefore,letting(^x,^y)betheoptimalHPPsolutionfoundthesecondtimeCPAencountersx=^x,wemusthavethat(^x,^y)isbilevelfeasible.ThisimpliesthatLB=UBatStep3,andthealgorithmwouldhaveterminated,whichleadstoacontradiction.Thiscompletestheproof. CPAmightslowlyconvergetoanoptimalsolution,particularlywhenthefaceofconv()inducedby 3)-222()]TJ /F4 11.955 Tf 21.26 0 Td[(4 onlyconsistsofthepoint(x,y),wherexwasthepointusedtogeneratethevalidinequality 3 andyisanoptimalsolutiontoFx.Asaresult,HPPmaybegraduallyaugmentedwithalargenumberofweakcuts,whichimpairsitssolvability.Becausethefaceonconv()inducedby 3 ispossiblyonlyonepoint(i.e.,a0-dimensionalface),werefertothemas0-cuts. Inordertondstrongercutsthatinducefacesofatleastq1dimension,whichwecallq-cuts,westatethefollowingcorollariesofProposition 3.1 50

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Corollary2. LetQ=fx1,...,x(2q)gX,forsomeq1,suchthat xi6=xk1i
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wherethelastequalityistruebecausex2Q. Ourmotivationforusingthetermq-cutstemsfromthefactthatiftheinequality 3 isbindingforatleastq+1pointsinQ0,thentheresultinginequality 3 mustbebindingonatleastq+1afnelyindependentbilevelfeasiblepoints,implyingthatitinducesaq+1(orhigher)dimensionalfaceofconv().Asaresult,Corollaries 2 and 3 permitustoobtainstrongercutswithinCPA(whichremainscorrectandnitelyconvergentbythesameargumentintheproofofTheorem 3.2 ).Forinstance,let(x1,y1)beabilevelinfeasiblesolutionobtainedbysolvingHPP,andsupposethatweseeka2-cutbasedonthissolution.WestartbyconstructingsetQ0fromCorollary 2 forq=2,whichcontains(xi1,xi2,xi),8i=1,...,4.Nextwendanyinequality1x1+2x2+thatisbindingat(x11,x12,x1)andtwooftheotherthreepoints,andisvalidwithrespecttotheremainingpointinQ0.ThisinequalitysatisesthenecessaryassumptionsforCorollaries 2 and 3 ,andhence,a2-cutisgeneratedoftheform 3 basedon,1,and2. 3.3.2FollowerandSeparationSubproblem Inthissection,wepresentanalternativeapproachtogeneratea(q+1)-cut,givensomestartingq-cutthatisknowntobevalid.ThemotivationforthisapproachstemsfromtheobservationthatemployingCorollary 2 requiresexcessivecomputationaleffortforlargervaluesofq.Forexample,toobtaina3-cutthatisviolatedby(x1,y1),wemustidentifyaninequalityinR4thatisbindingat(x11,x12,x13,x1)andthreepointsofthesetQ0=f(xi1,xi2,xi3,xi):i=2,...,8g,andisvalidwithrespecttotheotherfourpointsinQ0.Thisrequiresthesolutionofeightinstancesofthefollowerproblem.Inaddition,wemayalsohavetoexamine)]TJ /F9 7.97 Tf 5.48 -4.38 Td[(73possiblehyperplanesintheworstcasetoobtainaninequalitysatisfyingtheconditionsofCorollary 2 Giventheinitialq-cut,andasetofpointsfx1,...,xq+1gQbindingonthisinequality(whereQisdenedasinCorollary 2 ),ourstrategyconstructsasinglenew 52

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point~xasfollows. ~xj=1)]TJ /F3 11.955 Tf 11.95 0 Td[(x1jj=q+1~xj=x1jj6=q+1,j2f1,...,ng(3) Itiseasytoverifythat~xremainsafnelyindependentfromtheinitialsetofq+1bindingpoints.Denotingtheinequalitythatisbindingattheseq+2pointsbyPq+1j=1jxj+,wehaveacandidatefor 3 thatcanbepotentiallyusedtoobtaina(q+1)-cut.However,inequality 3 generatedbasedon(,)maynotbevalidforallx-vectors.Therefore,weseekanefcientwayofverifyingthevalidityofthe(q+1)-cut.Considerthefollowingseparationproblem: min0Tx+x)]TJ /F8 11.955 Tf 11.95 0 Td[(0 (3a)s.t.x2X, (3b) where0isthecoefcientvectorofx-variablesin 3 basedon,0istheconstanttermof 3 ,andxisdenedasearlier.Asolutionx2Xhasanegativeobjectivefunctionvaluetoproblem 3 ifandonlyifitviolatesinequality 3 generatedaccordingto(0,0). However,notethatproblem 3 isatwo-stagemixed-integerprogram,becausecomputingxrequiressolvingFx,whichisembeddedin 3 asaninneroptimizationproblem.Hence, 3 isdifculttosolveduetothenonconvexinnerproblem.Ourstrategyistosubstitutetheinnerproblemwithaconvexrestriction(whichyieldsarelaxationof 3 ).Iftherelaxedproblemhasanonnegativevaluefortheobjectivefunctionatoptimality,thentheproposed(q+1)-cutmustbevalid.Otherwise,iftherelaxedversionof 3 hasanegativeoptimalobjectivefunctionvalue,the(q+1)-cutmayormaynotbevalid.Themeritofusingaconvexrestrictionfortheinnerproblemin 3 isthatitcanbesubstitutedwithitsdual,allowingustostatetherelaxationof 3 asonemixed-integerprogram.Weseekarelaxationof 3)-222()]TJ /F4 11.955 Tf 21.25 0 Td[(10 whoseoptimalobjective 53

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valueisnotmuchsmallerthantheoptimalobjectivevalueof 3 ,inordertoverifythatthegeneratedinequalityisvalid. Beforedescribingourrelaxationsof 3 ,wesummarizebysubstitutingStep3withthefollowingtwosteps. Step3a(Termination) IfLB=UB,terminatewiththeincumbentsolutionbeingoptimal.Otherwise,generateacuttingplane 3 forsomevalueofqandproceedtoStep3b. Step3b(CutRoutine) Createanewpointusing 3 ,generateanewcandidateinequality,andverifythevalidityoftheinequalitybysolvingarelaxationof 3 .Iftheobjectiveof 3 isnegative,addtheq-cuttoC,andreturntoStep1.Otherwise,storethenewly-generated(q+1)-cutasthelastidentiedcut,incrementqbyone,andrepeatStep3b. Thefollowingfoursubsectionsconsideralternativeconvexrestrictionsoftheinnerproblem,andtheresultingmathematicalprogramfortheseparationproblemrelaxation. 3.3.2.1Thesameactionrestriction Considerthesettinginwhichthefollowerisrestrictedtoselectthesamesetofproductsselectedbytheleader.Thenforagivenx,thefollowerproblemnowbecomes: )]TJ /F3 11.955 Tf 9.3 0 Td[(cTx+maxXi2MXj2Oizij (3a)s.t.zijrijxj8i2M,j2Oi (3b)zijrij(1)]TJ /F8 11.955 Tf 11.96 0 Td[(ijxj)8i2M,j2Oi (3c)zijrij(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)8i2M,j2Oi,k2Hij. (3d) Letij,ij,8i2M,j2Oi,bethedualvariablesassociatedwithconstraints 3b and 3c ,respectively.Also,letijk,8i2M,j2Oi,k2Hij,bethedualvariablesassociatedwithconstraints 3d .Bysubstitutingthefollowerproblemwiththedualof 54

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3 andcombiningwith 3)-222()]TJ /F4 11.955 Tf 21.25 0 Td[(10 ,theleaderproblembecomes: min()]TJ /F3 11.955 Tf 11.96 0 Td[(c)Tx+Xi2MXj2Oi0@rijxjij+rij(1)]TJ /F8 11.955 Tf 11.96 0 Td[(ijxj)ij+Xk2Hijrij(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xk)ijk1A)]TJ /F8 11.955 Tf 11.95 0 Td[( (3a)s.t.ij+ij+Xk2Hijijk=18i2M,j2Oi (3b)ij,ij08i2M,j2Oi (3c)ijk08i2M,j2Oi,k2Hij (3d)x2X. (3e) Problem 3 hasquadratictermsijxjandijkxk.Wepresentagenericsetofinequalities(introducedin[ 40 ])thatservetolinearizetheseterms.Givenabinaryvariableandacontinuousvariable2[0,r],wereplacet=andrestricttviathepolyhedralset: P1(,,r)=ft2R+:tr,t,t+r)]TJ /F3 11.955 Tf 11.96 0 Td[(rg.(3) Weusetheinequalitiesdeningthispolyhedrontointroducevariables1ij=xjij,2ij=xjij,and3ijk=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xk)ijk,whichlinearizeallquadratictermsin 3a min()]TJ /F3 11.955 Tf 11.96 0 Td[(c)Tx+Xi2MXj2Oi0@rij1ij+rijij)]TJ /F8 11.955 Tf 11.96 0 Td[(ijrij2ij+Xk2Hijrij3ijk1A)]TJ /F8 11.955 Tf 11.95 0 Td[( (3a)s.t.ij+ij+Xk2Hijijk=18i2M,j2Oi (3b)1ij2P1(xj,ij,1)8i2M,j2Oi (3c)2ij2P1(xj,ij,1)8i2M,j2Oi (3d)3ijk2P1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xk,ijk,1)8i2M,j2Oi,k2Hij (3e)ij,ij08i2M,j2Oi (3f)ijk08i2M,j2Oi,k2Hij (3g)x2X. (3h) 55

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3.3.2.2Thesingleproductrestriction Consideranalternativerestrictionthatlimitsthefollowertochooseatmostoneproduct.Wedene: cj=)]TJ /F3 11.955 Tf 9.29 0 Td[(cj+Xi2M0@rij(1)]TJ /F8 11.955 Tf 11.96 0 Td[(ijxj)Yk2Hij(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xk)1A,8j2N.(3) Notethatcjistheleader'sprotassociatedwithproductjforagivenleader'sdecisionvectorx2X.Therestrictedfollowerproblemisstatedasfollows. maxXj2Ncjyj (3a)s.t.Xj2Nyj1 (3b)yj08j2N. (3c) Byintroducinguasthedualvariableassociatedwithconstraint 3b andsubstitutingtheinnerproblemof 3 withthedualof 3 ,problem 3 becomes: minTx+u)]TJ /F8 11.955 Tf 11.96 0 Td[( (3a)s.t.ucj8j2N (3b)u0 (3c)x2X. (3d) Tolinearizeproblem 3 ,letV=fv1,...,vjVjgbeasetofbinaryvariables.Wedenethefollowingpolyhedralset,whichenforcest=QjVji=1vi: P2(V)=ft2R+:tv,8v2V,t+jVj)]TJ /F4 11.955 Tf 29.56 0 Td[(1jVjXi=1vig.(3) 56

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DeneVij=f1)]TJ /F3 11.955 Tf 12.38 0 Td[(xk:k2Hijg,8i2M,j2Oi,andlet4ij=Qk2Hij(1)]TJ /F3 11.955 Tf 12.38 0 Td[(xk).Also,let5ij=xj4ij.Wethenobtainthefollowingrelaxationof 3 minTx+u)]TJ /F8 11.955 Tf 11.96 0 Td[(, (3a)s.t.u)]TJ /F10 11.955 Tf 11.95 11.35 Td[(Xi2M)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(rij4ij)]TJ /F3 11.955 Tf 11.95 0 Td[(rijij5ij)]TJ /F3 11.955 Tf 21.92 0 Td[(cj8j2N (3b)4ij2P2(Vij)8i2M,j2Oi (3c)5ij2P1(xj,4ij,1)8i2M,j2Oi (3d)u0 (3e)x2X. (3f) 3.3.2.3Thesameactionwithonefewerproductrestriction Inthisrestriction,thefollowerisconstrainedtoselectallbutatmostoneoftheproductsofferedbytheleader,andnonethathavenotalreadybeenselectedbytheleader.Hence,wedene: ^cj=cj)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xi2M0@rij(1)]TJ /F8 11.955 Tf 11.96 0 Td[(ijxj)Yk2Hij(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xk)1A,8j2N,(3) asthedifferenceinthefollower'sprotiftheyselectalloftheleader'sproductsexceptforj,andtheprotiftheyselectalloftheleader'sproducts.Then,givenx2X,thefollowingisarestrictionofthefollowerproblem. )]TJ /F3 11.955 Tf 9.3 0 Td[(cTx+maxXi2MXj2Oizij+Xj2N^cjyj (3a)s.t.Constraints( 3b ){( 3d ) (3b)Xj2Nyj1 (3c)yjxj8j2N (3d)yj08j2N. (3e) 57

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Problem 3 canbeseparatedintotwosubproblemsgivenx:oneintermsofz-variables(referredtoasthez-subproblem),andtheotheroneintermsofy-variables(referredtoasthey-subproblem).Notethatthez-subproblemisexactlythesameproblemas 3 (excludingtheterm)]TJ /F3 11.955 Tf 9.3 0 Td[(cTx).Letuandvj,j2N,bethedualvariablesassociatedwithConstraints 3c and 3d ,respectively.Weobtainamixed-integerprogrambydualizingthez-subproblemsimilartoProblem 3 ,anddualizingthey-subproblemusingu-andv-variables,whichresultsinpresenceofquadratictermsxjvj.Inordertolinearizetheseterms,weneedupperboundsontheoptimalvaluesofv-variables.Observethatthedualofthey-subproblemisasfollows. minu+Xj2Nxjvj (3a)s.t.u+vj^cj8j2N (3b)u0 (3c)vj08j2N. (3d) Weclaimthatinanyoptimalsolution(u,v),vjmaxf0,^cjg,8j2N.Toseethis,consideranysolution(u,v)withvj>^cjforsomej2N.Let(u,v)bethesolutionobtainedbylettingvj=minfvj,^cjgif^cj0,andvj=0if^cj<0.Itiseasytoverifythat(u,v)remainsfeasible,becauseu0.Moreover,ityieldsanobjectivefunctionvaluenoworsethanthevaluecomputedfromsolution(u,v),becausex0andu0.Thus,vjmaxf0,^cjg,8j2N,whichimpliesvjcjfrom 3 .Usingthisresult,weobtainthefollowingrelaxationofProblem 3 58

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min'+u+Xj2N 1j (3a)s.t.u+vj+Xi2M)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(rij4ij)]TJ /F3 11.955 Tf 11.96 0 Td[(rijij5ijcj8j2N (3b) 1j2P1(xj,vj,cj)8j2N (3c)u0 (3d)vj08j2N (3e)Constraints( 3b ){( 3h ),( 3c ),( 3d ), (3f) where 1-variablesareintroducedtolinearizethequadratictermsvjxj,'isdenedastheobjectivefunctionofProblem 3 ,andvariables4ijand5ijaredenedsimilartoProblem 3 3.3.2.4Thesameactionwithonemoreproductrestriction Thefourthrestrictionrestrictsthefollowertoofferalloftheproductsselectedbytheleader,whilehavingtheoptionofselectingoneadditionalproduct.Foragivenvectorx2X,thefollowersolves: )]TJ /F3 11.955 Tf 9.3 0 Td[(cTx+maxXi2MXj2Oizij+Xj2Ncjyj (3a)s.t.Constraints( 3b ){( 3d ) (3b)Xj2Nyj1 (3c)yj1)]TJ /F3 11.955 Tf 11.96 0 Td[(xj8j2N (3d)yj08j2N. (3e) Letuandvj,8j2N,bethedualvariablesassociatedwithConstraints 3c and 3d ,respectively.UsingananalysissimilartoProblem 3 ,andlinearizingthe 59

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nonlinearterms,weobtainthefollowingrelaxationofProblem 3 min'+u+Xj2N 2j (3a)s.t.u+vj)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xi2M)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(rij4ij)]TJ /F3 11.955 Tf 11.96 0 Td[(rijij5ij)]TJ /F3 11.955 Tf 21.92 0 Td[(cj8j2N (3b)u0 (3c)vj08j2N (3d) 2j2P1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xj,vj,cj)8j2N (3e)Constraints( 3b ){( 3h ),( 3c ),( 3d ), (3f) where 2-variablesareintroducedtolinearizethequadratictermsvj(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xj),'isdenedastheobjectivefunctionofProblem 3 ,andvariables4ijand5ijaredenedsimilarto 3 3.4ComputationalResults Inthissection,weexaminetheefcacyofourapproachonrandomlygeneratedtestinstances.Westartbydescribingdifferentimplementationdetailsofthealgorithmspresentedinthischapter,andthenshowhowtotailorthealgorithmpresentedin[ 49 ]toourproblem.WeconductdifferentcomputationalstudieswithrespecttotheproductintroductionandfacilitylocationapplicationsdescribedinSection 3.1 3.4.1ImplementationDetailsandInstanceGeneration TherstthreeimplementationsofCPAthatweexamine,denotedbyCPA1,CPA2,andCPA3,donotuseanyoftheseparationprocedurespresentedinSection 3.3.2 .ForCPA1,weimplementCPAwithq=2.CPA2generatesa2-cut(anditscorrespondingQ0),andthenattemptstocomputea3-cutbyconstructinganewpointaccordingto 3 .CPA2thenveriesthevalidityoftheinequalityforallpointsinQ0,whichnowcontainseightpointscorrespondingto3-cut.Ifthecandidateinequalityisvalid,itisaddedtoC;otherwiseCPA2augmentsCbytheinitiallygenerated2-cut.CPA3isimplemented 60

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similarly.Itstartswitha3-cut,aimingtoconvertthatinequalitytoa4-cutusingthesameprocessasgiveninCPA2. WealsostudyfourimplementationsofCPAaugmentedbytheseparationproblemspresentedinSection 3.3.2 .WedenotetheaugmentedCPAimplementationsbyACPA1,ACPA2,ACPA3,andACPA4,correspondingtothefourseparationsubproblemsgiveninSection 3.3.2 .Foreachimplementation,Step3aintheaugmentedCPAstartswitha2-cut.Using 3 ,anewcandidateinequalityisidentiedandthevalidityoftheinequalityisveriedviathecorrespondingseparationsubproblem. Finally,wealsoexaminetheefcacyofusingahybridstrategy,denotedbyHYB,whichrstidentiesa2-cutandemploysProblem 3 toobtaina3-cut.IfHYBcannotverifythevalidityofthecandidateinequalitybysolvingProblem 3 ,itexplicitlyteststhevalidityofthecandidateinequalityforallpointsinthecorrespondingQ0.Ifthecandidateinequalityisnotvalid,theinitial2-cutisaddedtoC.Otherwise,theHYBapproachsetsq=3,andexecutesStep3boftheaugmentedCPA. Finally,weexplainhowtoimplementtheproposedcutting-planealgorithmbyMitsos[ 49 ],denotedbyMITS,inordertocomparetheperformanceofouralgorithmtotheexistingworkintheliterature.Asacutting-planealgorithm,MITSisdesignedtocalculateaglobaloptimalsolutiontogeneralnonlinearmixed-integerbilevelprograms.Toobtainupperbounds(formaximizationproblems),MITSsolvesHPPaugmentedwithcuttingplanesthataredifferentfromthoseproposedinourapproach.Moreprecisely,let(xk,yk)beabilevelinfeasibleoptimalsolutionofHPP(possiblyaugmentedwithvalidinequalities)atiterationk1,andletyksolveFxk.MITSseeksanewset,denotedbyXk,suchthatsolutions(x,yk),8x2Xk,remainfeasibletoFx.LettingfFbetheobjectivefunctionof 3 ,thefollowingisavalidinequalityforProblem 3 : fF(x,y)fF(x,yk),8x2Xk. 61

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Fortheproductintroductioninstances,wegeneratedfourtestsets,denotedbyS1,S2,S3,andS4byvaryingjMj2f6,9gandjNj2f12,15g.Eachtestsetcontainstenrandomlygeneratedinstances.Table 3-1 showstheparametersandthecorrespondinglowerandupperboundsforeachparameter.Allparametersarerandomlygeneratedasintegersdrawnfromauniformdistributionoverthestatedranges,exceptforthe-values,whichareuniformlygeneratedoverthecontinuousinterval[0.1,0.9].NotethatinTable 3-1 ,Bmin=0.75Sb=jNjandBmax=1.25Sb=jNj,whereSbisthesumofallgeneratedb-values. Table3-1. Productintroduction:parametersusedtogeneratetestinstances ParameterNameValue Leader'sbudget(B)[Bmin,Bmax]Leader'sproductselectioncosts(bj)[70,190]Follower'sproductselectioncosts(cj)[90,170]Customerpreferencelistsizes(jOij)[1,n]Revenues(rij)[20,130]Revenuesharecoefcients(ij)[0.1,0.9] Forthefacilitylocationapplication,wegeneratedtestsetsS5,S6,S7,andS8byvaryingjMj2f12,15gandjNj2f9,12g.TheparametersarereportedinTable 3-3 .WerstrandomlygeneratedjMjasthenumberofcustomersandjNjasthenumberofpotentialfacilitylocationswithinarectangleoflength120andwidthof90.Next,basedonsomerandomlygenerateddistancethreshold,d,wedeterminedthepotentialfacilitiesthatcanserveeachcustomer.Foreachcustomeri,wethenplaceallpotentialfacilitylocationsinOiinnondecreasingorderoftheirdistancefromcustomeri,omittingthosewhosedistancefromiexceedsd.Notethatrij=rikforallj,k2Nforthisapplication,becausethedemandfromeachcustomerisassumedtobeindependentoftheselectedfacility.Finally,BminandBmaxaredenedinasimilarwaytoTable 3-1 WeimplementedallvariantsinVisualC++8.0equippedwithCPLEX12.2ConcertTechnologyonanIntelCorei5PCwith4GBofmemory.Foreachimplementation,wesetthemaximumallowablerunningtimetobe1200seconds. 62

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Table3-2. Facilitylocation:parametersusedtogeneratetestinstances ParameterNameValue Leader'sbudget(B)[Bmin,Bmax]Leader'sfacilitylocationselectioncosts(bj)[50,200]Follower'sfacilitylocationselectioncosts(cj)[10,100]Distancethresholdforcustomers(d)[50,70]Revenues(rij)[30,150]Revenuesharecoefcients(ij)[0.1,0.9] 3.4.2Results WenowdiscusstheperformanceoftheCPAvariantsonthetwofeaturedapplications.Table 3-3 illustratestheaveragetimeandnumberofcutsnecessaryforeachCPAvarianttoreachanoptimalsolution.AccordingtoTable 3-3 ,CPA1ismoreefcientthanCPA2andCPA3insolvinginstancesofproductintroductionapplication.Ontheotherhand,CPA1isoutperformedbyCPA2andCPA3forthefacilitylocationapplication,particularlyonlargerinstances.NotethatCPA1requiresmorecutstoreachanoptimalsolutionincomparisontoCPA2andCPA3.ThisisduetothefactthatCPA2andCPA3arecapableofgeneratingstrongercuts.Theresultshowsthatforthefacilitylocationapplication,generatingstrongercutsarebenecialinthattheoverallalgorithmtakeslesstimetoreachanoptimalsolution.However,thereductionintimerequiredtosolveHPPforproductintroductioninstancesdoesnotcompensatefortheextratimerequiredbyCPA2andCPA3togeneratestrongercuts. Table3-3. ComparisonofCPAimplementations CPA1CPA2CPA3 ApplicationSetAvgTimeAvgCutsAvgTimeAvgCutsAvgTimeAvgCuts ProductintroductionS1:(jNj=12,jMj=6)7.5468.923911.0234S2:(jNj=12,jMj=9)20.4513123.7710924.0397S3:(jNj=15,jMj=6)14.9110912.417615.2566S4:(jNj=15,jMj=9)40.9527542.0122545.34201FacilitylocationS5:(jNj=9,jMj=12)7.34486.68316.4826S6:(jNj=12,jMj=12)101.0738190.9125089.82207S7:(jNj=9,jMj=15)9.81469.34309.4326S8:(jNj=12,jMj=15)149.36419131.37262133.79214 Similarly,weinvestigatetheefcacyofthefouraugmentedCPAimplementations.Table 3-4 indicatesthatforbothapplications,ACPA2outperformsACPA1,ACPA3, 63

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andACPA4,butnotthebestavailableCPAvariants.Theseparationsubproblemstendtobesubstantiallydifculttosolve,andtheyeitherfailtogenerateastrongercut,orthegeneratedcutisnotstrongenoughtooffsetthetimespentsolvingtheseparationsubproblem.Notethatfortheproductintroductioninstances,ACPA2employsslightlyfewercutscomparedtootherimplementations,whereasforthefacilitylocationapplication,ACPA4requiresthefewestnumberofcutstoreachanoptimalsolution.Infact,ACPA4computesstrongercutsattheexpenseofspendingextratimetosolvemorefollowerprobleminstancesorharderseparationproblems,whichresultsinlongercomputationtimesfortheoverallalgorithm.ItisalsoworthnotingthatthedifferencebetweentheperformanceofACPA3andACPA4isindistinguishableontheproductintroductiontestinstances,buttheadvantageofusingACPA4insteadofACPA3isstronglypronouncedonthefacilitylocationinstances. Table3-4. ComparisonofaugmentedCPAimplementations ACPA1ACPA2ACPA3ACPA4 SetAvgTimeAvgCutsAvgTimeAvgCutsAvgTimeAvgCutsAvgTimeAvgCuts S120.444112.513019.724119.5841S268.4911537.8110169.3211569115S341.758926.638956.208953.1789S4152.4824478.93242251.27244241.06251S520.194012.4240264023.0127S6278.86312143.97261370.18312295.81217S725.333816.143733.833826.6319S8307.89331226.67331379.23331372.50259 Asaresult,relaxation 3 outperformstheotherproposedrelaxationsinverifyingthegeneratedcandidatevalidinequalities.Theresultsalsoshowthatthelessrestrictedversionsof 3 presentedinSections 3.3.2.3 and 3.3.2.4 donotyieldmorepromisingrelaxationsthan 3 ingeneral. BasedontheresultsfromTables 3-3 and 3-4 ,wealsoexaminetheefcacyofHYB.Inparticular,weemployrelaxation 3 intheimplementationofHYBduetothefactthatitturnedouttobethebestproposedrelaxationinSection 3.3.2 .WereporttheresultsofemployingHYBandMITSinTable 3-5 ,alongsidetheresultsfromusingthe 64

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bestCPAvariantforeachapplication(CPA1forproductintroductionandCPA2forfacilitylocation)andtheresultsofACPA2previouslystatedinTables 3-3 and 3-4 ,respectively. Table3-5. PerformancecomparisonforthebestCPA,ACPA2,HYB,andMITS BestCPAACPA2HYBMITS SetAvgTimeAvgCutsAvgTimeAvgCutsAvgTimeAvgCutsAvgTimeAvgCuts#Opt S17.54612.51308.742344.725910S220.4513137.8110126.3581497.381798S314.9110926.638915.9863545.771598S440.9527578.9324251.83196988.742793S56.683112.42406.992637.884110S690.91250143.9726189.34185569.821546S79.343016.14379.9425105.536310S8131.37262226.67331134.73214893.982384 NotethatMITSconsumesalargeamountofcomputationtimeforlargerinstances,andmayfailtoreachanoptimalsolutionwithinthemaximumallowablerunningtime.Therefore,wehavereportedthenumberoftimesthatMITSterminateswithin1200secondsinthecolumn#OptinTable 3-5 .(ACPUtimeof1200secondsisrecordedforthoseinstancesthatdonotterminatewithinthetimelimit,andthenumberofcutsgeneratedforMITSwithinthistimelimitisfactoredintotheAvgCutscolumn.) BasedontheresultsinTable 3-5 ,MITSisalwaysoutperformedbyallvariantsofourproposedalgorithm.Notethatonproductintroductioninstances,CPA1remainsthebestvariantofourproposedalgorithm.However,thedifferencebetweenCPA2andHYBforthefacilitylocationtestinstancesisindistinguishable.ThemeritofusingHYBisgenerallyobservableinfewercutsnecessarytosolvetheproblem,butthismayhappenattheexpenseofspendingextratimeonsolvingmorefollowerprobleminstancesorseparationproblemsthatmaynotbebenecialoverall(seeresultsofusingCPA2andHYBforsetsS2andS4). 65

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CHAPTER4ACUTTING-PLANEALGORITHMFORSOLVINGAWEIGHTEDINFLUENCEINTERDICTIONPROBLEM 4.1IntroductionandLiteratureStudy Weconsiderascenarioinwhichtwoplayers,adefenderandanattacker,competeonadirectednetworkG(V,A),whereVisthesetofnodesandAisthesetofarcs.Initially,thedefenderownseverynodeinthenetwork,andcanprotectasubsetofnodesagainstanimpendingactionbytheattacker.Theattackerthenacts,withfullknowledgeofthedefender'saction,tocaptureasetofunprotectednodes.Forconsistencywithpriorrelatedstudies,wesaythatcapturednodeshavebeeninuencedbytheattacker.Thisinitialactiontakesplaceattime0,andthegamecontinuesforT(discrete)timeperiodsaccordingtothefollowingrules. 1. Aninuencednoderemainsinuencedfortheremainderofthetimehorizon. 2. Anodethatwasprotectedbythedefendercannotbeinuencedatanytime. 3. Consideranunprotectednodej2Vthatisnotinuencedattimet2f0,...,T)]TJ /F4 11.955 Tf -420.04 -14.45 Td[(1g.Thennodej2Vbecomesinuencedattimet+1ifandonlyiftherearesomeQnodesi2Vsuchthatiisinuencedattimet,and(i,j)2A. 4. Theattackerearnsarewardofrtiifnodeiisinuencedattimet(butnotattimet)]TJ /F4 11.955 Tf 11.95 0 Td[(1,ift1). Thegoalofthedefenderistominimizethemaximumsumofrewardsthattheattackercanearn(e.g.,minimizingthemaximumamountofdamagethattheattackercouldpossiblyinictonthedefender'snetwork). Figures 4-1 and 4-2 illustrateaprobleminstanceinwhichQ=3andT=2.Ther-valuesarestatedforeachtimeperiodnexttoeachnode.Considerthecaseinwhichnonodesareinitiallyprotected,andtheattackerinuencesnodes1,2,6,8,and9att=0(Figure 4-1 a).Astheresultofthisaction,nodes3and5becomeinuencedatt=1,becausenodes1,2,and8areinuencedatt=0(Figure 4-1 b).Nodes4and7becomeinuencedatt=2(Figure 4-1 c).Hence,theattackerearnsarewardof480.Next,supposethatthedefenderprotectsnodes6and9.Thenanoptimalresponse 66

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Figure4-1. AninstancewithQ=3andT=2,intheabsenceofprotectednodes. Figure4-2. AninstancewithQ=3andT=2,withnodes6and9protectedbythedefender. fromtheattackeristoinuencenodes1,2,and8(Figure 4-2 a).Althoughnodes3and5becomeinuencedatt=1(Figure 4-2 b),onlynode7willbeinuencedatt=2(Figure 4-2 c).Inthiscase,theattacker'srewardreducesto310. ThisproblembelongstotheclassofStackelbergleader-followergames[ 73 ],becausethetwoplayersmaketheiractionsinturns,wherethefollower(attacker)operateswithfullknowledgeoftheleader's(defender's)decision.Apopularapproachforsolvingtheseproblemsmodelsthemastwo-stageinterdictionproblems,whichare 67

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thensolvedviabilevelprogrammingmethods.Brownetal.[ 13 14 ]providesummariesonthevariousapplicationsofinterdiction,mostlyfromahomelandsecurityperspective(seealsorecentsurveysanddiscussionsin[ 66 67 ]). Whilewerefertothosesurveysforatreatmentofthehistoryofinterdictiondevelopment,wenotethatacommonmethodofapproachingthesolutionofinterdictionproblemstransformsthebilevelmin-maxproblemintoanonlinearminimizationproblembydualizingthe(attacker's)innermaximizationproblem,e.g.,asdonebyWood[ 75 ].However,thisapproachassumestheexistenceofastrongdualformulationfortheattacker'soptimizationproblem,which(aswewillshow)isnoteasilyobtainableinthiscase,becausetheattacker'sproblemisNP-hardinthestrongsense.Hence,thecommonmethodologyusedtosolvethesedefender-attackerproblemsisnotapplicabletotheinuenceinterdictionproblemthatweconsider,whichnecessitatesanewapproachthatwewillexploreinthischapter. Theattacker'sproblemthatweconsiderinthischapterisrelatedtotheclassicaldominatingsetproblem[ 62 ].Inthedominatingsetproblem,aminimum-cardinalitysubsetofnodesDissoughtinanundirectedgraphG=(V,E)suchthateverynodeinVnDisadjacenttoatleastoneofthenodesinD[ 35 ].Severalvariantsofdominatingsetproblemhavebeenstudiedintheliterature,includingtheconnecteddominatingsetproblem,inwhichthesubsetDneedstobeaconnectedgraph[ 12 ],andthepowerdominatingset,inwhichnodesinDmustdominatenodesandarcsinG[ 39 77 ]. Inthecontextoftheproblemswestudyinthischapter,dominationisaspecialcaseofinuenceinwhichthereisasingletimeperiod,andasinglenodecaninuenceallofitsadjacentnodes.Thisconceptcanbeextendedtodominationviamultiplelinksinanetworkaswell.Forinstance,Wuetal.[ 76 ]studyanextendedversionofthedominatingsetprobleminwhichanodeisinuencedeitherbyitsdominatingneighbororbysomekdominatingnodesthatcanreachthenodeintwohops.Kempeetal.[ 42 ]considertheproblemofidentifyingasetofsomeknodestoinitiallyinuence,with 68

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theaimofinuencingasmanynodesaspossible(untilnomorenodescanbecomeinuenced,e.g.,TjVj).Theauthorsconsidertwodiffusionmodels,whichdictatehowinuencespreadsacrossthenetwork.Oneisathresholdmodelsimilartotheonewedescribeabove,andtheotherisanindependencecascademodel,whereeachinuencednodehasoneopportunitytoinuenceaneighbor,anddoessoinaprobabilisticmanner.Theauthorsshowthattheirinuencefunctionissubmodular,whichenablesthemtoprovidea(1)]TJ /F4 11.955 Tf 12.15 0 Td[(1=e)greedyapproximationalgorithm(see[ 52 ]forapproximationalgorithmtheoryasappliedtosubmodularfunctions). Leskovecetal.[ 44 ]consideramodelinwhichinuencespreadstoalladjacentnodes,asinformationspreadsthroughoutasetofnetworkedblogs.Thegoalistomonitorasetofblogs(nodes)thatdetectspreadinginformationasquicklyaspossible.Chenetal.[ 19 ]provideimprovedscalablealgorithmsforapproximatingthemaximuminuenceproblemaddressedin[ 42 ].Fromadifferentperspective,theinuenceproblemcanalsobeformulatedasndingaminimumcardinalitysetofnodes,whichwheninitiallyinuenced,willeventuallyleadtotheinuenceofallnodesinthenetwork.Dinhetal.[ 28 ]showthatthenumberofinitiallyinuencednodesis(n),andprovideanO(1)-approximationalgorithminpower-lawnetworksandO(logn)-approximationalgorithmingeneralnetworks.Finally,Shenetal.[ 64 ]investigatethisprobleminmultiplexnetworks,inspiredbythescenarioinwhichuserscansimultaneouslyspreadinuenceintomultiplenetworks. Theprecedingworksfocusonmaximizationofinuencewithoutthepresenceofnodeprotection.Conversely,insteadofmaximizinginuence,someresearchhasrecentlybeenproposedtocontainthespreadofmisinformation.Shenatal.[ 63 ]examineanodedeletionproblemwiththeaimofminimizingthemaximumcomponentsizeofagraph.Viewingdeletednodesasprotectednodes,thisproblemtherebylimitsthemaximumnumberofnodesthatcouldbeinuencedfromasinglesourcewhen 69

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Q=1.Fromacomputernetworkperspective,[ 55 70 ]explorethedeploymentofbenigncomputerwormstocounteractmaliciouscodeinanactivefashion. Relevanttoourstudy,Budaketal.[ 15 ]addresstheproblemofinuencelimitation,givenasingleinitialpointofinuence.Asinourstudy,theirproblemprotectsnodesagainstinuence,butprotectednodesalsoinjecttheirowngoodinuenceintothenetwork(asopposedtotheadversary'sbadinuence).Theauthorsassumethatifgoodandbadinformationsimultaneouslyarriveatanode,thegoodinformationwillbeadopted,andthatthesetofnodesspreadingmisinformationisknownapriori.Nguyenetal.[ 54 ]studytheproblemofndingasmallestsetofnodesfromwhichgoodinuenceservestocontainthespreadofmisinformation.Theyinvestigateboththecaseinwhichtheoriginatingnodesthatspreadmisinformationareknown,andthecaseinwhichtheyareunknown. Thereareseveralkeydifferencesbetweenthestudyproposedinthischapterandthosethatprecedeit.Forone,weseekanoptimalsolutiontotheinuenceinterdictionprobleminlieuofanapproximationscheme.Moreover,therewardfunctionearnedbytheattackerismoregeneral,andcancapturethecaseinwhichtheattacker'sbenetininuencingnodesisdiscountedasafunctionoftime.Aswewilldescribeinthenextsection,theonlyassumptionmadeontherewardfunctionisthattheattacker'srewardforinuencinganodeisanonincreasingfunctionofthetimeatwhichitisrstinuenced:Thisfunctionneedbeneitherconcavenorconvex.Assuch,ourproblemcharacterizationcapturesdifferentproblemclassesthanthosethathavebeenstudiedintheliterature. Exactoptimizationalgorithmsforthisproblemwillnaturallyrequireconsiderablymorecomputationaleffortthantheaforementionedapproximationalgorithms,especiallythosethatarescalabletolarge-scalesocialnetworks.Ofcourse,exactalgorithmsareusefulincreatingbenchmarksforheuristicschemes,sothatonecanempirically(onsmallernetworks)determinetheeffectivenessofsuchalgorithmsinndingnear-optimal 70

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solutions.Still,theinterdictionofinuencehasmanyapplicationsonsmallerscalenetworks,whereexactalgorithmsmaybeappliedinpracticalsituations.Forinstance,ifQ=1andT=1,withallr-valuesequalling1,thentheproblemseekstominimizethemaximumnumberofnodesthatcouldbedominatedbyasetofunprotectednodes.AsTgrows,thenotionofdominationisrelaxed.Defensenetworksmayseektofortifyphysicalpositionsagainstattack,wherethesepositions(representedasnodes)arevulnerableifsomeQlocationsdecidetosimultaneouslyattack.Notethatthespreadofinuenceinthiscasemayrefertoadvancingmilitaryunitsthatcapturepositionsastheyattack,usingthemasforwardpointsforfurtherattacks. Theremainderofthischapterisorganizedasfollows.WeformallydenetheinuenceinterdictionprobleminSection 4.2 andprovideatwo-stagemathematicalformulationthatmodelstheproblem.InSection 4.3 ,weexamineasetofalternativecutting-planeapproachestosolvingthisproblem.Section 4.4 revisitstheattacker'sproblemformulatedinSection 4.2 ,exploringformulationsthatemployfewerbinaryvariablesthanthenaturalformulationfortheproblem.Finally,weinvestigatetheefcacyofouralgorithmsonrandomlygeneratedinstancesinSection 4.5 4.2ProblemFormulation Foreachnodei2V,denethesetofincomingneighborsofiasV)]TJ /F4 11.955 Tf 7.09 -4.33 Td[((i)=fj2V:(j,i)2Ag,andthesetofoutgoingneighborsofiasV+(i)=fj2V:(i,j)2Ag.LetT=f1,...,Tgbethesetoftimeperiods.Recallthatanunprotectednodei2Vthatisnotinuencedattimet)]TJ /F4 11.955 Tf 12.13 0 Td[(1becomesinuencedattimet2TifatleastQnodesinV)]TJ /F4 11.955 Tf 7.09 -4.33 Td[((i)areinuencedattimet)]TJ /F4 11.955 Tf 12.64 0 Td[(1,wherewerefertoQasthethresholdinuenceparameter.Also,recallthattheattackerearnsarewardofrtiifnodei2Visinuencedattimet2T[f0gforthersttime,wherer0irTi.Thereexistsacostofci,i2V,fortheattackertoinuencenodeiattimezero.Similarly,thedefenderincursacostofbi,i2V,toprotectnodei.Inourmodel,thedefender(attacker)hasabudgetofB(D)toprotect(initiallyinuence)nodes. 71

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Inordertoformulatethisproblem,werstdenetwosetsofbinarydecisionvariablesxi,i2V,andy0i.Inourmodel,xi=1ifthedefenderprotectsnodei2V,andxi=0otherwise.Also,y0i=1ifnodei2Visinuencedbytheattackerattimezero,andy0i=0otherwise.Additionally,weintroducebinarydecisionvariablesyti=1,t2T,ifnodei2Visinuencedattimet,andyti=0otherwise.Notethatwehaveseparatedy0-variablesfromyt-variablestoemphasizethedifferencebetweeninuenceatt=0andt>0,becausethelatterresultsfromthespreadofinuence.Thedefender'sproblemcanbeformulatedasfollows. minz(x) (4a)s.t.bTxB (4b)xi2f0,1g8i2V, (4c) wherez(x)istheoptimalobjectivevalueoftheattacker'sproblem,whichcanbecomputedbysolvingthefollowingintegerprogramgivensomexedvalueofx=x: ATT1(x):z(x)=maxXi2V r0iy0i+TXt=1rti(yti)]TJ /F3 11.955 Tf 11.95 0 Td[(yt)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)! (4a)s.t.yti1)]TJ /F4 11.955 Tf 12.13 0 Td[(xi8i2V,t2T[f0g (4b)QytiQy0i+Xj2V)]TJ /F9 7.97 Tf 6.25 -2.27 Td[((i)yt)]TJ /F9 7.97 Tf 6.58 0 Td[(1j8i2V,t2T (4c)cTy0D (4d)y0i2f0,1g8i2V (4e)yti2f0,1g8i2V,t2T. (4f) Theobjectivefunction 4a reectsthedefender'sgoalofminimizingthemaximumrewardearnedbytheattacker(computedbysolving 4 ).Thedefender'sbudgetlimitandthebinarinessofthex-variablesareenforcedbyConstraints 4b and 4c ,respectively.Theobjectivefunction 4a representstheattacker'sreward,notingthat 72

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y0i=1ifnodei2Visinitiallyinuenced,andyti)]TJ /F3 11.955 Tf 12.18 0 Td[(yt)]TJ /F9 7.97 Tf 6.59 0 Td[(1i=1ifnodei2Visinuencedforthersttimeattimet2T.Constraint 4b impliesthataprotectednodecanneverbeinuencedbytheattacker.Constraints 4c governsthespreadofinuence:Ifnodei2Visinitiallyinuenced,thentheright-hand-side(RHS)ofConstraints 4c willbeatleastQforallt2T,implyingthatnodeiwillremaininuencedatalltimeperiods.Now,supposethatnodei2Visnotinitiallyinuenced,andconsiderConstraint 4c fornodeiandtimet2T.TheconstraintimpliesthatnodeicanbeinuencedattimetifandonlyifPj2V)]TJ /F9 7.97 Tf 6.26 -2.27 Td[((i)yt)]TJ /F9 7.97 Tf 6.59 0 Td[(1jQ,i.e.,ifandonlyifatleastQnodesadjacenttonodeiareinuencedattimet)]TJ /F4 11.955 Tf 12.14 0 Td[(1.Notethatforanynode-timepairi2Vandt2T,ytiispresentwithanonnegativecoefcientintheobjectivefunction(duetononincreasingvaluesfornodei'srewardsovertime).Therefore,thereexistsanoptimalsolutioninwhichyti=1ifandonlyifeithernodeiisinitiallyinuenced,oratleastQnodesadjacenttonodeiareinuencedattimet)]TJ /F4 11.955 Tf 11.95 0 Td[(1.Finally,Constraint 4d enforcestheattacker'sbudgetlimit,andConstraints 4e and 4f restrictthey-variablestobebinary-valued. Intherestofthischapter,wedeneX=fx2f0,1gjVj:bTxBgasthesetofpossibleactionsforthedefender.Givenx2X,wealsodeneY(x)=fy02f0,1gjVj:cTy0D,y0i1)]TJ /F4 11.955 Tf 12.51 0 Td[(xi,8i2Vgasthesetofavailableactionsfortheattackerattimezerowhenthedefenderchoosesx. Wemayalsowishtoconsiderthecaseinwhichtheinuencethresholdvaluedependsonthenodebeinginuenced,andsonodeibecomesinuencedattimetifsomeQinodesinV)]TJ /F4 11.955 Tf 7.08 -4.34 Td[((i)areinuencedattimet)]TJ /F4 11.955 Tf 12.71 0 Td[(1.Thiscasecanbetransformedtothecaseinwhichallnodeshaveacommonthresholdvalue,Q.Toseethis,letQ=maxi2VfQig.CreateasetofQdummynodesthatareimpossibletoprotectandfreefortheattackertoinitiallyinuence,andlettherewardforinuencingthesenodesequal0atalltimeperiods.Foreachi2V,createanarcfromQ)]TJ /F3 11.955 Tf 12.56 0 Td[(Qiofthedummynodestonodei.Becauseanoptimalsolutionexistsinwhichallofthesedummynodeswouldbeinitiallyinuenced,onlyQimorenodesinV)]TJ /F4 11.955 Tf 7.08 -4.34 Td[((i)fromtheoriginalgraphmust 73

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beinuencedinordertoinuencenodei,asdesired.Hence,forsimplicity,weusethecommonthresholdvalueofQinthischapter. Wenishthissectionbyobservingthattheattacker'sproblemisstronglyNP-hard.Toseethis,wesketcha(polynomial)reductionfromthedominatingsetproblem[ 35 ]toavariantoftheattacker'sproblem.Inthedominatingsetproblem,weseekasubsetDofnodesinanundirectedgraphG(V,E)suchthateachnodeinVnDisadjacenttoatleastonenodeinD,andsuchthatjDjforsomegivenpositiveinteger.Now,considertheattacker'sproblemwithQ=1andT=1.Lettheattacker'sproblemnetworkG(V,A)consistofthesamenodesetasinthedominatingsetinstance,andletAcontaintwodirectedarcs,(i,j)and(j,i),foreach(i,j)2E.Also,deneB=andbi=r0i=r1i=1,8i2V.Thereexistsadominatingsethavingnodesifandonlyifthereexistsasolutiontotheattacker'sproblemwithrewardjVj.Hence,theattacker'sproblemisstronglyNP-hard.Moreover,thedefender'sproblemisalsoNP-hard,becauseevaluatingz(x)cannotbedoneinpolynomialtimeunlessP=NP. 4.3ExactSolutionMethod Inthissection,weprovideacutting-planeschemetosolvetheproblemconsideredinthischapter.InSection 4.3.1 ,westateareformulationto 4 thatisamenabletoacutting-planealgorithm,andprovideobjectivefunctionboundsthatwillbeusefulinouralgorithm.InSection 4.3.2 ,wedevelopasetofvalidinequalitiesfortheproblem,andstateourcutting-planealgorithm.Then,inordertoimprovetheefcacyoftheproposedalgorithm,wedeviseastrongerclassofcuttingplanesinSection 4.3.3 4.3.1ReformulationandObjectiveBounds TheinherentdifcultyinsolvingProblem 4 isduetothenonconvexityoftheattacker'sproblem,whichprohibitsusfromreadilyobtainingastrong(minimization)dualtoProblem 4 andemployingstandardinterdictionmodelsasusedin[ 75 ].Inordertodeviseacutting-planealgorithm,westartbyproposingareformulationoftheproblemconsideredinthischapter. 74

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Wereformulate 4 byintroducingavariablez,andminimizingzsubjecttotherestrictionthatx2Xandzz(x).Werefertoapair(x,z)asatwo-stagefeasiblesolutionifx2Xandzz(x).Deningasthesetofalltwo-stagefeasiblesolutions,weobtainthefollowingreformulationforthedefender'sproblem: DEF:minz (4a)s.t.x2X (4b)(x,z)2. (4c) Notethatanoptimalsolution(x?,z?)toProblem 4 satisesz?=z(x?),becauseProblem 4 isaminimizationprogram. Letbeafeasibleregioninducedbyasetofafneinequalities,anddeneDEF-RastherelaxationofProblem 4 obtainedbyreplacingwith.ThemotivationforintroducingDEF-Rstemsfromthefactthatexponentiallymanyinequalitiesmayberequiredfortheexplicitdenitionof.Hence,ourapproachstartswithaninitialdenedbyasmall(polynomial-size)setofinequalities.Ifanoptimalsolution(x,z)toDEF-Ristwo-stagefeasible,thenitmustbealsooptimalto 4 ,becauseDEF-RisarelaxationofProblem 4 .Otherwise,wecanaugmentDEF-Rwithacuttingplane(asdiscussedinSections 4.3.2 and 4.3.3 )andre-solveDEF-Rinaniterativefashionuntilatwo-stagefeasiblesolutionisfound.Westartbycomputinglowerandupperboundsfortheoptimalobjectivevalueoftheattacker'sproblem. Lemma5. Leti(j),1jjVj,bethenodehavingthejthlargestrewardatt=0.Denotebyq1thelargestintegersuchthatPi2V0ciD,8V0V:jV0j=q1.Also,denotebyp2thelargestintegersuchthatPi2V0biBforsomeV0V:jV0j=p2.Dene:zmin=minfjVj,p2+q1gXj=p2+1r0i(j). 75

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Wehave: z(x)zmin8x2X.(4) Proof. LetM=minfjVj,p2+q1g,anddeneJ=f1,...,Mg.Observethatp2isthemaximumnumberofnodesthatcanbeprotectedbythedefender,andthattheattackercaninuenceanysetofq1nodesattime0.Thus,theremustexistasubsetJJ,jJj=minfjVj)]TJ /F3 11.955 Tf 18.24 0 Td[(p2,q1g,thattheattackercaninitiallyinuence.TheattackercanthusalwaysachieveaninitialrewardgivenbythesumoftheMsmallestr0-valuesinJ,whichequalszmin.Thiscompletestheproof. Lemma6. Letq2bethelargestintegersuchthatPi2V0ciDforsomeV0V:jV0j=q2.Givenadefender'sdecisionvectorx,anupperboundontheattacker'soptimalobjectivevalueisobtainedbysolvingthefollowingproblem. zmax(x)=maxXi2V(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(r0iy0i+r1i(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y0i) (4a)s.t.Xi2Vy0iq2 (4b)0y0i18i2V (4c) Proof. Iftheattackeradoptsaninitialattack,y02Y(x),thentheattacker'srewardfromnodei2Vis0ifnodeiwasprotected,r0iifnodeiwasnotprotectedandy0i=1,andisnomorethanr1iifnodeiwasnotprotectedandy0i=0.Thelatterboundisvalidbecauseifanunprotectednodei2Visnotinitiallyinuenced,thenitcannotbeinuencedearlierthantime1,andr1irti,8t=2,...,T.Therefore z(x)maxXi2V(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(r0iy0i+r1i(1)]TJ /F4 11.955 Tf 12.25 0 Td[(y0i), overally02Y(x).BecausethefeasibleregiondenedbyConstraints 4b 4c containsY(x),wehavethatz(x)zmax(x),andthiscompletestheproof. 76

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NotethatProblem 4 canbeoptimizedinO(jVjlog(jVj))stepsbysortingthe(1)]TJ /F4 11.955 Tf 12.42 0 Td[(xi)(r0i)]TJ /F3 11.955 Tf 12.24 0 Td[(r1i)-valuesinnonincreasingorder,andsettingy0i=1foreachnodei2Vcorrespondingtotheq2-largestsuchcoefcients. Theboundz(x)zmax(x)isvalidforanyx2X,andsoonestrategymayenumerateseveralcandidatesolutionsx2X,computezmax(x)foreachvector,andobtaintheminimumsuchvalueasavalidupperbound.Additionally,wecansolvethefollowingoptimizationproblem. zmax=minzmax(x) (4a)s.t.x2X. (4b) ObservethatProblem 4 isatwo-stageprograminwhichthefeasibleregionoftheinnerproblemisindependentofx-variables.ThisallowsustostateProblem 4 asalinearmixed-integerprogrambydualizingtheinnerproblem.Letandi,i2V,bethedualvariablescorrespondingtoConstraints 4b and 4c ,respectively.Weobtainthefollowingreformulationof 4 zmax=Xi2Vr1i+minq2+Xi2V)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(i)]TJ /F3 11.955 Tf 11.95 0 Td[(r1ixi (4a)s.t.+i+(r0i)]TJ /F3 11.955 Tf 11.96 0 Td[(r1i)xi(r0i)]TJ /F3 11.955 Tf 11.96 0 Td[(r1i)8i2V (4b)0 (4c)i08i2V (4d)x2X. (4e) NotethatProblem 4 hastobesolvedonlyonceinordertoobtainanupperboundfortheproblem,andhencewillnotlikelyrepresentasubstantialportionofthetimerequiredtosolvetheoverallmodelweinvestigatehere. Analternativestrategyistoheuristicallyselectsomex2X,e.g.,byusingthefollowinggreedyalgorithm.Initializexi=0,8i2V,andsetaremainingbudgetvalue 77

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B=B.Findanindexisuchthatr0iismaximizedoveralli2Vsuchthatxi=0andbiisnotmorethanB.Ifnosuchindexiexists,thensettheupperboundtozmax(x).Else,setxi=1,reduceBbybi,andreiterate.Inourinitialcomputationalexperiments,wecomparetheeffectivenessoftheexactformulation 4 versustheuseofthisgreedyalgorithm,andemploythemosteffectiveoneinourcomputationalstudy. 4.3.2Cutting-PlaneAlgorithm Inthissection,weprovidevalidinequalitiesforDEF-R,andweproposeacutting-planeschemeforidentifyinganoptimalsolutionto 4 Considerx2Xandsupposethaty=(y0,...,yT)isoptimaltoATT1(x).Wedenei,i2V,astheearliesttimethatnodeiisinuencedinthesolutiony.Weusetheconventioni=T+1ifnodei2Visneverinuencedbytheattacker,andweletrT+1i=0.Considerthevector=(1,...,jVj),andforalli2V,deneRiasthesetofallunprotectednodesj2VsuchthatthereexistsadirectedpathfromnodeitonodejusingjorfewerarcsinA(andhence,i2Ri). Lemma7. Considerasolutionx2Xinwhichxi=0forsomenodei2V.Lety=(y0,...,yT)beanoptimalsolutiontoATT1(x),withcorrespondingvector.Supposethatthesolution^x,whichisidenticaltoxwiththeexceptionofsetting^xi=1,isfeasibleto 4 .Thenwehave: z(^x)z(x))]TJ /F10 11.955 Tf 13.1 11.36 Td[(Xj2Rirjj.(4) Proof. Westartbyconstructingasolution^y=(^y0,...,^yT)toATT1(^x)asfollows.Let^ytj=0,8j2Ri,t2T[f0g,and^ytj=ytj,8j2VnRi,t2T[f0g.Werstprovethat^yisfeasibletoATT1(^x). Notethat^ytjytj,8j2V,t2T[f0g,andinparticular,^yti=0(becausei2Ri).Hence,^ydoesnotviolateConstraints 4b and 4d .Moreover,all^y-valuesremainbinarytosatisfy 4e and 4f .Constraints 4c correspondingtonodej2V,with^y0j=1or^yTj=0areclearlysatised.ForConstraints 4c correspondingtonodej2Vforwhich^y0j=0and^yTj=1,observethat^ytj=ytj=0,8t=0,...,j)]TJ /F4 11.955 Tf 12.38 0 Td[(1,and 78

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^ytj=ytj=1,8t=j,...,T.Itissufcienttoshowthat^yjj=1satisesConstraint 4c fornodejandtimej.Notethatthisconstraintwassatisedinthesolutiony,andhence,if^yj)]TJ /F9 7.97 Tf 6.59 0 Td[(1k=yj)]TJ /F9 7.97 Tf 6.59 0 Td[(1k,8k2V)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((j),thentheresultholds.RecallthatthelengthofanyshortestpathfromnodeitonodejinGexceedsj(orelse,j2Riimplying^ytj=0,8t2T[f0g).Consideranyk2V)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((j)suchthatyj)]TJ /F9 7.97 Tf 6.59 0 Td[(1k=1.Theshortest-pathlengthfromnodeitonodekmustexceedj)]TJ /F4 11.955 Tf 12.26 0 Td[(1(orelse,theshortest-pathlengthfromnodeitonodejwouldnotexceedj).Becauseyj)]TJ /F9 7.97 Tf 6.59 0 Td[(1k=1,wehavethatkj)]TJ /F4 11.955 Tf 12.12 0 Td[(1,andbecausej)]TJ /F4 11.955 Tf 12.2 0 Td[(1<(shrotest-pathlengthfromnodeitonodek),wehavethatk2VnRi.Thisimpliesthat^ytk=1,8t=k,...,T,andinparticular,^yj)]TJ /F9 7.97 Tf 6.58 0 Td[(1k=1,i.e.,^yj)]TJ /F9 7.97 Tf 6.59 0 Td[(1k=1ifyj)]TJ /F9 7.97 Tf 6.58 0 Td[(1k=1,8k2V)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((j).Therefore,^yremainsfeasible. Second,theattacker'sobjective,^z,ofthissolutionisgivenby:^z=Xj2Vr0j^y0j+Xj2VXt2Trtj(^ytj)]TJ /F4 11.955 Tf 12.25 0 Td[(^yt+1j)=Xj2VnRir0jy0j+Xj2VnRiXt2Trtj(ytj)]TJ /F4 11.955 Tf 12.24 0 Td[(yt+1j)=z(x))]TJ /F10 11.955 Tf 11.95 24.03 Td[(0@Xj2Rir0jy0j+Xj2RiXt2Trtj(ytj)]TJ /F4 11.955 Tf 12.24 0 Td[(yt+1j)1A=z(x))]TJ /F10 11.955 Tf 13.1 11.35 Td[(Xj2Rirjj. Becausez(^x)^z,thelemmaholds. Foranygivenx2X,denePxasthesetofprotectednodesinx.Furthermore,lety=(y0,...,yT)beanoptimalsolutiontoATT1(x),anddeneVxasthesetofallinuencednodesinsolutiony.Weintroduceourrstvalidinequalityfor 4 inthenexttheorem. Theorem4.1. Let(x,~z)beanoptimalsolutiontoDEF-R,andsupposethat~z
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inequality: zz(x))]TJ /F10 11.955 Tf 12.35 11.36 Td[(Xi2Vx0@min8<:z(x))]TJ /F3 11.955 Tf 11.95 0 Td[(zmin,Xj2Rirjj9=;1Axi,(4) isvalidto 4 andcutsoff(x,~z). Proof. First,consideranysolution^x2Xsuchthat^xi=0,8i2Vx.Theattacker'ssolutionyisstillfeasibletoATT1(^x).Hence,z(^x)z(x)inthiscase,andso 4 isvalid.Inparticular,setting^x=xsatisesthecondition^xi=0,8i2Vx,requiringthatzz(x)atthispoint.Hence, 4 cutsoff(x,~z)bytheassumptionthat~z
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Step2(UpperBound). SolveATT1(x).Ifz(x)
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andhence,i2Oi.Next,considerany^x2Xanddene: I^x=fi2Vx:Oi\P^x6=;g, i.e.,I^xisthesetofallnodesi2Vxthatareeitherprotectedin^x,orsuchthatthereexistsadirectedpathonGxfromsomenodeh2P^x\Vxtonodei.Inthefollowinglemmaandtheorem,wederivealternativevalidinequalitiesfor 4 byusingtheideaofthespreadnetwork. Figure4-3. TwopossiblespreadnetworksforFigure 4-2 Lemma8. Givenx2X,let(y0,...,yT)beoptimaltoATT1(x)withcorrespondingvector.Forany^x2X,wehave: z(^x)Xj2VxnI^xrjj.(4) Proof. Supposethattheattackerchoosesaninitialattack,^y0,byletting^y0i=1ify0i=1and^xi=0,and^y0i=0otherwise,foralli2V.Weprovethislemmabyshowingthatifj2VtxnI^x,thennodejwillstillbeinuencedattimetwhenthedefenderchooses^x.First,notethatnodejisinitiallyinuencedinsolution^yforanynodej2V0xnI^x.Byinduction,supposethatforsomet2f0,...,T)]TJ /F4 11.955 Tf 12.87 0 Td[(1g,allnodesinfStt0=1Vt0xgnI^xareinuencedattimet0,whenthedefenderchooses^xandtheattackerchooses^y0. 82

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Consideranynodej2Vt+1xnI^x.ThereexistsnodirectedpathfromanynodeinI^xtonodej,orelsejwouldbelongtoI^xaswell.Thus,therealsoexistsnodirectedpathfromanynodeinI^xtoanynodeisuchthat(i,j)2Ax.ForeachoftheQnodesi2Stt0=1Vt0xsuchthat(i,j)2Ax,wehavebyinductionthatnodeiisinuencedattimetorearlierinthesolutiongivenby^y0.Thisimpliesthatnodejwouldbeinuencedattimet+1given^xand^y0,asdesired.Therefore,theRHSof 4 establishesalowerboundfortheattacker'soptimalobjectivevaluewhenthedefenderchooses^x. Theorem4.2. Let(x,~z)beanoptimalsolutiontoDEF-R,andsupposethatz(x)>~z.Also,letGbeany(undirected)acyclicgraphthatisconstructedoverVx,anddenotebyAitssetofarcs.Finally,deneAj,8j2Vx,asthesetofarcs(u,v)2Asuchthatu2Ojandv2Oj.Then,thefollowinginequality: zXj2Vxrjj0@1)]TJ /F10 11.955 Tf 12.14 11.36 Td[(Xi2Ojxi+X(u,v)2Ajxuxv1A,(4) isvalidto 4 andcutsoff(x,~z). Proof. First,notethatinequalities 4 forx=xreducetozPj2Vxrjj=z(x).Hence,inequalities 4 cutoff(x,~z)bytheassumptionthat~z0forsomenodej2Vx,thenP(u,v)2Aj^xu^xvcanbeatmostk)]TJ /F4 11.955 Tf 12.04 0 Td[(1.Otherwise,therewouldexistatleastkarcs(u,v)2AjthataredenedoverknodesinG,whichcontradictstheacyclicpropertyforG.Therefore,weobtainj0inthiscase. Giventhesetwocases,weobtain:Xj2Vxjrjj=Xj2VxnI^xjrjj+Xj2I^xjrjjXj2VxnI^xrjjz(^x), 83

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wherethelastinequalityisvalidbyLemma 8 .Thiscompletestheproof. Corollary4. Let(x,~z)beanoptimalsolutiontoDEF-R,andsupposethatz(x)>~z.IfA=;inTheorem 4.2 ,thenthefollowinginequality: zz(x))]TJ /F10 11.955 Tf 12.35 11.36 Td[(Xi2Vx0@min8<:z(x))]TJ /F3 11.955 Tf 11.95 0 Td[(zmin,Xj2Vx:i2Ojrjj9=;1Axi,(4) isvalidto 4 andcutsoff(x,z). Proof. Observethatvalidinequalities 4 reducetothefollowinginequalitywhenA=;:zXj2Vxrjj)]TJ /F10 11.955 Tf 12.47 11.36 Td[(Xj2VxrjjXi2Ojxi, orequivalently, zz(x))]TJ /F10 11.955 Tf 12.34 11.36 Td[(Xi2VxM0ixi,(4) whereM0i=Pj2Vx:i2Ojrjj.Hence, 4 isavalidinequalitythatcutsoff(x,z).Furthermore,becauseM0i0,8i2Vx;z(x)zmin,8x2X;andxi2f0,1g,8i2Vx; 4 canbestrengthenedbyreplacingM0iwithminfz(x))]TJ /F3 11.955 Tf 11.96 0 Td[(zmin,M0igusingasimilarargumentgiveninTheorem 4.1 .Thismodicationleadsto 4 andcompletestheproof. UsingTheorem 4.2 andCorollary 4 ,wecanemployCPAequippedwithvalidinequalitiesoftheform 4 or 4 insteadof 4 inStep3ofCPA.Themotivationforusingtheseinequalitiesstemsfromthefollowingtheoremthatcompares 4 to 4 Theorem4.3. Inequality 4 isatleastasstrongas 4 Proof. Denei=minfz(x))]TJ /F3 11.955 Tf 12.09 0 Td[(zmin,Pj2Vx:i2Ojrjjgandi=minfz(x))]TJ /F3 11.955 Tf 12.09 0 Td[(zmin,Pj2Rirjjgfori2Vx.Weprovetheclaimbyshowing Xj2Vx:i2OjrjjXj2Rirjj,8i2Vx,(4) 84

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whichimpliesii,8i2Vx.DeneO0i=fj2Vx:i2Ojg,8i2Vx.ItsufcestoprovethatO0iRifori2Vx.Ifj2O0i,thenthereexistsadirectedpathonGxfromnodeitonodej.BecausenodejbelongstoVjx,andAxA,theremustexistadirectedpathonGfromnodeitonodejthatconsistsofjorfewerarcs.Hencej2Ri.Thiscompletestheproof. Asimilartheoremcannotbestatedthatcomparesthestrengthofinequalities 4 and 4 .Ifinequality 4 wasweakenedbyreplacingthecoefcientsofxiwithPj2Vx:i2Ojrjj(insteadoftheminimumofthattermandz(x))]TJ /F3 11.955 Tf 12.11 0 Td[(zmin),then 4 wouldbeatleastasstrongas 4 duetothesubtractionofquadratictermspresentin 4 AfurtherconsiderationinimplementingCPAwithvalidinequalities 4 regardsthelinearizationofthequadratictermsintheseinequalities.ByrestrictingthesetofarcsthatcanbelongtothesetA,overallgeneratedinequalities 4 ,wecanlimitthenumberofquadratictermsthatmustbelinearized.WelinearizeeachquadratictermxixjbysubstitutingitwithacontinuousvariablexLij0,andincludingtheinequalityxLijxi+xj)]TJ /F4 11.955 Tf 12.87 0 Td[(1inDEF-R.(TheinequalitiesxLijxiandxLijxjusuallyrequiredtolinearizethisquadratictermarenotnecessary,becauseoptimizationforceseachxLij-variabletotakeitssmallestvalueallowedbyxiandxj.) 4.3.3.2Spreadnetworkmodicationstrategy Recallthatmultiplespreadnetworkscanbederivedforagivenx2Xanditsoptimalresponsey=(y0,...,yT),eachofwhichmightcorrespondtoadifferentvalidinequalityoftheform 4 or 4 .Givenacandidatespreadnetwork,Gx,correspondingtoxandy,weseekamechanismformodifyingGxtoanalternativespreadnetwork,G0x,suchthattheinequality 4 generatedcorrespondingtoG0xisatleastasstrongastheonecorrespondingtoGx. Theorem4.4. ConsideraspreadnetworkGx(Vx,Ax)forwhichthereexistnodesi,j,k2Vxsuchthati2V)]TJ /F4 11.955 Tf 7.08 -4.33 Td[((k),(i,k)=2Ax,(j,k)2Ax,andapathexistsfromitojonGx.LeteGxbeamodiedspreadnetworkobtainedbyreplacingarc(j,k)inGxwitharc 85

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(i,k).Thevalidinequality 4 inducedbyeGxisatleastasstrongasthatinducedbyGx. Proof. ConsiderthespreadnetworksGxandeGxwrittenwithrespecttox2X,asdenedinthetheorem.Foreachh2VxweagaindeneO0h=fj2Vx:h2OjgasinTheorem 4.3 ,withrespecttospreadnetworkGx,andeO0hanalogouslyforeGx.WeprovethateO0hO0hforallh2Vx.Asaresult,thexh-coefcientfor 4 generatedaccordingtoGxisatleastaslargeasthecorrespondingcoefcientin 4 accordingtoeGx,whichissufcienttoprovethetheorem. Notethatk2O0iduetotheassumptionthatthereexistsapathfromitojinGx,andthatarc(j,k)2Ax.Therefore,theadditionofarc(i,k)toAxdoesnotchangeO0i,andbyextensiondoesnotaffectanyothersetO0h,8h2Vx.Next,considerthedeletionofarc(j,k)fromAx,whichthenyieldseGxanditscorrespondingeO0isets.ThisarcdeletioncanonlydecreasemembershipwithintheO0-sets,andsoeO0hO0h,forallh2Vx.Thiscompletestheproof. Figure 4-4 illustratesaninstanceoftheproblemwithT=2andQ=3inwhichallrewardsequal1andzmin=1.Foragivenx,letGxbeaspreadnetworkpresentedinFigure 4-4 a.Notethatz(x)=7.Then,thefollowinginequalityz7)]TJ /F4 11.955 Tf 11.95 0 Td[(3x1)]TJ /F4 11.955 Tf 11.95 0 Td[(4x2)]TJ /F4 11.955 Tf 11.95 0 Td[(4x3)]TJ /F4 11.955 Tf 11.95 0 Td[(3x4)]TJ /F4 11.955 Tf 11.96 0 Td[(2x5)]TJ /F4 11.955 Tf 11.96 0 Td[(2x6)]TJ /F3 11.955 Tf 11.96 0 Td[(x7, isinducedbyGxfromCorollary 4 .Next,supposethatthereexistsanarc(4,7)2Aandnotethat(4,6)2Ax.UsingTheorem 4.4 ,weobtainamodiedspreadnetworkeGxfromGxbyaddingarc(4,7)andremovingarc(6,7).(SeeFigure 4-4 b.)ByusingCorollary 4 foreGx,weobtaintheinequalityz7)]TJ /F4 11.955 Tf 11.95 0 Td[(3x1)]TJ /F4 11.955 Tf 11.95 0 Td[(4x2)]TJ /F4 11.955 Tf 11.96 0 Td[(4x3)]TJ /F4 11.955 Tf 11.96 0 Td[(3x4)]TJ /F4 11.955 Tf 11.96 0 Td[(2x5)]TJ /F3 11.955 Tf 11.96 0 Td[(x6)]TJ /F3 11.955 Tf 11.96 0 Td[(x7, whichisstrongerthantheinequalityinducedbyGxduetothex6-coefcientsintheseinequalities. 86

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Itisworthnotingthattheinequality 4 inducedbyeGxmaynotnecessarilybeasstrongasthatinducedbyGxingeneral.LetA=f(2,6),(3,6)gbethesetofarcsusedtogeneratethequadratictermsin 4 .Inequalities z7)]TJ /F4 11.955 Tf 11.95 0 Td[(3x1)]TJ /F4 11.955 Tf 11.95 0 Td[(4x2)]TJ /F4 11.955 Tf 11.95 0 Td[(4x3)]TJ /F4 11.955 Tf 11.95 0 Td[(3x4)]TJ /F4 11.955 Tf 11.96 0 Td[(2x5)]TJ /F4 11.955 Tf 11.96 0 Td[(2x6)]TJ /F3 11.955 Tf 11.96 0 Td[(x7+2x2x6+2x3x6,(4) and z7)]TJ /F4 11.955 Tf 11.96 0 Td[(3x1)]TJ /F4 11.955 Tf 11.96 0 Td[(4x2)]TJ /F4 11.955 Tf 11.96 0 Td[(4x3)]TJ /F4 11.955 Tf 11.95 0 Td[(3x4)]TJ /F4 11.955 Tf 11.95 0 Td[(2x5)]TJ /F3 11.955 Tf 11.95 0 Td[(x6)]TJ /F3 11.955 Tf 11.96 0 Td[(x7+x2x6+x3x6,(4) areinducedbyGxandeGx,respectively,fromTheorem 4.2 .Toseethat 4 and 4 donotdominateoneanother,weshowthattheRHSforoneconstraintneednotalwaysbelargerthantheRHSfortheotherconstraint.Forx0=(0,0,0,0,0,1,0),notethattheRHSofinequality 4 is5,whiletheRHSfor 4 is6.However,forx00=(0,1,1,0,0,1,0),theRHSof 4 is1,andtheRHSof 4 is0. Figure4-4. SpreadnetworkmodicationusingTheorem 4.4 Algorithm 2 describesourmethodformodifyingagivenspreadnetworkusingtheideaofTheorem 4.4 inordertostrengthenvalidinequality 4 Algorithm 2 examinesallcandidatesfornodek(asdenedinTheorem 4.4 )fromamongthenodesinVTx,...,V2x,inthatorder.Givenachoiceofk,thealgorithmstartsbycreatingalistLk,containingallnodesj2V)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((k)thatareinuencedatsometime 87

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Algorithm2RevisinganexistingspreadnetworkusingTheorem 4.4 1: LetGx=(Vx,Ax)beaspreadnetworkwithcorrespondingvector. 2: DeneADJasajVxjjVxjmatrix,whereADJ(i,j)=1ifthereexistsapathfromnodeitonodejonGx,andADJ(i,j)=0otherwise. 3: fort=0toT)]TJ /F4 11.955 Tf 11.95 0 Td[(2do 4: forallnodesk2VT)]TJ /F7 7.97 Tf 6.58 0 Td[(txdo 5: InitializeLkasanarrayofallnodesj2V)]TJ /F4 11.955 Tf 7.08 -4.34 Td[((k)suchthatj
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NotethatasAlgorithm 2 proceeds,thespreadnetworkmightbemodiedbyaddorremoveoperationsperformedinStep 15 .However,theelementsofmatrixADJareneverupdatedthroughouttheexecutionofAlgorithm 2 .ThisisduetothefactthatADJ(i,j)correctlyindicatestheexistenceofapathbetweennodesiandjwheneveritisexaminedinStep 14 ,eventhoughthespreadnetworkmighthavebeenmodiedinearlierstagesofAlgorithm 2 .Toseethis,supposethatatsomestageofAlgorithm 2 ,thevalueofADJ(i,j)isexaminedwhilevisitingnodek2Vxinthefor-loopatStep 4 .Becausethisfor-loopexaminescandidatenodeskinnonincreasingorderoftheir-values,Algorithm 2 couldhaveonlymodiedthespreadnetworkbyaddingarcs(i0,k0)forsomenodei02Vxandk02Vtx,tk,orremovingarcs(j0,k0)forsomenodej02Vxandk02Vtx,tk.Recallthatthespreadnetworkcontainsnoarcs(u,v)suchthatuv.Therefore,theadditionordeletionofarcsinStep 15 cannotcreateanewpath,ordisconnectanexistingpath,fromnodeitonodej. ToanalyzethecomplexityofAlgorithm 2 ,observethattheconstructionofmatrixADJtakesO(QjVxj2)steps.Foreachnodekexaminedinthefor-loopatSteps 3 and 4 ,Algorithm 2 performsonesortingoperation(Step 6 ),whichisO(jVxjlogjVxj).Foreachnodejexaminedinthefor-loopinStep 7 ,Algorithm 2 executesO(jVxj)operationsinthewhile-loopatStep 12 correspondingtoeachcandidatenodei.(Notethatanarc(i,k)thatisaddedtoAxafterremovingsomearc(j,k)mightbereplacedlaterbysomeotherarc(i0,k)asthealgorithmproceeds,whichimpliesthatatotalofO(jVxj)nodesmightbeexaminedinthefor-loopinStep 7 ).Therefore,foreachnodekexaminedinthefor-loopsatSteps 3 and 4 ,Algorithm 2 performsO(jVxj2)operations,andhence,theoverallcomplexityofAlgorithm 2 isO(QjVxj2+jVxj3).Infact,thecomplexitycanbemorespecicallystatedasO(jVxj3):IfQjVxj,thenthisisobviouslytrue,andifQ>jVxj,theneverynodeinthespreadnetworkwasinuencedattime0,Axwouldnecessarilybeempty,andthealgorithmwouldterminateinconstanttime. 89

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4.4Attacker'sProblemSolutionApproach InordertogeneratethevalidinequalitiesintroducedinSection 4.3 ,wemustsolvethe(NP-hard)attacker'sproblem.Therefore,theefciencyofoursolutionmethodishighlydependentonthetimerequiredtosolveinstancesoftheattacker'sproblem.Thismotivatesfurtherinvestigationoftheattacker'sproblemwiththeaimofdevisingalternativeformulationsthatcanbemoreefcientlysolvedbymathematicaloptimizationtechniques. Notethatoncethey0-variablesarexed,theoptimalvalueofeachyti-variablefort2Tcanbereadilydeterminedviaapolynomial-timeprocedure,whichstartsfromtime1andidentiesthenumberofinuencednodesadjacenttonodeiattime0.Then,y1i=1ifxi=0andatleastQnodesadjacenttonodeiareinuencedattime0,andy1i=0otherwise.Byrepeatingthesameoperationforallothertimeperiods,theoptimalvalueofeachyti-variablewillbeeitherzeroorone. Thisobservationsuggeststhatamathematicalprogrammingformulationfortheattacker'sproblemthatincludesonlyjVjbinaryvariablesmaybeattainable.However,theConstraint 4f cannotberelaxedinProblem 4 ,whichindeedrequiresO(TjVj)binaryvariables.Inthissection,weinvestigatetwoalternativeformulationsfortheattacker'sproblemthatallowustorelaxthebinarinessrestrictiononyti-variablesfort2T. 4.4.1Reformulation1:ExponentialSetModel InSection 4.4.1.1 ,weproposeareformulationforProblem 4 thatrequiresO(T)binaryvariables.InSection 4.4.1.2 ,wedemonstratehowtoefcientlyimplementBenders'decompositiontosolvethisformulation. 90

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4.4.1.1Model Foreachi2V,deneSi=fS:SV)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((i),jSj=jV)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((i)j)]TJ /F4 11.955 Tf 18.01 0 Td[((Q)]TJ /F4 11.955 Tf 12 0 Td[(1)g.Thefollowingisareformulationforproblem 4 ATT2(x):z(x)=maxXi2V r0iy0i+TXt=1rti(yti)]TJ /F3 11.955 Tf 11.96 0 Td[(yt)]TJ /F9 7.97 Tf 6.58 0 Td[(1i)! (4a)s.t.ytiy0i+Xj2Syt)]TJ /F9 7.97 Tf 6.59 0 Td[(1j8i2V,t2T,S2Si (4b)yti2f0,1g8i2V,t2T (4c)Constraints( 4b ),( 4d ),and( 4e ). (4d) Notethattheonlydifferencebetweenmodels 4 and 4 liesintheconstraintsthatgovernthespreadofinuence.AccordingtoConstraints 4b ,ifnodei2Visinitiallyinuenced,thentheRHSofConstraints 4b willbeatleastoneforallt2TandS2Si,implyingthatnodeiwillremaininuencedatalltimeperiods.Now,supposethatnodei2Visnotinitiallyinuenced,andexamineConstraints 4b attimet2T.IffewerthanQnodesadjacenttonodeiareinuencedattimet)]TJ /F4 11.955 Tf 12.31 0 Td[(1,thenthereexistsasubsetS2SisuchthatnonodeinSisinuencedattimeperiodt)]TJ /F4 11.955 Tf 12.38 0 Td[(1.Inthiscase,yti=0duetoConstraint 4b correspondingtoS.Otherwise,theRHSofConstraints 4b forallS2Siwillbeatleastonefornodeiattimeperiodt,andhence,yti=1atoptimalityifxi=0.ThefollowingtheoremdemonstratesthatConstraint 4c canequivalentlyberelaxedtotakecontinuousvalues. Theorem4.5. ConsiderProblemATT2(x)foranyx2X,inwhichConstraints 4c arereplacedwith0yti1,8i2V,t2T.Thereexistsanoptimalsolution(^y0,...,^yT)tothisrelaxationinwhich^yti2f0,1g,8i2V,t2T. Proof. Consideranyfeasiblesolutiony=(y0,...,yT)totherelaxedversionofATT2(x)inwhichConstraints 4c arereplacedwith0yti1,8i2V,t2T,andsupposethat0
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theexceptionofsetting^yt0i0=1.First,notethat^ydoesnotviolateConstraints 4b 4d 4e ,andtherelaxedversionof 4c .Also,ourdenitionoft0impliesthattheRHSofConstraints 4b fornodei0,timet0,andallS2Si,mustbeatleastone,andthus,increasingyt0i0fromyt0i0doesnotviolatetheseconstraints.Additionally,theRHSofallConstraints 4b correspondingtotimet0+1willnotdecreasewhenyt0i0increases.Hence,^ymustalsobefeasibletoProblem 4 .Finally,eachyti-variablehasanonnegativeobjectivecoefcientrti)]TJ /F3 11.955 Tf 12.38 0 Td[(rt+1i.Itfollowsthat^ycannotyieldaworseobjectivevaluethany.Byrepeatingthesameapproachforallyti2(0,1),i2V,t2T,weobtainafeasiblesolutioninwhichyti2f0,1g,8i2V,t2T,withanobjectivevaluenotworsethantheobjectivevaluefory.Thiscompletestheproof. UsingTheorem 4.5 ,wehenceforthrelaxConstraint 4c to0yti1,8i2V,t2T,inATT2(x). 4.4.1.2Benders'decomposition Inthissection,weinvestigatetheapplicationofBenders'decompositioninsolvingmodel 4 .Observethatmodel 4 reducestothefollowinglinearprogramforgivenvectorsxandy0: maxXi2V T)]TJ /F9 7.97 Tf 6.58 0 Td[(1Xt=1(rti)]TJ /F3 11.955 Tf 11.95 0 Td[(rt+1i)yti+rTiyTi! (4a)s.t.y1iy0i+Xj2Sy0j8i2V,S2Si (4b)yti)]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xj2Syt)]TJ /F9 7.97 Tf 6.59 0 Td[(1jy0i8i2V,t2Tnf1g,S2Si (4c)yti1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi8i2V,t2T (4d)yti08i2V,t2T. (4e) Let1i,S,ti,S,andtibethedualvariablesassociatedwithConstraints 4b 4c ,and 4d ,respectively.DeningSj,i=fS:S2Sj,i2Sgasthesetofallsets 92

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S2Sjthatincludenodei,weobtainthedualproblemto 4 givenxandy0: minXi2VTXt=2XS2Siy0iti,S+Xi2VXS2Si(y0i+Xj2Sy0j)1i,S+Xi2VTXt=1(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi)ti (4a)s.t.XS2Siti,S)]TJ /F10 11.955 Tf 18.63 11.36 Td[(Xj2V+(i)XS2Sj,it+1j,S+tirti)]TJ /F3 11.955 Tf 11.95 0 Td[(rt+1i8i2V,t2TnfTg (4b)XS2SiTi,S+TirTi8i2V (4c)ti,S08i2V,S2Si,t2T (4d)ti08i2V,t2T. (4e) NotethatanoptimalsolutionmustexisttoProblem 4 ,becausetheobjectivefunctionvalueisalwaysnonnegative,andafeasiblesolutioncanbeobtainedbysettingti=rti)]TJ /F3 11.955 Tf 12 0 Td[(rt+1i,8i2V,t2T,withall-variablesequaltozero.LettingdenotethesetofallextremepointstoProblem 4 ,theBenders'masterproblemisgivenas: max (4a)s.t. Xi2V(r0i)]TJ /F3 11.955 Tf 11.95 0 Td[(r1i)y0i+Xi2VXS2Si1i,S(y0i+Xj2Sy0j)+Xi2VTXt=2XS2Siti,Sy0i+Xi2VXt2T(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi)ti8(,)2 (4b)y02Y(x), (4c) withProblem 4 beingtheBenders'subproblem.Therestrictedmasterproblem(RMP)isgivenby 4 withonlyalimitedsetofdualextremepoints,denotedby,andcorrespondingConstraints 4b Note,however,thatProblem 4 hasanexponentialnumberofConstraints 4c .Therefore,weaimtosolvethissubproblemusingmethodsotherthanlinearprogrammingtechniques. Lety=(y1,...,yT)beanoptimalsolutiontoProblem 4 ,andsupposeyti=0forsomeunprotectednodei2Vandtimet2T.Then,theremustexistsomeSti2Sisuch 93

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thatnonodeinStiisinuencedattimet)]TJ /F4 11.955 Tf 12.36 0 Td[(1.WerefertoStiasasafesubsetfornodei2Vandtimet2T.Itisworthnotingthatasafesubsetfornodeisuchthatyt0i=0isasafesubsetforalltimett0.Wewillthusdiscardthet-indexfromSti,andsimplyrefertoSiasasafesubsetfornodeiforalltimeperiodstsuchthatyti=0.Weprovideadualrecoveryalgorithm(DRA)foridentifyinganoptimalsolutiontoProblem 4 asfollows. Step1 Foralli2V,ifx1=1oryTi=1,thensetTi=rTiandTi,S=0,8S2Si.Otherwise,setTi=0,Ti,Si=rTi,andTi,S=0,8S2SinfSig.Initializet=T. Step2 Ift=0,thenterminate.Otherwise,sett=t)]TJ /F4 11.955 Tf 11.95 0 Td[(1andproceedtoStep3. Step3 Foreveryi2V: a. Ifxi=1,thenti=rti)]TJ /F3 11.955 Tf 12.95 0 Td[(rt+1i+Pj2V+(i)PS2Sj,it+1j,Sandti,S=0,8t2TnfTg,S2Si. b. Ifxi=0andyti=0,thensetti=0,ti,Si=rti)]TJ /F3 11.955 Tf 11.96 0 Td[(rt+1i+Pj2V+(i)PS2Sj,it+1j,S,andti,S=0,8S2SinfSig. c. Ifxi=0andyti=1,thensetti=rti)]TJ /F3 11.955 Tf 11.96 0 Td[(rt+1iandti,S=0,8S2Si. ProceedtoStep2. Inthefollowinglemma,weestablishtheoptimalityofthesolutionidentiedbyDRA. Lemma9. Supposethatthesolutiony=(y1,...,yT)isoptimaltoProblem 4 .LetSibeasafesubsetforeachunprotectednodei2Vsuchthatyti=0forsomet2T.Then,DRAconstructsanoptimalsolution(,)toProblem 4 Proof. First,weshowthat(,)asconstructedbyDRAisfeasibletoProblem 4 .Lett=T.Notethattheleft-hand-side(LHS)ofConstraints 4c correspondingtoanynodei2VwillberTifromStep1.Therefore,(,)satisesConstraints 4c .Next,lett=T)]TJ /F4 11.955 Tf 12.71 0 Td[(1.FromStep3a,theLHSofConstraint 4b forprotectednodei2VandtimeT)]TJ /F4 11.955 Tf 12.08 0 Td[(1willberT)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F3 11.955 Tf 12.08 0 Td[(rTi.Similarly,foreachunprotectednodei2VsuchthatyT)]TJ /F9 7.97 Tf 6.59 0 Td[(1i=0,Step3bguaranteesthattheLHSofConstraint 4b correspondingtonodeiandtimeT)]TJ /F4 11.955 Tf 12.37 0 Td[(1willequalrT)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F3 11.955 Tf 12.37 0 Td[(rTi.Now,consideranyinuencednodei2V 94

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attimeT)]TJ /F4 11.955 Tf 12.53 0 Td[(1,i.e.,yti=1.Notethatnodeicannotbeinthesafesubsetofanynodej2V+(i)andtimeT,i.e.,Pj2V+(i)PS2Sj,it+1j,S=0inthecorrespondingConstraint 4b .Therefore,bysettingT)]TJ /F9 7.97 Tf 6.59 0 Td[(1i=rT)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)]TJ /F3 11.955 Tf 12.79 0 Td[(rTiandti,S=0,8S2Si,inStep3c,Constraint 4b issatised.Hence,thesolution(,)satisesConstraints 4b .Byinductivelyrepeatingasimilarargumentforallnodesi2VandalltimesT)]TJ /F4 11.955 Tf 11.99 0 Td[(2,...,1,weconcludethat(,)isfeasibleto 4 Next,wemustshowthatthedualobjectivecomputedat(,)matchestheoptimalvalueoftheprimalobjectivefunction,whichisgivenbyPi2V:xi=0rii.Consideranynodei2V.Ify0i=1,thenyti=1,8t2T,implyingthatti,S=0,8t2T,S2Si.Hence,weobtain:Xi2VTXt=2XS2Siy0iti,S=0. Also,notethatif1i,S>0forsomeS2Si,thenSmustbeasafesubsetfornodeiandtimet=1implyingthaty0i=0andy0j=0,8j2S.Hence,Xi2VXS2Si(y0i+Xj2Sy0j)1i,S=0. Thus,theobjectivefunction 4a evaluatestoPi2VPt2T(1)]TJ /F4 11.955 Tf 12.8 0 Td[(xi)tiat(,).FromSteps1,3b,and3c,thistermreducestotheprimaloptimalobjectivefunctionvalue,i.e.,Pi2V:xi=0rii.Thiscompletestheproof. AnimmediateresultfromLemma 9 isthatweonlyneedtoidentifyasinglesafesubsetSiforeachunprotectednodei2Vthatisnotinuencedattime1.ThisallowsustodiscardS-indicesfromthe-variableswhenreferringtoProblem 4 ,andtorewriteConstraints 4b as: Xi2V(r0i)]TJ /F3 11.955 Tf 11.96 0 Td[(r1i)y0i+Xi2V1i(y0i+Xj2Siy0j)+Xi2VTXt=2tiy0i+Xi2VXt2T(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi)ti, 95

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orequivalently, Xi2V0@r0i)]TJ /F3 11.955 Tf 11.96 0 Td[(r1i+1i+Xj:i2Sj1j+TXt=2ti1Ay0i+Xi2VXt2T(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi)ti.(4) Inequality 4 canbestrengthenedbyastandardcoefcienttighteningprocedureasfollows.Letibethecoefcientofvariabley0i,i2V,in 4 .Thefollowinginequality: Xi2V minfi,zmax(x))]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xi2VXt2T(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi)tig!y0i+Xi2VXt2T(1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi)ti,(4) isvalidto 4 ,becausei0andy0i2f0,1g,8i2V;zmax(x))]TJ /F10 11.955 Tf 10.41 8.97 Td[(Pi2VPt2T(1)]TJ /F4 11.955 Tf 10.6 0 Td[(xi)ti0(notingthatPi2VPt2T(1)]TJ /F4 11.955 Tf 10.66 0 Td[(xi)tiistheoptimalattacker'sobjectiveinthelastinequality,whichisnomorethanzmax(x));and zmax(x).Thus,whensolvingtheBenders'masterproblem,wereplace 4b with 4 4.4.2Reformulation2:CompactModel Inthissection,weprovideanalternativecompact(polynomial-size)formulationfortheattacker'sproblem,inwhichthebinaryrestrictionsontheyti-variablescanberelaxed.Foreachi2V,arbitrarilyorderthenodesinV)]TJ /F4 11.955 Tf 7.08 -4.34 Td[((i)asfi1,...,ijV)]TJ /F9 7.97 Tf 6.25 -2.27 Td[((i)jg.Denevtimk=1ifatleastkofrstmnodesinV)]TJ /F4 11.955 Tf 7.08 -4.34 Td[((i)areinuencedattimet)]TJ /F4 11.955 Tf 12.26 0 Td[(1,andvtimk=0otherwise.Byconvention,weletvtimk=0,k>m.LettingNp=f1,...,pgforanypositive 96

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integerp,thefollowingisareformulationformodel 4 ,givenx: aATT3(x):maxXi2V r0iy0i+TXt=1rti(yti)]TJ /F3 11.955 Tf 11.95 0 Td[(yt)]TJ /F9 7.97 Tf 6.59 0 Td[(1i)! (4a)s.t.vtimkvti,m)]TJ /F9 7.97 Tf 6.59 0 Td[(1,k)]TJ /F9 7.97 Tf 6.59 0 Td[(18i2V,t2T,m2NjV)]TJ /F9 7.97 Tf 6.25 -2.27 Td[((i)j,k2Nminfm,Qg (4b)vtimkvti,m)]TJ /F9 7.97 Tf 6.59 0 Td[(1,k+yt)]TJ /F9 7.97 Tf 6.59 0 Td[(1im8i2V,t2T,m2NjV)]TJ /F9 7.97 Tf 6.25 -2.27 Td[((i)j,k2Nminfm,Qg (4c)ytiy0i+vti,jV)]TJ /F9 7.97 Tf 6.25 -2.27 Td[((i)j,Q8i2V,t2T (4d)yti1)]TJ /F4 11.955 Tf 12.14 0 Td[(xi8i2V,t2T (4e)vtimk2f0,1g8i2V,t2T,m2NjV)]TJ /F9 7.97 Tf 6.25 -2.27 Td[((i)j,k2Nminfm,Qg (4f)yti2f0,1g8i2V,t2T (4g)y02Y(x). (4h) Theobjectivefunction 4a isthesameastheobjectivefunctioninmodel 4 .Constraints 4b and 4c enforcethedenitionofvtimk.Toseethis,supposethatfewerthankoftherstmnodesinV)]TJ /F4 11.955 Tf 7.08 -4.34 Td[((i)areinuencedattimet)]TJ /F4 11.955 Tf 12.87 0 Td[(1.Ifnodeimisinuencedattimet)]TJ /F4 11.955 Tf 13.15 0 Td[(1,thenatmostk)]TJ /F4 11.955 Tf 13.15 0 Td[(2ofrstm)]TJ /F4 11.955 Tf 13.15 0 Td[(1nodesinV)]TJ /F4 11.955 Tf 7.08 -4.34 Td[((i)canbeinuencedattimet)]TJ /F4 11.955 Tf 12.47 0 Td[(1.Thisimpliesthatvti,m)]TJ /F9 7.97 Tf 6.58 0 Td[(1,k)]TJ /F9 7.97 Tf 6.58 0 Td[(1=0,andConstraints 4b forcevtimk=0.Otherwise,ifimisnotinuencedattimet)]TJ /F4 11.955 Tf 12.43 0 Td[(1,thenatmostk)]TJ /F4 11.955 Tf 12.43 0 Td[(1oftherstm)]TJ /F4 11.955 Tf 12.61 0 Td[(1nodesinV)]TJ /F4 11.955 Tf 7.09 -4.33 Td[((i)areinuencedattimet)]TJ /F4 11.955 Tf 12.62 0 Td[(1,i.e.,vti,m)]TJ /F9 7.97 Tf 6.58 0 Td[(1,k=0,andConstraints 4c forcevtimk=0.Ontheotherhand,ifatleastkoftherstmnodesinV)]TJ /F4 11.955 Tf 7.09 -4.34 Td[((i)areinuencedattimet)]TJ /F4 11.955 Tf 12.33 0 Td[(1,thenatleastk)]TJ /F4 11.955 Tf 12.34 0 Td[(1oftherstm)]TJ /F4 11.955 Tf 12.33 0 Td[(1nodeswereinuencedattimet)]TJ /F4 11.955 Tf 12.55 0 Td[(1.Furthermore,eitheratleastkoftherstm)]TJ /F4 11.955 Tf 12.55 0 Td[(1nodeswereinuenced,ornodeimitselfwasinuencedattimet)]TJ /F4 11.955 Tf 10.54 0 Td[(1.Hence,vti,m)]TJ /F9 7.97 Tf 6.58 0 Td[(1,k)]TJ /F9 7.97 Tf 6.59 0 Td[(1=1andvti,m)]TJ /F9 7.97 Tf 6.59 0 Td[(1,k)]TJ /F9 7.97 Tf 6.59 0 Td[(1+yt)]TJ /F9 7.97 Tf 6.59 0 Td[(1im1,whichallowsvtimk1(aswillbethecaseatoptimality).Constraints 4d implythatnodeicannotbeinuencedattimetunlesseitherithasbeeninitiallyinuencedorat 97

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leastQofitsadjacentnodesareinuencedattimet)]TJ /F4 11.955 Tf 12.29 0 Td[(1.Thebinarinessofthev-andy-variablesandthebudgetlimitareenforcedbyConstraints 4f 4h .However,thefollowingtheoremdemonstratesthatConstraints 4f and 4g canequivalentlyberelaxedtotakecontinuousvalues. Theorem4.6. ConsidertherelaxedversionofATT3(x)inwhichConstraints 4f and 4g arereplacedwiththefollowingconstraints: 0vtimk18i2V,t2T,m2NjV)]TJ /F9 7.97 Tf 6.26 -2.27 Td[((i)j,k2Nminfm,Qg (4a)0yti18i2V,t2T. (4b) Thereexistsanoptimalsolution(^y,^v)totherelaxedprobleminwhich^vtimk2f0,1g,8i2V,t2T,m2NjV)]TJ /F9 7.97 Tf 6.26 -2.27 Td[((i)j,k2Nminfm,Qg,and^yti2f0,1g,8i2V,t2T. Proof. Consideranoptimalsolution(y,v)totherelaxationofATT3(x)describedinthetheoreminwhichyti2(0,1)forsomei2V,t2T,and/orvtimk2(0,1)forsomei2V,t2T,m2NjV)]TJ /F9 7.97 Tf 6.26 -2.27 Td[((i)j,k2Nminfm,Qg.Lett0bethesmallestindexforwhicheither0
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Table4-1. Sizecomparisonofattacker'sproblemformulations. ModelBinaryVariablesContinuousVariablesConstraints ATT1O(TjVj)0O(TjVj)ATT2O(jVj)O(TjVj)O(TjVj)]TJ /F12 7.97 Tf 5.48 -4.38 Td[(jVjQ)ATT3O(jVj)O(TQjVj2)O(TQjVj2) Werepeattheargumentgivenaboveuntilallfractionaly-andv-variablevaluesbecomebinary.Thesolutionidentiedattheendofthisprocessremainsfeasibleandhasanobjectivefunctionvaluethatisatleastaslargeasthatfor(y,v).Thiscompletestheproof. Intheremainderofthechapter,wethusreplace 4f and 4g with 4a and 4b ,respectively.NotethatwhileonlyjVjbinaryvariablesareneededinmodel 4 a,theformulationrequirestheadditionofO(TQjVj2)continuousvariables.Table 4-1 comparesthesizeoftheproposedthreeformulationsfortheattacker'sproblem. 4.5ComputationalResults Inthissection,westudytheperformanceofourproposedmethodsonrandomlygeneratedtestinstances.WestartbyintroducingtheparametersusedtogeneratethetestinstancesanddiscussingtheimplementationdetailsinSection 4.5.1 .InSection 4.5.2 ,weinvestigatetheefciencyofemployingCPLEXinsolvingtheattacker'sproblemusingformulationsATT1,ATT2,andATT3.Finally,weexaminetheefcacyofCPAequippedwithvariouscuttingplanesinSection 4.5.3 4.5.1ImplementationDetails WeimplementedallalgorithmsinC++equippedwithCPLEX12.3ConcertTechnologyonanIBMx3650systemwithtwoIntelE5640Xeonprocessorsand24gigabytesofmemory.Forstudyingthemethodsfortheattacker'sproblem,weset600secondsasthemaximumallowablerunningtimeforCPLEX.Wesetthemaximumrunningtimeto1800secondswhenstudyingourcutting-planealgorithmsforthedefender'sproblem. 99

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Weconsidertwotypesofrandomlygeneratednetworks:1)networkswitharbitrarydegreedistribution(denotedbyADDnetworks),and2)scale-freenetworks(denotedbySFnetworks).WhilethedegreedistributionforADDnetworksisdeterminedbyarbitrarily-chosendensityvalues,SFnetworksareassociatedwithpower-lawdegreedistributions[ 2 ].SFnetworksarewidelyknownforreectingthetopologyofvariousreal-worldlarge-scalecommunicationandsocialnetworks. Table 4-2 showstheparametersandthecorrespondingvaluesthatweusedtogeneratethetestinstancesforbothtypesofnetworks.Allparametersarerandomlygeneratedasintegersderivedfromauniformdistributionoverthestatedrange,exceptforthebudget.Forthedefender'sandtheattacker'sbudget,wedenecoefcientslandf,respectively,whichareuniformlygeneratedoverthecontinuousinterval[0.35,0.65].Next,weletB=blSbcandD=bfScc,whereSbandScarethesumofallgeneratedb-andc-values,respectively. Table4-2. Parametersusedtogeneratetestinstances ParameterNameValue Defender'sprotectioncost(b)[30,100]Attacker'sinitialattackcost(c)[100,200]Infectionrewards(r)[70,180]Defender'sbudgetcoefcient(l)[0.35,0.65]Attacker'sbudgetcoefcient(f)[0.35,0.65] Inordertogenerate(directed)SFnetworkinstances,weemploythe~-preferentialattachmentschemesuggestedbyChungandLu[ 21 ].TheirmethodstartswithaninitialgraphG0formedbyonevertexhavingoneloop.Ateachsteps>0,GsisconstructedfromGs)]TJ /F9 7.97 Tf 6.59 0 Td[(1byaddinganewnodewithanoutgoingarctoanexistingnodeinGs)]TJ /F9 7.97 Tf 6.58 0 Td[(1withprobabilityp1,addinganewnodewithanincomingarcfromanexistingnodeinGs)]TJ /F9 7.97 Tf 6.59 0 Td[(1withprobabilityp2,oraddingadirectededgebetweentwoexistingnodesinGs)]TJ /F9 7.97 Tf 6.58 0 Td[(1withprobability1)]TJ /F3 11.955 Tf 12.68 0 Td[(p1)]TJ /F3 11.955 Tf 12.68 0 Td[(p2.AnexistingnodefromGs)]TJ /F9 7.97 Tf 6.59 0 Td[(1ischosentobeatail(orahead)nodebasedonaprobabilityproportionaltosumofthenumberofthenode'soutgoing(orincoming)arcsandachosenparameter~.Theprocessstopswhenthedesired 100

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numberofnodesexistsinGs.Notethatsmallervaluesofp1+p2resultingraphshavinglowerdensity.Forourcomputationalstudy,wechoosethevaluesofallotherparametersaccordingtothevaluesstatedinTable 4-2 NotethatCPAreliesonsolvingpossiblymanyinstancesoftheattacker'sproblem.Therefore,itiscrucialtopickthemostefcientsolutionapproachtosolvetheattacker'sproblem.Inordertocomparethecomputationalefciencyofvariousattacker'sformulationsdiscussedinSections 4.2 and 4.4 ,wesolvemodels 4 (denotedbyATT1)and 4 (denotedbyATT3)directlyusingCPLEX.WesolveATT2usingtheBenders'decompositionmethodpresentedinSection 4.4.1.2 ,wherewecomputezmax(x)(usedin 4 )bysolvingProblem 4 Forthedefender'sproblem,weconsidersixvariantsofCPA.Fortherstvariant,denotedbyCPA1,weimplementCPAequippedwithvalidinequality 4 .Next,weconsiderdifferentvariantsofCPA,denotedbyCPA2-1,CPA2-2,CPA2-3,andCPA2-4,inwhichweusevalidinequalities 4 .ThesevariantsdifferinthewaythattheundirectedacyclicgraphG(statedinTheorem 4.2 )isconstructed.ForCPA2-1,weletGbeastargraphforwhichwerandomlypickacenternode.CPA2-2istheimplementationinwhichGisconstructedasarandomspanningtree.ForCPA2-3andCPA2-4,weconstructacyclicsubgraphsthathavedjVxj=2eanddjVxj=10erandomlyselectededges,respectively.Finally,weinvestigateavariantofCPA,denotedbyCPA3,whichisequippedwithvalidinequality 4 .Inparticular,ateachiterationofCPA3,wearbitrarilyconstructaspreadnetworktoderiveitscorrespondingvalidinequality 4 ,andemployAlgorithm 2 tostrengthentheinequalities. Finally,recallthatmodel 4 canbesolvedtocomputeanupperboundfortheoptimalobjectivevalueofthedefender'sproblem.Wealsoproposedagreedyalgorithmforthispurpose,butinitialcomputationalexperimentsindicatethatthetimerequiredtosolveProblem 4 isinsignicantcomparedtotheoveralltimerequiredbyCPAvariants. 101

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Hence,allCPAvariantscomputezmaxbysolvingProblem 4 ratherthanemployingtheproposedheuristic. 4.5.2Resultsfortheattacker'sproblem Forourcomputationalstudyoftheattacker'sproblem,westartbyexamininginstancesgeneratedbasedonADDnetworks.Weconsidertwoscenariosfortheseinstances.Intherstscenario,westudytheattacker'sproblemexactlyasstatedinSection 4.2 .Forthesecondscenario,weinvestigatethecaseinwhichsomenodescannotbeattackedattimezero,butcanpossiblybecomeinuencedaftertimezero.Thesenodesarevulnerabletoattack,butnotdirectlyaccessibletotheattacker(foreachsuchnodei2V,wesimplyxy0i=0). Fortherstscenario,westartbyconsideringeightvaluesetsforparametersjVj,T,andQ.Moreover,weconsiderthreegraphdensityvalues,d,as0.05,0.2,and0.4,resultingin24instancesets.Finally,wegenerateteninstancesforeachset.Table 4-3 illustratestheaveragetime(inseconds)requiredbyeachimplementationtosolvetestinstances,whereatimeof600secondsisrecordedforeachinstancethatdoesnotsolvewithinthecomputationallimits.ForanycombinationofjVj,T,Q,anddinwhichthealgorithmcannotidentifyanoptimalsolutionwithin600secondsforatleastoneinstance,wealsorecordtheaverageoptimalitygapproducedbythealgorithmoverallsuchinstances. AccordingtoTable 4-3 ,ATT1outperformsATT2andATT3.Notethattheinstanceswithd=0.05aresignicantlymoredifculttosolvethancasesforwhichd=0.2ord=0.4.ATT2isgenerallyoutperformedbytheothertwovariantsontheseinstances.ThisstemsfromthefactthatthereductionincomputationaltimeduetotheDRAmethodcannotcompensateforthetimerequiredtosolvetheBenders'masterproblem.Finally,itisworthnotingthattheefcacyofATT3signicantlydeclinesforlargerinstanceshavingdensergraphs.OnepossiblereasonforthisbehaviorisduetothefactthatthenumberofConstraints 4b and 4c signicantlyincreasesfordensergraphs, 102

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resultinginlongerrunningtimesforATT3.Asaresult,ATT3isalsooutperformedbyATT2oninstanceswithd=0.4. Table4-3. Computationalresultsfortherstscenariooftheattacker'sproblemonADDnetworks ATT1ATT2ATT3 Set(jVj,T,Q)dAvgTimeAvgGapAvgTimeAvgGapAvgTimeAvgGap (25,2,3)0.050.0300.0500.0300.20.0806.6700.0900.40.0300.0400.070(50,3,4)0.050.290200.541.3%0.2800.20.08064.211.0%0.5700.40.0300.0200.360(75,4,5)0.052.970528.995.7%2.2700.20.140180.251.0%6.0300.40.0500.0401.790(100,5,6)0.05196.572.0%60014.3%203.841.5%0.20.1004.9604.9100.40.0800.0804.450(125,6,8)0.05475.3210.9%60026.0%480.987.0%0.20.11023.083.0%8.3900.40.1100.15013.50(150,7,9)0.0560012.7%60031.8%60013.5%0.20.27060.9110.1%76.2400.40.1500.23022.890(175,7,10)0.0560024.7%60038.1%60028.6%0.20.20055.611.1%89.8000.40.2400.35063.070(200,8,10)0.0560018.1%60034.6%60020.0%0.20.2100.56047.4200.41.9300.970579.5130.2% Inordertogeneratetestinstancesforthesecondscenario,weconsiderfourvaluesetsforparametersjVj,T,Q,andd.LetbetheratioofthenumberofnodesthatcannotbeinitiallyattackedtojVj.Byvarying2f40%,70%g,wegenerateatotalofeightsets,eachhavingtenrandomlygeneratedinstances.Table 4-4 reportstheresultsofthisexperimentusingthesamecolumndenitionsasinTable 4-3 RecallthatATT2isdesignedtocombatthegrowthofmathematicalprogrammingmodelsasafunctionofT,whereintheDRAmethodexecutesalow-polynomial-time 103

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Table4-4. Computationalresultsforthesecondscenariooftheattacker'sproblemonADDnetworks ATT1ATT2ATT3 Set(jVj,T,Q,d)AvgTimeAvgGapAvgTimeAvgGapAvgTimeAvgGap (100,75,9,0.05)40%0.62060.3803.44070%0.2800.0302.80(150,100,11,0.05)40%61.090300.041.8%69.256070%0.0800.0508.6370(200,125,13,0.05)40%2.850480.067.6%24.45070%2.5700.11020.980(250,175,15,0.05)40%8.570480.474.0%138.091.5%70%7.3600.321087.110 routinetocalculatetheimpactofanattacker'sactionandgenerateaBenders'cut.ThetradeoffisthatATT2requiresthesolutionofa(mixed-integer)masterproblemthatmayrequiretheadditionofmanycuts.Weobservethatwhen=40%,ATT1stilloutperformstheothertwovariants.However,ATT2outperformsATT1andATT3for=70%.Evidently,whenchangesfrom40%to70%,theBenders'masterproblembecomeslessdifculttosolve,andsolvingATT2becomesthemostefcientapproach. Wealsostudytheattacker'sproblemfortheSFnetworkinstances.WestartbyconsideringtenvaluesetsforparametersjVj,T,andQ.InordertogeneraterelativelydenseandsparseSFnetworks,wealsoconsidertwovaluesetsfortheprobabilityvalues,p1andp2,resultingin20instancesets.Finally,wegenerateteninstancesforeachset.Table 4-5 illustratestheaveragetime(inseconds)requiredbyeachimplementationtosolvetestinstances.NotethatwehavenotreportedthetimeandgapinformationforATT2oninstanceshaving3000ormorenodes,becauseATT2isclearlyinferiortoATT1andATT3ontheseinstances. AccordingtoTable 4-5 ,ATT1outperformsATT3,withATT2beinganimpracticalmethodtosolvethisclassofinstances.TheSFnetworkinstancesare,ingeneral,verysparsecomparedtotheADDnetworkinstances.Asaresult,ATT1andATT3areabletosolvelargerinstancesoftheattacker'sproblemonSFnetworkscomparedtoADD 104

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Table4-5. Computationalresultsfortheattacker'sproblemonSFnetworks ATT1ATT2ATT3 Set(jVj,T,Q)(p1,p2)AvgTimeAvgGapAvgTimeAvgGapAvgTimeAvgGap (250,3,2)(0.2,0.2)0.150600.111%0.170(0.4,0.4)0.120600.010%0.150(500,4,3)(0.2,0.2)0.150600.119%0.280(0.4,0.4)0.180600.117%0.300(750,5,3)(0.2,0.2)0.280600.319%0.820(0.4,0.4)0.230600.019%0.530(1000,5,4)(0.2,0.2)0.340600.221%0.770(0.4,0.4)0.270600.118%0.600(3000,8,7)(0.2,0.2)3.650--20.720(0.4,0.4)3.620--12.70--(5000,9,8)(0.2,0.2)12.160--97.910(0.4,0.4)12.210--42.330--(7000,10,8)(0.2,0.2)28.750--116.480(0.4,0.4)29.880--71.760--(9000,10,9)(0.2,0.2)58.140--216.590(0.4,0.4)59.250--201.80--(11000,11,9)(0.2,0.2)97.220--286.430(0.4,0.4)83.830--213.920--(13000,12,10)(0.2,0.2)171.050--449.020(0.4,0.4)170.720--392.760--(15000,13,11)(0.2,0.2)374.650--718.71%(0.4,0.4)364.530--717.261% networks.NotethatwhileATT1spendsroughlythesameamountoftimeondenseSFnetworks(p1=p2=0.2)andsparseSFnetworks(p1=p2=0.4),ATT3requiressignicantlylesstimeinsolvingsparseSFnetworks.ThisbehaviorisduetothefactthatthenumberofconstraintsinATT1doesnotdependonthesparsityofthenetwork,whereasATT3requiresfewerConstraints 4d forsparseSFnetworks. 4.5.3Resultsforthedefender'sproblem Forthedefender'sproblem,weonlygenerateADDnetworkinstances.(Ourpreliminarycomputationalstudyoninstancesofthedefender'sproblemonSF 105

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networksdidnotleadtosignicantlydifferentresultscomparedtotheinstancesthataregeneratedonADDnetworks.) RecallthatStep2ofCPArequiresthesolutionoftheattacker'sproblemgivenadefender'sdecisionvector.BasedontheresultsfromSection 4.5.2 ,weemployformulationATT1tosolvetheattacker'sproblem.WegeneratesixvaluesetsforparametersjVj,T,andQ.Foreachofthevaluesets,weconsiderthreedensityvaluesof0.05,0.2,and0.4.Finally,wegenerateteninstancesforeachset.InTable 4-6 ,wereporttheaveragetime(inseconds)requiredbyeachimplementationtosolvetestinstances,aswellastheaverageoptimalitygapproducedbythealgorithm.SimilartoTable 4-3 ,wecomputetheaverageoptimalitygapoverallinstancesthatwerenotsolvedwithin1800seconds. Table4-6. ComputationalresultsofCPAimplementations CPA1CPA2-1CPA2-2CPA2-3CPA2-4CPA3 Set(jVj,T,Q)dTimeGapTimeGapTimeGapTimeGapTimeGapTimeGap (15,4,3)0.0555.408.501.801.801.901.600.258.004.901.401.501.601.400.4122.308.102.702.702.602.40(17,5,4)0.05169.5013.705.104.705.104.100.2268.6011.704.203.903.903.400.4490.9022.809.609.409.708.20(19,6,5)0.051074.120.7%48.6022.4021.6021.5016.900.21190.220.4%44.7019.8020.3020.1015.600.41723.421.3%74.2035.9033.2034.9025.20(21,7,6)0.051754.619.8%132.9075.8088.1072.4057.100.21754.722.4%203.60121.90116.70122.6095.800.41800.022.4%244.90168.60160.40169.90119.10(23,8,7)0.051800.022.9%1261.87.6%1127.84.1%1105.23.4%1107.24.2%993.43.2%0.21800.021.1%1259.86.7%1030.14.5%1051.54.9%1031.15.2%902.14.9%0.41800.021.2%1101.47.2%925.79.6%929.29.5%942.09.6%772.87.2%(25,9,8)0.051800.024.2%1514.16.6%1443.15.3%1468.66.0%1478.25.9%1156.54.2%0.21800.019.1%1643.17.3%1596.38.6%1589.85.5%1588.76.6%1472.88.0%0.41800.022.8%1753.29.8%1719.07.6%1742.67.1%1742.89.4%1680.85.5% Table 4-6 indicatesthatCPA3outperformsallothervariants.Inparticular,CPA3outperformsCPA1aspredictedbyTheorem 4.3 .Infact,CPA1failedtoterminatewithin1800secondsonanyinstancehavingjVj=23or25nodes.NotethatCPA3isfasterthanalloftheCPA2variants,whichemployvalidinequalities 4 .Moreover, 106

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theaverageoptimalitygapproducedbyCPA3isnotworsethanthatproducedbyCPA2variantsingeneral.ThismaystemfromthefactthattheproblemDEF-RinCPA2variantsisaugmentedwithextravariables(duetothepresenceofxL-variables),resultinginamixed-integerproblemthatishardertosolve.Furthermore,recallthatvalidinequalities 4 arenotnecessarilystrongerthan 4 duetothefactthatwereducesomecoefcientsofxi-variablesin 4 thatcannotbereducedin 4 .Thisobservation,alongwiththemodesttighteningstepaffordedbyAlgorithm1,alsoexplainswhyCPA3outperformstheCPA2variations.Also,notethatCPA2-1isoutperformedbyotherCPA2variants,althoughthedifferencebetweenCPA2-2,CPA2-3,andCPA2-4isinsignicantespeciallyforsmallerinstances. 107

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CHAPTER5CONCLUSIONSANDFUTURERESEARCH InChapter 2 ,weaddressedanite-horizonoptimalstoppingproblemfromtheseller'sperspective.Webeganbydemonstratingthatwhenthecustomerisoptimal,thesellercanoptimizeprotfromsellingitemsinO(nlogn)time,wherenisthenumberofitemsforsale.Thevastliteratureinexperimentalresearchonstoppingproblemshasshownthathumandecision-makers,actingasthecustomer,tendtostopsearchtoosoon,andinanycasecannotbeassumedtobeoptimaldecision-makers.Wemodeledtheunpredictabilityofhumandecision-makingbehaviorbyanalyzingsituationsinwhichtheitems'values,prots,andcustomerstoppingthresholdsareuncertain.Werstexaminedamax-mincaseinwhichthesellerwishestomaximizetheminimumprotthatcanbemadegivensomeuncertaintysetinwhichthedatavaluesmustreside.Aspecialcaseofthismax-minproblemthatwestudiedinChapter 2 remainspolynomiallysolvable.Next,weexaminedthecaseinwhichthesellerwishestomaximizeexpectedprot.ThisproblemturnsouttobeNP-hard,evenwhenuncertaintyisconnedtotheitems'values. Weprovidedaformulationforsolvingtheproblemofmaximizingexpectedprot(inwhichuncertaintycanbeappliedtoanypartofthedataexceptforn).However,wedidnotexploresolutiontechniquestailoredforthisproblem,beyondtheuseofstandardmixed-integerprogrammingsolvers.WhennorjQjislarge,itisnotlikelythatformulation( 2 )willbetractable.Oneareaoffutureresearchmayinsteadfocusoncustomsolutiontechniquesforsolving( 2 )withinreasonablecomputationallimits.Anotherareaofinterestiscertainlyinlaboratorytestingofthesemodels.Conservativemodels(suchasthosepresentedinSection 2.3 )tendtosacricepotentialprotinfavororguaranteeingminimumprots.Itwouldbeofinteresttodemonstratehowconservativethesellershouldbeinpracticegivenahumandecision-maker.Furthermore,wehaveassumedthattheitems'valuesandprotsareindependentin 108

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general.Asanextensiontoourwork,thescenariosinwhichthereexistadegreeofcorrelationbetweenvaluesandprotscanalsobeconsideredforfuturestudies.Finally,anexpandedversionofthisproblemmayattempttoobservethisgameinarepeatedsetting,inwhichthecustomeradaptsthepurchasingstrategybasedonthetendenciesofaprot-motivatedseller. InChapter 3 ,westudiedaversionofthesetcoveringprobleminwhichitemsareusedtocoverclauses,andwhereeachclausehasaprioritizedlistonwhichitemswouldbeusedtocovertheclause.Theclauseisthensatisedbytheselecteditemhavingthehighestpriority.Weconsideredatwo-playerStackelberggameinwhichplayersintroduceitemsinturn,andthenearnarewardforeachclausethattheysatisfy.Thekeyassumptionsarethatthefolloweractswithknowledgeoftheleader'sdecision,andthatthefolloweractstomaximizeitsownobjective(ratherthan,e.g.,minimizingtheleader'sobjective).Weformulatedamixed-integerbilevelprogrammingmodelfortheproblem,alongwithacutting-planealgorithmforsolvingtheproblem.Weshowedthatourfamilyofapproachesiscomputationallypreferabletogeneralbileveloptimizationapproachesthathavebeenpreviouslydeveloped. Forfutureresearch,therearemanyimplementationchallengesthatcanbeinvestigatedunderthisapproach.TheaugmentedCPAreliesonanapproachthatrestrictsthepossiblefolloweractionsin( 3 ),andassuchrelaxestheouteroptimizationprobleminthatformulation.Inourcomputationalexperiments,theACPAimplementationsoccasionallyshowsomepromisebutareinconsistentinsuccessfullyverifyingthevalidityofthecandidateinequalities.Therefore,itwouldappearthatthereexistsanopportunitytoinvestigatetighterrelaxationsof( 3 ),whichwouldallowthevalidationprocesstomoreaccuratelyassesswhetherornotacandidateinequalityisvalid,thusresultinginfasterimplementations.Anotherlineofresearchmightseektoderivelocallyvalidinequalitiesonrationalfollowerreactionswithinthebranch-and-boundtree,basedonbranchingdecisionsfortheleader'sdecisionsataparticularnodeofthetree.Finally, 109

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hybridizingbranch-and-boundsearchwithheuristicstrategiesforthefollowermayproveusefulinanear-optimalalgorithmicapproach,especiallyforthoseinstancesthatappeartoresistexactsolutionmethods. InChapter 4 ,weaddressedaStackelberggameonanetworkinwhichanattackerseekstospreadinuenceonthenodesoveranitenumberoftimestages.Thedefenderthusaimstoprotectthenetworkagainstthespreadoftheattacker'sinuence.BydevisingseveralvalidinequalitiesdiscussedinSection 4.3 ,weproposedanexactcutting-planealgorithmthatiscapableofnitelyidentifyinganoptimalsolutionforthedefender'sproblem.Wealsodevelopedalternativeformulationsfortheattacker'sprobleminSection 4.4 ,andstudieddifferentcharacteristicsofeachformulation.Inparticular,weproposedasolutionmethodfortheattacker'sproblembasedonBenders'decompositioninwhichthecutsarecalculatedusingapolynomial-timecutgeneratingschemethatdoesnotrequiresolvingalinearprogrammingsubproblem. SeveralextensionscanbeconsideredfortheresearchworkpresentedinChapter 4 .First,recallthatseveralspreadnetworksmaycorrespondtooneoptimalsolutionfortheattacker'sproblem.Ourapproachtogeneratingspreadnetworkinequalitiesstartswithsomespreadnetwork,andmodiesittogenerateastrongervalidinequality.Asaresult,thestrengthoftheidentiedvalidinequalityisdependentontheinitially-chosenspreadnetwork.Thus,afuturetaskmightfocusonoptimizingthestructureofaspreadnetworkinordertoproduceastrongestpossiblevalidinequality. AsecondareaofresearchmayconsiderotherdiffusionmodelsasidefromthethresholdmodelexaminedinChapter 4 .Forinstance,itisinterestingtoextendourmodeltocasesinwhichtheneighborsofeachnodemayhaveunequaleffectsonthenode.(Forexample,theinuenceofnodevoveranothernodewmayberepresentedbysomeparameterbvw,andnodewmaybecomeinuencedifPv2Vbvwexceedssomegiventhreshold.See[ 42 ]forfurtherdetails.)Anotherlineofresearchistoincorporateuncertaintyinthediffusionprocess.OnesuchprocessdiscussedbyGoldenbergetal. 110

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[ 38 ]addressesthesituationinwhichonceanodeisinuenced,itisgivenonechancetoinuenceitsneighborsaccordingtoaBernoullidistribution. AthirdlineofresearchmayinvestigateadifferentStackelberggameinwhicheachplayeraimstospreaditsowninuencebyseekingasubsetofnodestoinitiallyattack.Inthisgame,theobjectiveofeachplayeristomaximizetherewardobtainedfrominuencingnodeswithrespecttosomebudgetrestriction.Thesesettingsoftenleadtobilevelprograms,withthefollower'soptimizationproblemembeddedintheconstraintsoftheleader'sproblem.Forsuchproblems,ideassimilartothereformulationgiveninSection 4.3.1 forthedefender'sproblemmaybepromisingindevisingexactcutting-planesolutionmethods. 111

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APPENDIXAAPPENDIXONREPRESENTATIONOFTHRESHOLDVALUES Inordertopreciselydiscussthecomplexityoftheproblemsunderinvestigationhere,wemustaddressthesizeofthedatausedinourcomputations.Evenaftermakingthesimplifyingassumptionthatthecustomer'svaluesareuniformlydistributedontheinterval[0,100],itisnotclearthatthecustomercantrulysolvetheoptimalpurchasing(stopping)probleminpolynomialtime.Therecursionsin( 2a )and( 2b )allowthegenerationofthresholddatainO(n)timeprovidedthatcomputationsareperformedinconstanttime.However,notethat(afterdividingthemaximumcustomervaluesby100)tn=1=2,andthatti=(t2i+1+1)=2foreachi=1,...,n)]TJ /F4 11.955 Tf 12.68 0 Td[(1.Thismeansthattn)]TJ /F9 7.97 Tf 6.59 0 Td[(1=5=8,tn)]TJ /F9 7.97 Tf 6.58 0 Td[(2=89=128,andsoon:Theimplicationisthattn)]TJ /F7 7.97 Tf 6.59 0 Td[(j+1=j=(22j)]TJ /F9 7.97 Tf 6.59 0 Td[(1)forsomeintegernumeratorj,8j=1,...,n.Unfortunately,thisimpliesthatthenumberofbitsrequiredtostore-valuesisO(2n).Therefore,itisnottechnicallypermissibletolett?=(tn+1+tn+2)=2intheproofofTheorem 2.2 ,becausestoringthisvalueevidentlyrequiresanexponentialnumberofbits.(Infact,itismoreaccuratetosaythatwedonotknowhowtostorethisnumberusingapolynomialnumberofbits.) Asimplifyingassumptionwouldstatethatthecustomermakesallcomputationswithniteprecision,andthatthislevelofprecisionistreatedasaconstantvalueinourcomputationalanalysis.Butinterestingly,forthecaseinwhichthecustomerperceivesauniformdistributionofprobabilitydata,Theorem 2.2 holdstrueevenwhennoassumptionismadethatrestrictstheprecisionofthecustomer'scomputations.Wediscussthedetailsofthisargumentbelow. Considerthecustomer'soptimalstoppingproblemwithntotalitems.Weseekasequenceofvaluess1,...,snsuchthattn=0.5sn
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inequalitychainabove,afterdividingby8,16,32,and64,respectively.Forj=5,wecanselect5tobeanyvalueinf100,101,102g.Usingj=5asabasecase,wewillprovebyinductionthatforanyj5,thereexistatleastthreevaluesofjsuchthatsn+1)]TJ /F7 7.97 Tf 6.58 0 Td[(j=j=2j+2isvalid. Supposethatthispropertyholdsforagivenj5.Wethushave:tn+1)]TJ /F7 7.97 Tf 6.59 0 Td[(jj=2j+2<(j+2)=2j+23=4.Comparingthedifferenceinthenumeratorsin( A )and( A )beforerounding,wehave:)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(2j+4j+4+22(j+2))]TJ /F10 11.955 Tf 11.95 9.69 Td[()]TJ /F8 11.955 Tf 5.48 -9.69 Td[(2j+22(j+2) 2j+2=4j+4 2j+2>3, wherethelatterinequalityisduetothefactthatj=2j+2tn)]TJ /F9 7.97 Tf 6.59 0 Td[(4>3=4.Performingtheceilingandooroperationsonthenumeratorsof( A )and( A ),respectively,narrows 113

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thegapbetweenthesevaluestoatleasttwo,whichveriesourclaim.(Observenowthatwehaverequired0j+200jsothatweareguaranteedtohaveanonemptyinterval[0j+1,00j+1]whenusingtheaboveinductionargument.) Therefore,wecancomputethe-valuesasgivenbythebasecasesaboveforj=1,...,5,andthenbyrecursionusing( A )thereafter,usingapolynomialnumberofbits.Hence,intheuniformdistributioncase,Theorem 2.2 isstillvalidevenwhenthecustomerusesinniteprecision,whenweselectt?=sn+1inthattransformation,assumingthatj5(withthecaseofj4beingtrivial).Thisguaranteesthattn+1>t?>tn+2,andthatt?isencodableusingapolynomialnumberofbits. 114

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APPENDIXBPROOFOFTHEOREM3.1 WeprovethatFxisNP-hardinthestrongsensebyprovidingapolynomialreductionfromEXACTCOVERBYTHREESETS(X3C)[ 35 ]toadecisionversionofFx.X3CisdenedwithasetofelementsS=f1,...,3pg,andacollectionofq>psubsetsofS,A=fS1,...,Sqg,eachhavingacardinalityofthree. TotransformX3CtoadecisionversionofFx,weletM=f1,...,3pgandN=f1,...,qg.Thefollowerincursanintroductioncostof1foreachproduct.Eachpreferencelist,Oi,containsallproductsj:i2Sj,whichareorderedarbitrarily.Also,letrij=ij=1,8i2M,j2N.Theleaderchoosesnottointroduceanyproducts.Thenadecisionversionofthefollowerproblemisasfollows:doesthereexistasetofproductsthatyieldsaprotof2pforthefollower?WeshowthatthereexistsanexactcoverofSbyasubsetofAifandonlyifthefollower'smaximumprotis2p. FirstweproveifthereexistasolutiontoX3C,thefollowercanmakeaprotof2p.SupposeAAisanexactcover,andthatthefollowerintroducesproductsj:Sj2A.Foreveryi2M,notethatbecauseAisanexactcoverofA,exactlyoneproductj,forsomej:i2Sj,hasbeenintroduced.Hencethefollowerearnsarevenueof1fromall3pcustomers.BecausejAj=p,thefollowerspentpinintroducingproductsandearnedaprotof2p. Next,supposethatthefollowercanmakeaprotof2p.WeshowthatthesetofintroducedproductscorrespondstoasolutionforX3C.Thefollowermustintroduceexactlypproductstoobtainaprotof2p.Introducingp0pproductsincursacostofp0,withamaximumrevenueof3p;theprotwouldbenomorethan3p)]TJ /F3 11.955 Tf 12.4 0 Td[(p0<2p.Ifexactlypproductsareintroduced,aprotof2pisobtainedifandonlyifarevenueof3pisachievable,i.e.,everyproductispurchasedbythree 115

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customers.DeneA=fSj2A:productjisintroducedg. Sinceeveryintroducedproductwaspurchasedbythreecustomers,wemusthaveSk1\Sk2=;,8k1,k2:Sk1,Sk22A.HenceAsolvesX3C. Becauseallnumericaldatausedinthistransformationequals1,andbecausethenumberofcustomersandproductsispolynomiallyboundedbytheX3Cproblemsize,weconcludethatthedecisionversionofthefollowerproblemisstronglyNP-complete,andthatthefollower'soptimizationproblemisstronglyNP-hard. 116

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REFERENCES [1] Audet,C.,Savard,G.,andZghal,W.NewBranch-and-CutAlgorithmforBilevelLinearProgramming.JournalofOptimizationTheoryandApplications134(2007):353. [2] Barabasi,A.-L.andAlbert,R.Emergenceofscalinginrandomnetworks.Science286(1999):509. [3] Bard,J.F.PracticalBilevelOptimization:AlgorithmsandApplications.Boston:KluwerAcademicPublishers,1998. [4] Bartoszynski,R.andGovindarajulu,Z.Thesecretaryproblemwithinterviewcost.Sankhya:TheIndianJournalofStatistics40(1978):11. [5] Bayus,B.L.,Jain,S.,andRao,A.G.TruthorConsequences:AnAnalysisofVaporwareandNewProductAnnouncements.JournalofMarketResearch38(2001):3. [6] Bearden,J.N.andRapoport,A.Operationsresearchinexperimentalpsychology.TutorialsinOperationsResearch:EmergingTheory,Methods,andApplications.ed.J.C.Smith,vol.1.INFORMS,Linthicum,MD,2005.213. [7] Bearden,J.N.,Rapoport,A.,andMurphy,R.O.Sequentialobservationandselectionwithrank-dependentpayoffs:Anexperimentaltest.ManagementScience52(2006):1437. [8] Bellman,R.DynamicProgramming.Princeton,NJ:PrincetonUniversityPress,1957. [9] Ben-Tal,A.,ElGhaoui,L.,andNemirovski,A.RobustOptimization.PrincetonSeriesinAppliedMathematics.PrincetonUniversityPress,Princeton,NJ,2009. [10] Bialas,W.F.andKarwan,M.H.Two-LevelLinearProgramming.ManagementScience30(1984).8:1004. [11] Birge,J.R.andLouveaux,F.V.IntroductiontoStochasticProgramming.NewYork:Springer,1997. [12] Blum,J.,Ding,M.,Thaeler,A.,andCheng,X.ConnectedDominatingSetinSensorNetworksandMANETs.HandbookofCombinatorialOptimizationSupplementVolumeB.eds.D.DuandP.M.Pardalos.Springer,2005.329. [13] Brown,G.G.,Carlyle,W.M.,Salmeron,J.,andWood,K.DefendingCriticalInfrastructure.Interfaces36(2006).6:530. 117

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BIOGRAPHICALSKETCH MehdiHemmati(Soheil)wasborninTehran,Iran.HegraduatedfromAlborzHighSchoolatTehranin1998anddecidedtopursueanengineeringdegreeincollege.HewasadmittedtoSharifUniversityofTechnologyinfall1998andreceivedhisbachelor'sandmaster'sdegreesinindustrialandsystemsengineeringfromthesameuniversityin2003and2005,respectively.Infall2009,hereceivedalumnigraduateawardtostudyforPh.D.degreeintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.Hisresearchcenteredonmultileveldiscreteoptimizationandinterdictiontheorywithapplicationsthatinvolvecompetition,eitherbetweentwoagencies(e.g.,inmarket)oragainstuncertainexternalfactors.HereceivedhisPh.D.degreeinAugust2013. 123