The set is the set of nodes in the network The set is the set of directed links ij The set is the set of capacities ij of the links ij The problem is to determine the maximum amount of 64258ow that can be sent from the source node to the sink node T ID: 59062
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Lecture20 Outline Max-Flowproblem Description Someexamples Algorithmforsolvingmax- owproblem Augmentingpathalgorithm OperationsResearchMethods1 Lecture20 Max-FlowProblem:Multiple-SourcesMultiple-Sinks Wearegivenadirectedcapacitatednetwork(V;E;C)connectingmultiplesourcenodeswithmultiplesinknodes.ThesetVisthesetofnodesandthesetEisthesetofdirectedlinks(i;j)ThesetCisthesetofcapacitiescij0ofthelinks(i;j)2ETheproblemistodeterminethemaximumamountof owthatcanbesentfromthesourcenodestothesinknodes. Thisis Max-FlowProblem formultiple-sourcesandmultiple-sinks OperationsResearchMethods3 Lecture20 Multiple-sourcesmultiple-linksproblemcanbeconvertedtoasingle-sourceandsingle-sinkproblemby Introducingadummysourcenodethatisconnectedtotheoriginalsourcenodeswithinnitecapacitylinks Introducingadummysinknodethatisconnectedwiththeoriginalsinknodeswithinnitecapacitylinks OperationsResearchMethods4 Lecture20 Forsmallscaleproblems:Anotheralternativeistointroduceasingle\dummy"sourcenodeconnectedwithalltheoriginalsourcenodesBUT EachoutgoinglinkfromthedummysourcenodetoanoriginalsourcenodesgetsassignedacapacitythatisequaltothetotalcapacityoftheoutgoinglinksfromsSimilarly,iftherearemultiplesinknodes,weintroduceasingledummysinknodeBUT Eachincominglinkfromanoriginalsinknodettothedummysinknodegetsassignedacapacitythatisequaltothetotalcapacityoftheincominglinkstosinkt OperationsResearchMethods5 Lecture20 MaximumFlowProblem:MathematicalFormulation WearegivenadirectedcapacitatednetworkG=(V;E;C))withasinglesourceandasinglesinknode.Wewanttoformulatethemax- owproblem. Foreachlink(i;j)2E,letxijdenotethe owsentonlink(i;j), Foreachlink(i;j)2E,the owisboundedfromabovebythecapacitycijofthelink:cijxij0 Wehavetospecifythebalanceequations Allthenodesinthenetworkexceptforthesourceandthesinknodearejust\transit"nodes(in ow=out ow)Xf`j(`;i)2Egx`i�Xfjj(i;j)2Egxij=0foralli6=s;t OperationsResearchMethods7 Lecture20 maximizex12+x13subjecttox12�x25�x24=0balancefornode2x13�x35�x34=0balancefornode3x24+x34�x46=0balancefornode4x25+x35�x56=0balancefornode50x1250x13100x2440x2550x3450x3550x4680x5610Max-FlowisanLPproblem:wecoulduseasimplexmethod OperationsResearchMethods9 Lecture20 Max-FlowAlgorithms Foragivengraph:ndenotesthenumberofnodesmdenotesthenumberofedgesmaxjfjisthemaximumamountof ow Method Complexity Description Simplex { Constrainedbylegal ow Ford-Fulkersonalgorithm O(mmaxjfj) Weightshavetobeintegers Edmonds-Karpalgorithm O(nm2) BasedonFord-Fulkerson Dinitzblocking owalgorithm O(n2m) Buildslayeredgraphs Generalpush-relabelalgorithm O(n2m) Usesapre ow Ford-FulkersonAlgorithmisalsoknownasAugmentingPathalgorithmWewillalsorefertoitasMax-FlowAlgorithm OperationsResearchMethods10 Lecture20 Example:Augmentingpath Supposewechoosetosendthe owalongpath1!3!5!6Supposewechoosetosend4unitsof owalongthispathThentheresidualcapacitiesoflinks(1,2),(3,5)and(5,6)are6,1,and6respectivelyAbetterchoiceistosend5unitsalongthispath.Inthiscase,thecapacityofthelink(3,5)isreached(itsresidualcapacitybecomes0)Wesaythatlink(3,5)issaturated(nomore owcanbesent) OperationsResearchMethods12 Lecture20 ResidualCapacity Thelinksthathavebeenusedtosenda owgetupdatedtore ectthe owpushEverysuchlink(i;j)getsacapacitylabeloftheforma=bwhere aistheremainingcapacityofthelinkand bisthetotal owsentalongthatlink aisviewedasforwardcapacityofthelink bisviewedasbackwardcapacityofthelink(capacityifwechoosetotraversethelinkintheoppositedirection) NOTE:a+b=cijThesumofthesenumbersisequaltotheoriginalcapacityofthelink OperationsResearchMethods14