ExpertSystemsWithApplications225(2023)120151Maximumflowaccelerationbytraversingtreebasedtwo-boundarygraphcontractionWeiWeia,b,PengpengWanga,b,YaboDongc,∗aKeyLaboratoryofGrainInformationProcessingandControl(HenanUniversityofTechnology),MinistryofEducation,Zhengzhou450001,ChinabHenanProvincialKeyLaboratoryofGrainPhotoelectricDetectionandControl,HenanUniversityofTechnology,Zhengzhou450001,ChinacCollegeofcomputerscienceandtechnology,ZhejiangUniversity,Hangzhou310027,ChinaARTICLEINFOABSTRACTKeywords:Tree-cutmappingBoundarynodeMaximumflowGraphcontractionOverlayThemaximumflow(max-flow)problemissofundamentalthatthecorrespondingalgorithmshavemanyapplicationsinmanyscenarios.Acommonaccelerationstrategyistoreducegraphsizebycontractingsubgraphs,buttherearefewcandidatesubgraphswithnosizelimitthatcanguaranteeexactmax-flowsolutionsaftercontraction.Weaddanewsubgraphcontractionconditiontotheexistingconditionswithcorrespondingefficientlocatingalgorithm,andweproposeamax-flowaccelerationalgorithmusingtraversingtree-basedcontraction(TTCMax).TTCMaxcanworkefficientlyonlybyusingconnectivityinformation.Throughdepth-firsttraversing,itcancontracttwo-boundarygraphsofanysizeintoedges,andthenaccelerateexistingmax-flowalgorithmsusingcontractedgraphs.Large-scalerandomandreal-worldgraphexperimentstestitseffectandtheaccelerationratiocanbemorethan50timeswithlowgraphreductionoverhead,indicatingthattheproposedalgorithmcanbeusedasaneffectivecomplementtoexistingalgorithms.1.IntroductionAsafundamentalandwidelyinvestigatedproblem,themaximumflow(max-flow)problemanditsdualproblem,theminimumcut(min-cut)problemhavemanyapplicationsandareoftenusedassubroutinesinotheralgorithms(Sherman,2009).Manyadvanceshavebeenmadeinthedevelopmentofefficientalgorithmsforthisproblem(Goldberg&Rao,1998).Fastsolutionscanbefoundthroughalgorithm-orientedacceleration(Boykov&Kolmogorov,2004;Goldberg&Tarjan,2014;Verma&Dhruv,2012),orgraphdata-orientedacceleration(Gusfield,1990;Wei,Liu,&Zhang,2018;Zhang,Hua,Jiang,Zhang,&Chen,2011;Zhang,Xu,Hua,&Zhao,2012;Zhao,Su,Liu,&Zhang,2014;Zhao,Xu,Hua,&Zhang,2012).Asfordata-orientedalgorithms,thecoreideaistoreducethesizeoftheinput,suchasreducingthesizeofthegraphbycontractingsubgraphstonodes.Therearemanycontraction-basedalgorithmsthatexploitvarioussubgraphcontractionconditions.Unfortunately,mostofthesealgorithmscanonlyobtainapproximatedvaluewhenusingsubgraphcontractionconditionwithnosizelimit.Inthepaper,weproposealosslessalgorithmthatemploysasub-graphcontractionconditionwithnosizelimit,andatoyexampleisgiveninFig.1toclarifythedifferencebetweentheproposedmethodandexistingcontraction-basedmethods.Unlikeexistingnode-orientedcontractions,theproposedalgorithmcontractssubgraphsintoedges.Thecorepartisaroutinethatcanefficientlysearchforthetwo-boundarygraph(TBG),thatis,thesubgraphcontainingtwoboundarynodesthatseparateitfromotherpartsofthewholegraph.Experimentsvalidateitseffectivenessbyusinglarge-scalegraphscontainingupto107nodes.Theworkadvancesthestateoftheartbyintroducinganewlosslessandgeneralsize-freecontractionconditionwithahighlyefficientwaytolocatethecorrespondingsubgraphs.Specifically,thetheoreticalcon-tributionsinclude:(a)Weprovethatthecontractionoftwo-boundarygraphscanmaintaintheexactmax-flowresult,whichleadstothecontractionalgorithmthathasfewconstraints(e.g.,unlimitedsubgraphsize)andcanensureaccuratesolution...
