Table Of Content

SpringerBriefs in Computer Science Robert Kudelić Feedback Arc Set A History of the Problem and Algorithms SpringerBriefs in Computer Science SeriesEditors StanZdonik,BrownUniversity,Providence,RI,USA ShashiShekhar,UniversityofMinnesota,Minneapolis,MN,USA XindongWu,UniversityofVermont,Burlington,VT,USA LakhmiC.Jain,UniversityofSouthAustralia,Adelaide,SA,Australia DavidPadua,UniversityofIllinoisUrbana-Champaign,Urbana,IL,USA XueminShermanShen,UniversityofWaterloo,Waterloo,ON,Canada BorkoFurht,FloridaAtlanticUniversity,BocaRaton,FL,USA V.S.Subrahmanian,UniversityofMaryland,CollegePark,MD,USA MartialHebert,CarnegieMellonUniversity,Pittsburgh,PA,USA KatsushiIkeuchi,UniversityofTokyo,Tokyo,Japan BrunoSiciliano,UniversitàdiNapoliFedericoII,Napoli,Italy SushilJajodia,GeorgeMasonUniversity,Fairfax,VA,USA NewtonLee,InstituteforEducation,ResearchandScholarships,LosAngeles,CA, USA SpringerBriefs present concise summaries of cutting-edge research and practical applicationsacrossawidespectrumoffields.Featuringcompactvolumesof50to 125pages,theseriescoversarangeofcontentfromprofessionaltoacademic. Typicaltopicsmightinclude: (cid:129) Atimelyreportofstate-of-theartanalyticaltechniques (cid:129) A bridge between new research results, as published in journal articles, and a contextualliteraturereview (cid:129) Asnapshotofahotoremergingtopic (cid:129) Anin-depthcasestudyorclinicalexample (cid:129) A presentationofcoreconceptsthatstudentsmustunderstandinordertomake independentcontributions Briefsallowauthorstopresenttheirideasandreaderstoabsorbthemwithminimal time investment. Briefs will be published as part of Springer’s eBook collection, withmillionsofusersworldwide.Inaddition,Briefswillbeavailableforindividual print and electronic purchase. Briefs are characterized by fast, global electronic dissemination, standard publishing contracts, easy-to-use manuscript preparation and formatting guidelines, and expedited production schedules. We aim for pub- lication 8–12 weeks after acceptance. Both solicited and unsolicited manuscripts areconsideredforpublicationinthisseries. **Indexing:ThisseriesisindexedinScopus,Ei-Compendex,andzbMATH** Robert Kudelic´ Feedback Arc Set A History of the Problem and Algorithms RobertKudelic´ FacultyofOrganizationandInformation Science,TheoreticalandApplied FoundationsofInformationScience UniversityofZagreb Varaždin,Croatia ISSN2191-5768 ISSN2191-5776 (electronic) SpringerBriefsinComputerScience ISBN978-3-031-10514-2 ISBN978-3-031-10515-9 (eBook) https://doi.org/10.1007/978-3-031-10515-9 ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword Thefeedbackarcsetproblemisaoneofthequintessentialproblemsofalgorithmics and,moregenerally,ofcomputerscience.Theproblemiseasilystated:foragiven directedgraphG=(V,E),findthesmallestsubsetE(cid:2) ⊂EsuchthatG(cid:2) =(V,E\ (cid:2) E ) is acyclic. The problem is well-known and has been investigated by the best minds in computer science. It arises as a fundamental issue in many problems in industry,fromcircuitdesigntologistics.Yet,theproblemhasprovenverydifficult tosolve. Over the last fifty years, methods to solve the feedback arc set problem have spannedthelengthandbreadthofthetechniquesofcomputerscience:paradigmsof graphalgorithms,well-designeddatastructures,optimizationviaintegerprogram- ming, heuristics with proven performance guarantees, “soft computing” methods suchasantalgorithms,andstochasticmethodssuchasMonte-Carloalgorithms. TheauthorRobertKudelic´haspublishedpapersonseveraldisparateapproaches tothefeedbackarcsetproblem.Hisbooktakesthereaderchronologicallythrough the feedbackarc setliterature,andas suchitprovidesa historicalcompendiumof techniquesincomputerscience. Sydney,NSW,Australia PeterEades January2022 v Preface As is true for any subject in science, there comes a time when one needs to look back,andtakeagazeuponworkdonethusfar.Andifwearetochoosesuchatime foraclassicalcomputerscienceproblemknownasfeedbackarcset(FAS),andmore thanhalfa centuryhaspassedsince itsfledglingdays,thisis agoodtimeto work onthatgaze.Ageneralintroductiontotheproblemwillbepresented,hardnessand importancewillbeargued.Spanwillbegivenfromtheoriginandthename,through problemdefinitionandvariousversions,allthe wayto thedualversionandalink with the feedbackvertexset and the forcing problem—practicalrelevance will be coveredaswell. By building on that foundation,a review of the algorithms will be given, with special emphasis on FAS. The goal of this review is to give both depth, through coveringalgorithms,andbreath,throughspanninglifetimeoftheFASproblem.It istheintentionthat,throughthisbook,anyinterestedpartywillbeabletoquickly findrelevantinformation,fosteringbothlearningandresearch.1 Thereareanumberofpeoplethatwereofhelptomeduringpreparationofthis book, and therefore a few words of gratitude are in order. I would like to thank mycollegeforsuggestingthatthematerialwhichisfoundinthiswork,a workin progressatthattime,wouldbeagoodfitforabook—theideaquicklygrewinmy mind,andthepublisherwassoonfound. IwouldalsoliketoexpressmysinceregratitudetoPeterEades,wellknownfor hisexpertisein graphproblems,forbeingso kindandtakingthetime towritethe forewordtothisbook. 1Onlinesupplementarymaterialisavailableat:https://cs.foi.hr/fas/book/. vii viii Preface Aspecialappreciationwouldgotomyfatherforenablingmyacademiccareer. Gratitudeisinorderforacolleagueofmine,NikolaIvkovic´,whowasveryhelpful inacquiringanumberofpapersforme.AndIcan’tforgetWayneWheeler,forthe supportandforbeingsuchanicepersontotalktoo,itwaswonderful. Varaz˘din,Croatia RobertKudelic´ April2022 Contents PartI OverviewofFindings 1 FeedbackArcSet............................................................. 3 1.1 TheProblemandtheName ........................................... 3 1.2 Origin .................................................................. 3 1.3 Description ............................................................ 4 1.4 LinearProgrammingFormulation .................................... 5 1.5 InherentlyHardtoSolve .............................................. 5 1.6 HardtoApproximatebyExtension .................................. 6 1.7 FeedbackArcSetandTournament ................................... 6 1.8 FeedbackArcSetandEulerianDigraph ............................. 7 1.9 SubsetFeedbackArcSet ............................................. 8 1.10 GraphBipartization ................................................... 8 1.11 MaximumAcyclicSubgraph—Dual ................................. 8 1.12 FeedbackArcandVertexSet ......................................... 9 1.13 TheForcingProblem ................................................. 10 1.14 PracticalRelevance ................................................... 10 References .................................................................... 12 PartII FeedbackArcSetandAlgorithmsThereof 2 IntroductoryRemarks....................................................... 17 2.1 Methodology .......................................................... 17 2.2 CitedLiterature ....................................................... 18 2.3 AbouttheHistory ..................................................... 18 3 PapersandAlgorithms...................................................... 19 3.1 OnDirectedGraphsandIntegerPrograms .......................... 19 3.2 MinimumFeedbackArcSetsforaDirectedGraph ................. 19 3.3 MinimumFeedbackArcandVertexSetsofaDirected Graph .................................................................. 21 3.4 AHeuristicAlgorithmfortheMinimumFeedbackArcSet ........ 22 ix x Contents 3.5 ABranchandBoundAlgorithmfortheAcyclicSubgraph ......... 24 3.6 AcyclicSubdigraphsandLinearOrderings .......................... 25 3.7 FindingaMinimumFeedbackArcSetinReducibleFlow Graphs ................................................................. 27 3.8 AContractionAlgorithmforFindingSmallCycleCut-sets ........ 28 3.9 ApproximationAlgorithmsfortheMaximumAcyclic Subgraph .............................................................. 29 3.10 ExactandHeuristicAlgorithmsfortheWeightedFAS ............. 30 3.11 Approximationfor ConcurrentFlow with Uniform Capacities.............................................................. 31 3.12 MethodforFinding a MinimalFeedbackArc Set in Digraphs ............................................................... 35 3.13 AFastandEffectiveHeuristicfortheFeedbackArcSet ........... 35 3.14 ApproximationsfortheMaximumAcyclicSubgraph .............. 36 3.15 AHeuristicfortheFeedbackArcSet ................................ 37 3.16 NewResultsontheComputationofMedianOrders ................ 40 3.17 Approx.MinimumFeedbackSets and Multi-Cutsin Digraphs ............................................................... 42 3.18 AFastandEffectiveAlgorithmfortheFeedbackArcSet .......... 43 3.19 ComputingApproximateSolutionsofFSPsusingGRASP ........ 48 3.20 ANewRoundingProcedurefortheAssignmentProblem .......... 50 3.21 OnEnumeratingAllMinimalSolutionsofFeedback Problems ............................................................... 51 3.22 CombinatorialAlgorithmsforFeedbackProblemsin Digraphs ............................................................... 53 3.23 AContractionAlgorithmforFindingMinimalFeedback Sets .................................................................... 54 3.24 ExactExp.AlgorithmsforVertexBipartizationandOthers ........ 56 3.25 ParameterizedAlgorithmsforFeedbackSetProblems ............. 57 3.26 Branch and Bound Algo. for Linear Ordering in Tournaments ........................................................... 59 3.27 HowtoRankwithFewerErrors ...................................... 61 3.28 FeedbackArcSetinBipartiteTournaments ......................... 63 3.29 AggregatingInconsistentInformation:Rankingand Clustering ............................................................. 66 3.30 MinimumFeedbackArcSetsinRotatorGraphs .................... 69 3.31 RankingTournaments:LocalSearchandaNewAlgorithm ........ 70 3.32 DeterministicPivoting for ConstrainedRankingand Clustering ............................................................. 75 3.33 FasterFAST(FeedbackArcSetinTournaments) ................... 77 3.34 Fixed-Parameter Tractability Results for FSPs in Tournaments ........................................................... 78 3.35 Fast Local Search Algorithm for Weighted FAS in Tournaments ........................................................... 79 3.36 FasterAlgorithmsforFeedbackArcSetTournament ............... 80