Integer programming and combinatorial optimization 17th international conference ipco 2014 bonn germ

Integer Programming and

Combinatorial

Optimization

17th International Conference IPCO 2014

Bonn Germany June 23 25 2014

Proceedings 1st Edition Jon Lee

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Jon Lee Jens Vygen (Eds.)

Integer Programming and Combinatorial Optimization

17th International Conference, IPCO 2014 Bonn, Germany, June 23–25, 2014

Proceedings

LectureNotesinComputerScience8494

CommencedPublicationin1973

FoundingandFormerSeriesEditors: GerhardGoos,JurisHartmanis,andJanvanLeeuwen

EditorialBoard

DavidHutchison LancasterUniversity,UK

TakeoKanade CarnegieMellonUniversity,Pittsburgh,PA,USA

JosefKittler UniversityofSurrey,Guildford,UK

JonM.Kleinberg CornellUniversity,Ithaca,NY,USA

AlfredKobsa UniversityofCalifornia,Irvine,CA,USA

FriedemannMattern ETHZurich,Switzerland

JohnC.Mitchell StanfordUniversity,CA,USA

MoniNaor

WeizmannInstituteofScience,Rehovot,Israel

OscarNierstrasz UniversityofBern,Switzerland

C.PanduRangan IndianInstituteofTechnology,Madras,India

BernhardSteffen TUDortmundUniversity,Germany

DemetriTerzopoulos UniversityofCalifornia,LosAngeles,CA,USA

DougTygar UniversityofCalifornia,Berkeley,CA,USA

GerhardWeikum MaxPlanckInstituteforInformatics,Saarbruecken,Germany

JonLeeJensVygen(Eds.)

IntegerProgramming andCombinatorial

Optimization

17thInternationalConference,IPCO2014

Bonn,Germany,June23-25,2014

Proceedings

VolumeEditors

JonLee

UniversityofMichigan

DepartmentofIndustrialandOperationsEngineering

1205BealAvenue,AnnArbor,MI48109-2117,USA

E-mail:jonxlee@umich.edu

JensVygen

UniversityofBonn

ResearchInstituteforDiscreteMathematics

Lennéstr.2,53113Bonn,Germany

E-mail:vygen@or.uni-bonn.de

ISSN0302-9743e-ISSN1611-3349

ISBN978-3-319-07556-3e-ISBN978-3-319-07557-0

DOI10.1007/978-3-319-07557-0

SpringerChamHeidelbergNewYorkDordrechtLondon

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Preface

Thisvolumecontainsthe34extendedabstractspresentedatIPCO2014,the 17thConferenceonIntegerProgrammingandCombinatorialOptimization,held June23-25,2014,inBonn,Germany.

TheIPCOconferenceisundertheauspicesoftheMathematicalOptimization Society.Itisheldeveryyear,exceptfor thoseinwhichtheInternationalSymposiumonMathematicalProgrammingtakesplace.Theconferenceisaforumfor researchersandpractitionersworkingonvariousaspectsofintegerprogramming andcombinatorialoptimization.Theaimistopresentrecentdevelopmentsin theory,computation,andapplicationsintheseareas.Traditionally,IPCOconsistsofthreedaysofnon-parallelsessions,withnoinvitedtalks.MoreinformationonIPCOanditshistorycanbefoundatwww.mathopt.org/?nav=ipco.

Thisyear,therewere143submissions,twoofwhichwerewithdrawnbefore thereviewprocessstarted.Eachreviewedsubmissionwasreviewedbyatleast threeProgramCommitteemembers,often withthehelpofexternalreviewers. TheProgramCommitteemetinAussoisinJanuary2014and,afterthorough discussions,selected34paperstobepresentedatIPCO2014andincludedin thisvolume.Therecordnumberofsubmissions,theirhighquality,andthemore orlessconstantnumberofpapersthatcanbeacceptedmadethisIPCOeven morecompetitivethanpreviouseditions,withanacceptancerateoflessthan 25%.

Wewouldliketothank:

–

AllauthorswhosubmittedextendedabstractstoIPCO;itisapleasuretosee howactiveallareasofintegerprogrammingandcombinatorialoptimization are

–

ThemembersoftheProgramCommittee,whograciouslygavetheirtime andenergy

– Theexternalreviewers,whoseexpertisewasinstrumentalinguidingour decisions

– TheEasyChairdevelopersfortheirexcellentplatformmakingmanythings somucheasier

– Springerfortheirefficientcooperationinproducingthisvolume

– ThemembersoftheOrganizingCommitteeandallpeopleinBonnwho helpedtomakethisconferencepossible

– ThespeakersofthesummerschoolprecedingIPCO:G´erardCornu´ejols, Andr´asFrank,ThomasRothvoß,andDavidShmoys

– TheMathematicalOptimizationSocietyandinparticularthemembersof itsIPCOSteeringCommittee:AndreasSchulz,AndreaLodi,andDavid Williamson,fortheirhelpandadvice.

March2014JonLee JensVygen

Organization

ProgramCommittee

FlaviaBonomoUniversidaddeBuenosAires,Argentina SamBurerUniversityofIowa,USA

G´erardCornu´ejolsCarnegieMellonUniversity,USA

SatoruFujishigeKyotoUniversity,Japan

MichaelJ ¨ ungerUniversit¨atzuK¨oln

MatthiasK¨oppeUniversityofCalifornia,Davis,USA

JonLee(chair)UniversityofMichigan,USA

JeffLinderothUniversityofWisconsin,USA

Jean-PhilippeRichardUniversityofFlorida,USA

Andr´asSeb˝oCNRS,LaboratoireG-SCOP,Grenoble,France MaximSviridenkoUniversityofWarwick,UK

ChaitanyaSwamyUniversityofWaterloo,Canada

JensVygenUniversit¨atBonn,Germany

DavidP.WilliamsonCornellUniversity,USA LaurenceWolseyUniversit´ecatholiquedeLouvain,Belgium

OrganizingCommittee

StephanHeld(co-chair) StefanHougardy

BernhardKorte JensVygen(chair)

AdditionalReviewers

Aardal,Karen Aharoni,Ron Ahmadian,Sara Albers,Susanne An,Hyung-Chan Ando,Kazutoshi Argiroffo,Gabriela Badanidiyuru,Ashwinkumar Balasundaram,Baski Baldacci,Roberto Bansal,Nikhil Bhaskar,Umang B¨okler,Fritz

Bornstein,Claudson Boyar,Joan Brenner,Ulrich Breˇsar,Boˇstjan Buchbinder,Niv Buchheim,Christoph Butenko,Sergiy Byrka,Jaroslaw Chrobak,Marek Cornaz,Denis deKlerk,Etienne Dey,Santanu DiSumma,Marco

Dourado,Mitre DuartePinto,PauloEust´aquio

Dvorak,Zdenek Ehrgott,Matthias Ene,Alina Faenza,Yuri Fanelli,Angelo Feuerstein,Esteban Fiorini,Samuel Frank,Andr´as Friggstad,Zachary Gaspers,Serge Gester,Michael Gleixner,Ambros Goemans,Michel Grigoriev,Alex Groß,Martin Guenin,Bertrand G ¨ unl ¨ uk,Oktay Gupta,Anupam Gy´arf´as,Andr´as H¨ahnle,Nicolai Hajiaghayi,Mohammadtaghi Harks,Tobias Havet,Fr´ed´eric Held,Stephan Hirai,Hiroshi Hoefer,Martin Hungerlander,Philipp Im,Sungjin

Imai,Hiroshi Jansen,Klaus Jeronimo,Gabriela Jord´an,Tibor Kakimura,Naonori Kami´nski,Marcin Kamiyama,Naoyuki Karakostas,George Karpinski,Marek Kawahara,Jun Kelner,Jonathan Kir´aly,Tam´as Kitahara,Tomonari Klewinghaus,Niko Kobayashi,Yusuke

Kortsarz,Guy Krumke,Sven Kumar,Amit Letchford,Adam Li,Jian Li,Shi Liers,Frauke Lin,MinChih Luedtke,James Madry,Aleksander Maffray,Fr´ed´eric Mallach,Sven Manlove,David Margot,Fran¸cois Martin,Alexander Marx,D´aniel Matuschke,Jannik McCormick,S.Thomas Megow,Nicole Mestre,Julian Mirrokni,Vahab Miyazaki,Shuichi Moldenhauer,Carsten Moscardelli,Luca Moseley,Ben Muller,Dirk Murota,Kazuo Nagano,Kiyohito Nagarajan,Viswanath Newman,Alantha Niedermeier,Rolf Ochsendorf,Philipp Olver,Neil Oriolo,Gianpaolo Ostrowski,Jim Pferschy,Ulrich Pilipczuk,Marcin Pokutta,Sebastian Queyranne,Maurice Rautenbach,Dieter Ravi,R. Rinaldi,Giovanni R¨oglin,Heiko Rothvoß,Thomas Rotter,Daniel

Sanit`a,Laura Sassano,Antonio Sau,Ignasi Savelsbergh,Martin Sch¨afer,Till Schalekamp,Frans Scheifele,Rudolf Schieber,Baruch Schmidt,Daniel Schneider,Jan Schorr,Ulrike Shepherd,Bruce Shigeno,Maiko Shioura,Akiyoshi Silvanus,Jannik Silveira,Rodrigo Sitters,Ren´ e Skopalik,Alexander Skutella,Martin Smriglio,Stefano Soto,Jos´ e Spirkl,Sophie Spisla,Christiane vanStee,Rob Svensson,Ola

Szigeti,Zolt´an Takazawa,Kenjiro Tanigawa,Shin-ichi Trotignon,Nicolas Tun¸cel,Levent Uetz,Marc VanVyve,Mathieu V´egh,L´aszl´ o Ventura,Paolo Verschae,Jos´ e Vielma,JuanPablo Vishnoi,Nisheeth Vondr´ak,Jan Ward,Justin Wiese,Andreas Woeginger,GerhardJ. Wollan,Paul Wong,PrudenceW.H. Woods,Kevin Wotzlaw,Andreas Young,Neal Zenklusen,Rico vanZuylen,Anke vanderZwaan,Ruben Zwick,Uri

TheCyclingPropertyfortheClutterofOdd st-Walks ................

AhmadAbdiandBertrandGuenin

OnSimplexPivotingRulesandComplexityTheory

IlanAdler,ChristosPapadimitriou,andAviadRubinstein

AStronglyPolynomialTimeAlgorithmforMulticriteriaGlobal MinimumCuts

HasseneAissi,A.RidhaMahjoub,S.ThomasMcCormick,and MauriceQueyranne

IntegerProgramswithPrescribedNumberofSolutionsandaWeighted VersionofDoignon-Bell-Scarf’sTheorem ............................

IskanderAliev,Jes´usA.DeLoera,andQuentinLouveaux

CentralityofTreesforCapacitated k -Center .........................

Hyung-ChanAn,AdityaBhaskara,ChandraChekuri, ShalmoliGupta,VivekMadan,andOlaSvensson

SequenceIndependent,SimultaneousandMultidimensionalLiftingof GeneralizedFlowCoversfortheSemi-ContinuousKnapsackProblem withGeneralizedUpperBoundsConstraints

AlejandroAngulo,DanielEspinoza,andRodrigoPalma

GennadiyAverkovandAmitabhBasu

MaximumWeightedInducedBipartiteSubgraphsandAcyclic SubgraphsofPlanarCubicGraphs

MouradBaıouandFranciscoBarahona

n-StepCycleInequalities:FacetsforContinuous n-MixingSetand StrongCutsforMulti-ModuleCapacitatedLot-SizingProblem

ManishBansalandKiavashKianfar

NikhilBansalandViswanathNagarajan

AnandBhalgatandSanjeevKhanna

NetworkImprovementforEquilibriumRouting

UmangBhaskar,KatrinaLigett,andLeonardJ.Schulman

FindingSmallStabilizersforUnstableGraphs

AdrianBock,KarthekeyanChandrasekaran,JochenKonemann, BrittaPeis,andLauraSanit` a

TheTriangleSplittingMethodforBiobjectiveMixedInteger Programming

NatashiaBoland,HadiCharkhgard,andMartinSavelsbergh

PierreBonamiandFran¸coisMargot

A 5 4 -ApproximationforSubcubic2ECUsingCirculations

SylviaBoyd,YaoFu,andYuSun

Box-ConstrainedMixed-Integer PolynomialOptimizationUsing SeparableUnderestimators 198 ChristophBuchheimandClaudiaD’Ambrosio

SubmodularMaximizationMeetsStreaming:Matchings,Matroids,and More

AmitChakrabartiandSagarKale

TheAll-or-NothingFlowProbleminDirectedGraphswithSymmetric DemandPairs

ChandraChekuriandAlinaEne

MicheleConforti,AlbertoDelPia,MarcoDiSumma,and YuriFaenza

StrongLPFormulationsforSchedulingSplittableJobsonUnrelated Machines

Jos´eR.Correa,AlbertoMarchetti-Spaccamela,JannikMatuschke, LeenStougie,OlaSvensson,V´ıctorVerdugo,andJos´eVerschae

HowGoodAreSparseCutting-Planes? .............................

SantanuS.Dey,MarcoMolinaro,andQianyiWang

ShortToursthroughLargeLinearForests

UrielFeige,R.Ravi,andMohitSingh

LinearProgrammingHierarchies SufficeforDirectedSteinerTree 285 ZacharyFriggstad,JochenKonemann,YoungKunKo, AnandLouis,MohammadShadravan,andMadhurTulsiani

AnImprovedApproximationAlgorithmfortheStableMarriage ProblemwithOne-SidedTies ......................................

Chien-ChungHuangandTelikepalliKavitha

SimpleExtensionsofPolytopes

VolkerKaibelandMatthiasWalter

LowerBoundsontheSizesofIntegerProgramswithoutAdditional Variables .......................................................

VolkerKaibelandStefanWeltge

OntheConfigurationLPforMaximumBudgetedAllocation

ChristosKalaitzis,AleksanderM¸adry,AlanthaNewman, Luk´aˇsPol´aˇcek,andOlaSvensson

FatmaKılın¸c-KarzanandSercanYıldız

Coupledand k -SidedPlacements:GeneralizingGeneralized Assignment .....................................................

MadhukarKorupolu,AdamMeyerson,RajmohanRajaraman,and BrianTagiku

AUnifiedAlgorithmforDegreeBoundedSurvivableNetworkDesign

LapChiLauandHongZhou

MatthiasMnichandAndreasWiese

ImprovedBranch-Cut-and-PriceforCapacitatedVehicleRouting ......

DiegoPecin,ArturPessoa,MarcusPoggi,andEduardoUchoa Claw-Free t-PerfectGraphsCanBeRecognisedinPolynomialTime

HenningBruhnandOliverSchaudt

TheCyclingPropertyfortheClutter ofOdd st-Walks

AhmadAbdiandBertrandGuenin

DepartmentofCombinatoricsandOptimization,UniversityofWaterloo {a3abdi,bguenin}@uwaterloo.ca

Abstract. Abinaryclutteriscyclingifitspackingandcoveringlinear programhaveintegraloptimalsolutionsforalleulerianedgecapacities. Weprovethattheclutterofodd st-walksofasignedgraphiscycling ifandonlyifitdoesnotcontainasaminortheclutterofoddcircuits of K5 northeclutteroflinesoftheFanomatroid.Corollariesofthis resultinclude,ofmany,thecharacterizationforweaklybipartitesigned graphs[5],packingtwo-commoditypaths[7,10],packing T -joinswith small |T |,anewresultoncoveringoddcircuitsofasignedgraph,as wellasanewresultoncoveringoddcircuitsandodd T -joinsofasigned graft.

1Introduction

A clutter C isafinitecollectionofsets, oversomefinitegroundset E (C ),with thepropertythatnosetin C iscontainedin,orisequalto,anothersetof C . ThisterminologywasfirstcoinedbyEdmondsandFulkerson[2].A cover B is asubsetof E (C )suchthat B ∩ C = ∅,forall C ∈C .The blocker b(C )isthe clutteroftheminimalcovers.Itiswellknownthat b(b(C ))= C ([8,2]).Aclutter is binary if,forany C1 ,C2 ,C3 ∈C ,theirsymmetricdifference C1 C2 C3 contains,orisequalto,asetof C .Equivalently,aclutterisbinaryif,forevery C ∈C and B ∈ b(C ), |C ∩ B | isodd([8]).Itisthereforeimmediatethataclutter isbinaryifandonlyifitsblockeris.

Let C beaclutterand e ∈ E (C ).The contraction C /e and deletion C\ e are cluttersonthegroundset E (C ) −{e} where C /e isthecollectionofminimalsets in {C −{e} : C ∈C} and C\ e := {C : e/ ∈ C ∈C}.Observethat b(C /e)= b(C ) \ e and b(C\ e)= b(C )/e.Contractionsanddeletionscanbeperformedsequentially andtheresultdoesnotdependontheorder.Aclutterobtainedfrom C bya sequenceofdeletions Ed andasequenceofcontractions Ec (Ed ∩ Ec = ∅)is calleda minor of C andisdenoted C\ Ed /Ec Givenedge-capacities w ∈ ZE (C ) + considerthelinearprogram (P )

min (we xe : e ∈ E (C )) s.t. x(C ) ≥ 1,C ∈C xe ≥ 0,e ∈ E (C ),

J.LeeandJ.Vygen(Eds.):IPCO2014,LNCS8494,pp.1–12,2014. c SpringerInternationalPublishingSwitzerland2014

2A.AbdiandB.Guenin

anditsdual (D )

max (yC : C ∈C ) s.t. (yC : e ∈ C ∈C ) ≤ we ,e ∈ E (C ) yC ≥ 0,C ∈C .

Aclutterissaidtobe ideal if,foreveryedge-capacities w ∈ ZE (C ) + ,(P )hasan optimalsolutionthatisintegral.AbeautifulresultofLehman[9]statesthata clutterisidealifandonlyifitsblockeris.Edge-capacities w ∈ ZE (C ) + aresaid tobe eulerian if,forevery B and B in b(C ), w (B )and w (B )havethesame parity.Seymour[13]callsabinaryclutter cycling if,foreveryeulerianedgecapacities w ∈ ZE (C ) + ,(P )and(D )bothhaveoptimalsolutionsthatareintegral. Itcanbereadilycheckedthatifaclutteriscycling(orideal)thensoareall itsminors([13,14]).Therefore,onecancharacterizetheclassofcyclingclutters byexcludingminor-minimalcluttersthatarenotinthisclass.Inthispaper,we willonlyfocusonbinaryclutters.

O5 istheclutteroftheoddcircuitsof K5 .Let L7 betheclutterofthelines oftheFanomatroid,i.e. E (L7 )= {

,

and L

,

Let P10 bethecollectionofthepostmansetsofthePetersengraph,i.e.setsof edgeswhichinduceasubgraphwhoseodddegreeverticesarethe(odddegree) verticesofthePetersengraph.Observethatthefourclutters O5 ,b(O5 ), L7 , P10 arebinary,andmoreover,itcanbereadilycheckedthatnoneoftheseclutters iscycling.Hence,ifabinaryclutteriscyclingthenitcannothaveanyofthese cluttersasaminor.Thefollowingexcludedminorcharacterizationispredicted.

Conjecture1(CyclingConjecture). Abinaryclutteriscyclingif,andonly if,ithasnoneofthefollowingminors: O5 ,b(O5 ), L7 , P10

TheCyclingConjecture,asstated,canbefoundinSchrijver[12].However,this conjecturewasfirstproposedbySeymour[13]andthenmodifiedbyA.M.H. GerardsandB.Guenin.Itisworthmentioningthatthisconjecturecontains the fourcolortheorem [15].Noneofourresultsinthispaperhaveanyapparent bearingsonthistheorem.

Considerafinitegraph G,whereparalleledgesandloopsareallowed.A cycle of G istheedgesetofasubgraphof G whereeveryvertexhasevendegree.A circuit of G isaminimalcycle,anda path isacircuitminusanedge.Wedefine an st-path asfollows:if s = t thenitisapathwhere s and t arethedegreeone verticesofthepath;otherwise,when s = t thenitisjustthesingletonvertex s.Let Σ beasubsetofitsedges.Thepair(G,Σ )iscalleda signedgraph.We sayasubset S oftheedgesis odd (resp. even)in(G,Σ )if |S ∩ Σ | isodd(resp. even).Let s,t beverticesof G.Wecallasubsetoftheedgesof(G,Σ )an odd st-walk ifitiseitheranodd st-path,oritistheunionofaneven st-path P and anoddcircuit C where P and C shareatmostonevertex.Observethatwhen s = t thenanodd st-walkissimplyanoddcircuit.Itiseasytoseethatclutters

ofodd st-walksareclosedundertakingminors.Asisshownin[6]theclutterof odd st-walksisbinary,anditdoesnothaveaminorisomorphicto b(O5 )or P10 . Inthispaper,weverifytheCyclingConjectureforthisclassofbinaryclutters:

Theorem2. Aclutterofodd st-walksiscyclingif,andonlyif,ithasno O5 andno L7 minor.

2RestatingTheorem2

OnecanviewTheorem2asapackingandcoveringresult.Weneedthefollowing definition:twoedgesofasignedgraphare parallel iftheyhavethesameendverticesaswellasthesamesign.Nowlet(G =(V,E ),Σ )beasignedgraph withoutanyparalleledges,andchoose s,t ∈ V .Let C betheclutteroftheodd st-walks,overthegroundset E ,andchooseedge-capacities w ∈ ZE + .An odd st-walkcoverof (G,Σ )issimplyacoverfor C .Whenthereisnoambiguity,we refertoanodd st-walkcoverasjustacover.

Proposition3(Guenin[6]). Ifasubsetoftheedgesisaminimalcoverthen itiseitheran st-bond(aminimal st-cut)oritisoftheform Σ C ,where C isacutwith s and t onthesameshore.

Theminimalcoversofthelatterformabovearecalled signatures.Noticethatif Σ isasignature,then(G,Σ )and(G,Σ )havethesameclutterofodd st-walks. Reset(G,Σ )asfollows:replaceeachedge e of(G,Σ )with we paralleledges. The packingnumber ν (G,Σ )of(G,Σ )isthemaximumnumberofpairwise (edge-)disjointodd st-walks.Adualparametertothepackingnumberisthe coveringnumber τ (G,Σ ),whichrecordstheminimumsizeofacoverof(G,Σ ). Considerapackingof ν (G,Σ )pairwisedisjointodd st-walkandacoverofsize τ (G,Σ ).Asthecoverintersectseveryodd st-walkinthepacking,itfollowsthat τ (G,Σ ) ≥ ν (G,Σ ).Anaturalquestionarises:whendoesequalityhold?Theorem 2givessufficientconditionsforasignedgraphtosatisfy τ (G,Σ )= ν (G,Σ ).To elaborate,observethat τ (G,Σ )isthevalueof(P )and ν (G,Σ )isthevalueof (D ).For w tobeeulerianistosaythateverytwominimalcoversof(G,Σ )have thesameparity.Therefore,Proposition3impliesthefollowing.

Remark4. Edge-capacities w = 1 areeulerianif,andonlyif, (i) s = t andthedegreeofeveryvertexiseven,or (ii) s = t, deg(s) −|Σ | andthedegreeofeveryvertexin V −{s,t} areeven.

Wecallsuchsignedgraphs st-eulerian

Justlikehowwedefinedminoroperationsforclutters,wenowdefineminor operationsforsignedgraphs.Let e ∈ E .Thentheminoroperationsfor C correspondtothefollowingminoroperationsfor(G,Σ ):(1) delete e:replace(G,Σ ) by(G \ e,Σ −{e}),(2) contract e:replace(G,Σ )by(G/e,Σ ),where Σ isa signatureof(G,Σ )thatdoesnotusetheedge e.Observethatvertices s and t movetowherevertheedgecontractionstakethem,andif s and t areever

4A.AbdiandB.Guenin

identifiedthenwesay s = t.Asignedgraph(H,Γ ) isaminorof (G,Σ )ifitis isomorphictoasignedgraphobtainedfrom(G,Σ )byasequenceofedgedeletions,edgecontractions,andpossiblydeletionofisolatedverticesandswitching s and t.Notethatif(H,Γ )isaminorof(G,Σ ),thentheclutterofodd st-walks of(H,Γ )isaminoroftheclutterofodd st-walksof(G,Σ ). Thetwospecialclutters O5 and L7 thatappearinTheorem2havethefollowingrepresentations: O5 istheclutterofodd st-walksof K5 :=(K5 ,E (K5 ))where s = t isoneofthefivevertices,and L7 istheclutterofodd st-walksofthesigned graph F7 with s = t,asshowninFigure1.Observethat τ (K5 )=4 > 2= ν (K5 ) and τ (F7 )=3 > 1= ν (F7 ).WecannowrestateTheorem2asfollows,andin fact,wewillprovethisrestatementinsteadoftheoriginalone: s t

Fig.1. Signedgraph F7 :arepresentationof L7 .Boldedgesareodd.

Theorem5. Let (G,Σ ) beasignedgraphwith s,t ∈ V (G).If (G,Σ ) isan steuleriansignedgraphthatdoesnotcontain K5 or F7 asaminorthen τ (G,Σ )= ν (G,Σ ).

3ExtensionsofTheorem2

Let(G =(V,E ),Σ )beasignedgraphwith s,t ∈ V .Suppose(G,Σ )isan steuleriansignedgraphthatdoesnotcontain K5 or F7 asaminor.If s = t let τst bethesizeofaminimum st-bond,otherwiselet τst := τ (G,Σ ).Observethat τst ≥ τ (G,Σ )asevery st-bondisalsoacover.Add τst τ (G,Σ )oddloopsto (G,Σ )toobtainanother st-euleriansignedgraph(G ,Σ ).Sinceneither K5 nor F7 containanoddloop,itfollowsthat(G ,Σ )alsodoesnotcontain K5 or F7 asaminor.Observethat τ (G ,Σ )= τ (G,Σ )+(τst τ (G,Σ ))= τst andso byTheorem2,onecanfindapackingof τst pairwisedisjointodd st-walksin (G ,Σ ).In(G,Σ )thispackingcorrespondstoacollectionof τst pairwisedisjoint elements, τ (G,Σ )ofwhichareodd st-walksandtheremainingelementsareeven st-paths.Therefore,wegetthefollowingequivalent,andsharper,formulationof Theorem5.

Theorem6. Let (G,Σ ) beasignedgraphwith s,t ∈ V (G).Supposethat (G,Σ ) isan st-euleriansignedgraphthatdoesnotcontain K5 or F7 asaminor.Then thereexistsacollectionof τst (G,Σ ) pairwise(edge-)disjointelements, τ (G,Σ ) ofwhichareodd st-walksandtheremainingelementsareeven st-paths.

WecanobtainacounterparttoTheorem6asfollows:let τΣ bethesizeofaminimumsignature.Observethat τΣ ≥ τ (G,Σ )andthat τ (G,Σ )=min {τst ,τΣ }.In contrasttoabove,thistimeweadd τΣ τ (G,Σ )evenedgesbetween s and t to (G,Σ )toobtainanother st-euleriansignedgraph(G ,Σ ).Notice,however,that wecannolongerguaranteethat(G ,Σ )containsno K5 or F7 minor.Observe thatthisistrueif,andonlyif,(G,Σ )doesnotcontain K5 , K5 0 , K5 1 , K5 2 , K5 3 or F7 asaminor,where

(i)for i ∈{0, 1, 2, 3}, K5 i isthesignedgraphobtainedfromsplittingavertex, anditsincidentedges,of K5 intotwovertices s,t,where s hasdegree i and t hasdegree4 i,and (ii) F7 isthesignedgraphobtainedfrom F7 bydeletingtheedgebetween s and t.

Notethatifweaddanevenedgetoanyofthesesignedgraphs,thena K5 or an F7 appearsasaminor.Itcanbereadilycheckedthatif(G,Σ )doesnot containanyofthesefivesignedgraphsasaminor,then(G ,Σ )containsno K5 or F7 minor.Observenowthat τ (G ,Σ )= τ (G,Σ )+(τΣ τ (G,Σ ))= τΣ and sobyTheorem2,onecanfindapackingof τΣ pairwisedisjointodd st-walks in(G ,Σ ).In(G,Σ )thispackingcorrespondstoacollectionof τΣ pairwise disjointelements, τ (G,Σ )ofwhichareodd st-walksandtheremainingelements areoddcircuits.Thus,thefollowingcounterparttoTheorem6isobtained.

Theorem7. Let (G,Σ ) beasignedgraphwith s,t ∈ V (G).Supposethat (G,Σ ) isan st-euleriansignedgraphthatdoesnotcontain K5 , K5 0 , K5 1 , K5 2 , K5 3 or F7 asaminor.Thenin (G,Σ ) thereexistsacollectionof τΣ (G,Σ ) pairwise (edge-)disjointelements, τ (G,Σ ) ofwhichareodd st-walksandtheremaining elementsareoddcircuits.

4ApplicationsofTheorem2

Inthissection,wediscusssomeapplicationsofTheorem2.Observethatacycling clutterisalsoideal.Asacorollary,wegetthefollowingtheorem:

Corollary8(Guenin[6]). Aclutterofodd st-walksisidealif,andonlyif,it hasno O5 andno L7 minor.

When s = t anodd st-walkisjustanoddcircuit.Asignedgraphissaidto be weaklybipartite iftheclutterofitsoddcircuitsisideal.Theclutterofodd circuitsdoesnotcontainan L7 minor[6].Hence,wegetthefollowingtworesults ascorollariesofTheorem2:

Corollary9(Guenin[5]). Asignedgraphisweaklybipartiteif,andonlyif, ithasno K5 minor.

Corollary10(GeelenandGuenin[3]). Aclutterofoddcircuitsiscycling if,andonlyif,ithasno O5 minor.

Observethat2w iseulerianforany w ∈ ZE (G) + .Asaresult,thefollowingresult followsasacorollaryofTheorem2:

Theorem11. Supposethat C isaclutterofodd st-walkswithoutan O5 oran L7 minor.Then,foranyedge-capacities w ∈ ZE (G) + ,thelinearprogram (P ) has anoptimalsolutionthatisintegralanditsdual (D ) hasanoptimalsolutionthat ishalf-integral.

ToobtainmoreapplicationsofTheorem2,wewillturntoitsrestatementTheorem5,andnaturallytrytofindniceclassesofsignedgraphswithouta K5 or an F7 minor.

4.1SignedGraphswithout K5 and F7 Minor

Let(G,Σ )beasignedgraphwith s,t ∈ V .Observethatif s = t then(G,Σ ) hasno F7 minor,andtherearemanyclassesofsuchsignedgraphswithouta K5 minor.Forinstance,whenever G isplanaror |Σ | =2,(G,Σ )doesnotcontain a K5 minor.Otherclassesofsuchsignedgraphscanbefoundin[4,3].Inthis section,wefocusonlyonsignedgraphs(G,Σ )withdistinct s,t ∈ V . A blockingvertex isavertex v whosedeletionremovesalltheoddcycles,and a blockingpair isapairofvertices {u,v } whosedeletionremovesalltheodd cycles.

Remark12. Thefollowingclassesofsignedgraphswith s = t donotcontain K5 or F7 asaminor:

(1) signedgraphswithablockingvertex, (2) signedgraphswhere {s,t} isablockingpair, (3) planesignedgraphswithatmosttwooddfaces, (4) signedgraphsthathaveanevenfaceembeddingontheprojectiveplane,and s and t areconnectedwithanoddedge, (5) signedgraphswhereeveryodd st-walkisconnected,and (6) planesignedgraphswithablockingpair {u,v } where s,u,t,v appearona facialcycleinthiscyclicorder.

Observethatclass(5)contains(2)and(4).WewillapplyTheorem5tothe firstthreeclasses,andinthefirsttwocases,weobtaintwowell-knownresults. However,thethirdclasswillyieldanewandinterestingresultonpackingodd circuitcovers.NoticethatonecanevenapplyTheorem6totheseclasses.

Observefurtherthatthesignedgraphsin(1)and(2)donotcontain K5 0 , K5 1 , K5 2 , K5 3 or F7 asaminoreither,soonemayevenconsiderapplyingTheorem 7totheseclasses.WeleaveittothereadertofindoutwhatTheorems6and7 appliedtotheseclassesimply.

4.2Class(1):Packing T -joinswith |T | =4

Let H beagraphwithvertexset W ,andchooseanevenvertexsubset T .A T -join of H isanedgesubsetwhoseodddegree verticesare(all)theverticesin T .A T -cut of H isanedgesubsetoftheform δ (U )where U ⊆ W and |U ∩ T | isodd.Observethattheblockeroftheclutterofminimal T -joinsistheclutter ofminimal T -cuts.

WearenowreadytoprovethefollowingresultasacorollaryofTheorem2. However,itshouldbenotedthatthisresult(for T ofsizeatmost8,infact)is relativelyeasytoprovefromfirstprinciples,asisshownin[1].

Corollary13(CohenandLucchesi[1]).

Let H beagraphandchoosea vertexsubset T ofsize 4.Supposethateveryvertexof H notin T haseven degreeandthatalltheverticesin T havedegreesofthesameparity.Thenthe maximumnumberofpairwise(edge-)disjoint T -joinsisequaltotheminimum sizeofa T -cut.

Proof. Supposethat T = {s,t,s ,t }.Identify s and t toobtain G,andlet Σ = δH (s ).Thenthesignedgraph(G,Σ )containsablockingvertex s t ,and soitbelongstoclass(i).ByRemark4,(G,Σ )is st-eulerian.Theorem2then impliesthat τ (G,Σ )= ν (G,Σ ).However,observethatanodd st-walkof(G,Σ ) isa T -joinof H ,anda T -joinin H containsanodd st-walkof(G,Σ ).Hence, τ (G,Σ )= ν (G,Σ )impliesthatthemaximumnumberofpairwisedisjoint Tjoinsisequaltotheminimumsizeofa T -cut.

4.3Class(2):PackingTwo-commodityPaths

Corollary14(Hu[7],RothschildandWhinston[10]). Let H beagraph andchoosetwopairs (s1 ,t1 ) and (s2 ,t2 ) ofvertices,where s1 = t1 , s2 = t2 ,all of s1 ,t1 ,s2 ,t2 havethesameparity,andalltheotherverticeshaveevendegree. Thenthemaximumnumberofpairwise(edge-)disjointpaths,thatarebetween si and ti forsome i =1, 2,isequaltotheminimumsizeofanedgesubsetwhose deletionremovesall s1 t1 -and s2 t2 -paths.

Proof. Identify s1 and s2 ,aswellas t1 and t2 toobtain G,andlet Σ = δH (s1 ) δH (t2 ).Let s := s1 s2 ∈ V (G)and t := t1 t2 ∈ V (G).Thenthesignedgraph (G,Σ )has {s,t} asablockingpair,andsoitbelongstoclass(2).Againby Remark4(G,Σ )is st-eulerian.Therefore, byTheorem2wegetthat τ (G,Σ )= ν (G,Σ ).However,observethatanodd st-walkof(G,Σ )isan si ti -pathof H , forsome i =1, 2,andsuchapathin H containsanodd st-walkof(G,Σ ).Thus, τ (G,Σ )= ν (G,Σ )provesthecorollary.

4.4Class(3):PackingOddCircuitCovers

Theorem15. Let (H,Σ ) beaplanesignedgraphwithexactlytwooddfaces andchoosedistinct g,h ∈ V (H ).Let (G,Σ ) bethesignedgraphobtainedfrom identifying g and h in H ,andsupposethateverytwooddcircuitsof (G,Σ ) have thesamesizeparity.Thenin (G,Σ ) themaximumnumberofpairwisedisjoint oddcircuitcoversisequaltothesizeofaminimumoddcircuit.

8A.AbdiandB.Guenin

(Hereanoddcircuitcoverissimplyacoverfortheclutterofoddcircuits.)As thereadermaybewondering,whatistherationalebehindtheratherstrange constructionof(G,Σ )above?Interestingly,theclutterofminimaloddcircuit coversisbinary,andsotheCyclingConjecturepredictsanexcludedminorcharacterizationforwhenthisclutteriscycling.Aswedidwiththeclutterofodd st-walks,onecanrestatetheCyclingConjecturefortheclutterofoddcircuit coversasfollows:

(?)forsignedgraphs (G,Σ ) withouta K5 minorsuchthateverytwoodd circuitshavethesameparity,themaximumnumberofpairwisedisjoint oddcircuitcoversisequaltotheminimumsizeofanoddcircuit.(?)

Theconstructioninthe statementofTheorem15yieldsasignedgraph(G,Σ ) thathasno K5 minor,andTheorem15verifiestherestatementaboveforthese classesofsignedgraphs.

Proof. Let H ∗ betheplanedualof H ,andlet P beanodd gh-pathin(H,Σ ).Let s and t bethetwooddfacesof(H,Σ ).Considertheplanesignedgraph(H ∗ ,P ); notethatthissignedgraphhaspreciselytwooddfaces,namely g and h,andsoit belongsto(3).Inparticular,(H ∗ ,P )containsno K5 and F7 minor.Sinceevery twooddcircuitsof(G,Σ )havethesameparity,itfollowsfromRemark4that (H ∗ ,P )is st-eulerian.SoTheorem2appliesandwehave τ (H ∗ ,P )= ν (H ∗ ,P ).

Weclaimthatanoddcycleof(G,Σ )isanodd st-walkcoverof(H ∗ ,P ),and vice-versa.Let L beanoddcycleof(G,Σ ).If L isanoddcycleof(H,Σ )then L separatesthetwooddfaces s and t,andsoitisan st-cutin(H ∗ ,P ).Otherwise, L isanodd gh-pathandso L P isanevencycleof(H,Σ ).However,aneven cyclein(H,Σ )isacutin(H ∗ ,P )having s and t onthesameshore.Hence, L is oftheform P δ (U )where s,t ∈ U ⊆ V (H ∗ ).Therefore,ineithercases, L isan odd st-walkcoverof(H ∗ ,P ).Similarly,onecanshowthatanodd st-walkcover of(H ∗ ,P )isanoddcycleof(G,Σ ).Therefore,since b(b(C ))= C foranyclutter C ,itfollowsthatanoddcircuitcoverof(G,Σ )isanodd st-walkof(H ∗ ,P ), andvice-versa.

Hence, τ (H ∗ ,P )istheminimumsizeofanoddcircuitof(G,Σ ),and ν (H ∗ ,P ) isthemaximumnumberofpairwisedisjointoddcircuitcoversof(G,Σ ).Since τ (H ∗ ,P )= ν (H ∗ ,P ),theresultfollows.

4.5ClutterofOddCircuitsandOdd T -joins

Here,weprovideyetanotherapplicationofTheorem2.Thisresultgeneralizes Theorem15.Let(G =(V,E ),Σ )beasignedgraph,andlet T ⊆ V beasubsetof evensize.Wecallthetriple(G,Σ,T )a signedgraft.Let C betheclutteroverthe groundset E thatconsistsofoddcircuitsandminimalodd T -joinsof(G,Σ,T ). Thisminor-closedclassofcluttersisfairlylarge.Forinstance,if T = ∅ then C istheclutterofoddcircuits,andif Σ isa T -cutthen C istheclutterof T -joins.

Remark16. C isabinaryclutter.

Proof. Takeanythreeelements C1 ,C2 ,C3 of C .Ifanevennumberof C1 ,C2 ,C3 areoddcircuits,then C1 C2 C3 isanodd T -joinandsoitcontainsan elementof C .Otherwise,anoddnumberof C1 ,C2 ,C3 areoddcircuits,andso C1 C2 C3 isanoddcycleandsoitcontainsanelementof C .Sincethisis trueforall C1 ,C2 ,C3 in C ,itfollowsfromdefinitionthat C isbinary.

Remark17. Minimalcoversof C areoftheform Σ δ (U ),where U ⊆ V and |U ∩ T | iseven.

Proof. Let B beaminimalcoverof C .Then B intersectseveryoddcircuitof (G,Σ ),andso B Σ = δ (U )forsome U ⊆ V .Theprecedingremarkshowed C isbinary,andso B intersectseveryodd T -joininanoddnumberofedges,so |U ∩ T | mustbeeven.

Fig.2. Signedgraft F7 ,wherealledgesareoddandfilled-inverticesarein T .Forthis signedgraft,theclutterofoddcircuitsandminimalodd T -joinsisomorphicto L7

Theorem18. Let (G,Σ,T ) beaplanesignedgraftwithexactlytwooddfaces thathasnominorisomorphicto F7 ,depictedinFigure2.Let C betheclutter ofoddcircuitsandminimalodd T -joins,andsupposethateverytwoelements of C havethesamesizeparity.Thenthemaximumofpairwisedisjointminimal coversof C isequaltotheminimumsizeofanelementof C .

Proof. TheproofissimilartotheproofofTheorem15.Let G∗ betheplanedual of G,andlet P beanodd T -joinin(G,Σ,T ).Let s and t bethetwooddfaces of(G,Σ,T ).Since(G,Σ,T )hasnominorisomorphicto F7 ,itfollowsthatthe signedgraph(G∗ ,P )containsno F7 minor,andsinceitisplanar,ithasno K5 minoreither.Sinceeverytwoelementsof C havethesameparity,itfollowsthat (G∗ ,P )is st-eulerian.Hence,byTheorem5, τ (G∗ ,P )= ν (G∗ ,P ).

10A.AbdiandB.Guenin

Weclaimthat C istheclutterofodd st-walkcoversof(G∗ ,P ),andviceversa.Let C ∈C .If C isanoddcircuitof(G,Σ,T ),then C isan st-cutof G∗ . Otherwise, C isanodd T -joinandso C P isanevencycleof(G,Σ ).Thus, C = P δ (U )forsome U ⊆ V (G∗ ) −{s,t},i.e. C isasignatureof(G∗ ,P ).

Hence, τ (G∗ ,P )istheminimumsizeofanelementof C ,and ν (G∗ ,P )isthe maximumnumberofpairwisedisjointcoversof C .Since τ (G∗ ,P )= ν (G∗ ,P ), theresultfollows.

LetusexplainhowthisresultimpliesTheorem15.InthecontextofTheorem15, let T = {g,h}.Observethat(H,Σ,T )isaplanesignedgraftwithexactlytwo oddfaces,andithasnominorisomorphicto F7 (for |T | =2).However,the clutterofoddcircuitsandminimalodd T -joinsof(H,Σ,T )isisomorphictothe clutterofoddcircuitsof(G,Σ ).ItisnoweasilyseenthatTheorem18implies Theorem15.

5OverviewoftheProofofTheorem2

Acompleteproofwillappearinthefullversion.Inthissection,however,we provideanoverviewofourproofofTheorem5,whichisequivalenttoTheorem2. Theprooffollowsaroutinestrategy.Westartwithan st-euleriansignedgraph (G,Σ )thatdoesnot pack,i.e. τ (G,Σ ) >ν (G,Σ ),andwewilllookforeitherof the obstructions K5 , F7 asaminor.

Wesaythatasignedgraph(H,Γ )isa weightedminor of(G,Σ )if(H,Γ ) minussomeparalleledgesisaminorof(G,Σ ).(Twoedgesareparallelifthey havethesameendverticesaswellasthesameparity.)Observethatif K5 or F7 appearsasaweightedminorof(G,Σ ),thenitisalsopresentasaminorsince neitherof K5 ,F7 containparalleledges.

Amongall st-euleriannon-packingweightedminorsof(G,Σ ),wepickone (G ,Σ )withsmallest τ (G ,Σ ),smallest |V (G )| andlargest |E (G )|,inthis orderofpriority.Suchanon-packingweightedminorexists.Indeed,ifanedge hassufficientlymanyparalleledges,thenitmaybecontractedwhilekeeping (G ,Σ )non-packingand τ (G ,Σ )unchanged.Reset(G,Σ ):=(G ,Σ )and let τ := τ (G,Σ ), ν := ν (G,Σ ).Byidentifyingavertexofeach(connected) componentwith s,ifnecessary,wemayassumethat G isconnected.(Notice thatneitheroftheobstructions K5 , F7 hasacut-vertex.)

Remark19. Theredonotexist τ 1 pairwisedisjointodd st-walksin (G,Σ ).

Proof. Supposeotherwise.Removesome τ 1pairwisedisjointodd st-walksin (G,Σ ).Observethatwhatisleftisanodd {s,t}-joinbecause |Σ |,deg(s), deg(t) and τ allhavethesameparityandallverticesotherthan s,t haveevendegree. Hence,sinceeveryodd {s,t}-joincontainsanodd st-walk,onecanactuallyfind τ pairwisedisjointodd st-walksin(G,Σ ),contradictingthefactthat(G,Σ )is non-packing.

Let B beacoverof(G,Σ )ofsize τ .Chooseanedge Ω asfollows.If s = t thenlet Ω ∈ E B ,andsincelabel s isirrelevanttoourprobleminthiscase,wemayas wellassume Ω ∈ δ (s).Otherwise,when s = t,let Ω ∈ (δ (s) ∪ δ (t)) B .Indeed, ifsuchanedgedoesnotexist,then δ (s) ∪ δ (t)iscontainedintheminimum cover B ,implyingthat δ (s) ∪ δ (t)= δ (s)= δ (t),butthiscannotbethecase as G isconnectedandnon-packing.Again,wemayassumethat Ω isincident to s.Let s betheotherend-vertexof Ω .Addtwoparalleledges Ω1 ,Ω2 to Ω toobtain(K,Γ );this st-euleriansignedgraphmustpacksince τ (K,Γ )= τ as B isalsoaminimumcoverfor(K,Γ ), V (K )= V (G)but |E (K )| > |E (G)| Hence,(K,Γ )containsacollection {L1 ,L2 ,...,Lτ } ofpairwisedisjointodd st-walks.Observethatallof Ω,Ω1 and Ω2 mustbeusedbytheodd st-walks in {L1 ,L2 ,...,Lτ },sayby L1 ,L2 ,L3 ,sinceotherwiseonefindsatleast τ 1 disjointodd st-walksin(G,Σ ),whichisnotthecasebytheprecedingremark. Asaresult,thesequence(L1 ,L2 ,L3 ,...,Lτ )correspondstoan Ω -packingof odd st-walks in(G,Σ ),describedasfollows:

(i) L1 ,...,Lτ areodd st-walksin(G,Σ ), (ii) Ω ∈ L1 ∩ L2 ∩ L3 and Ω/ ∈ L4 ∪···∪ Lτ ,and (iii)(Lj −{Ω } :1 ≤ j ≤ τ )arepairwisedisjointsubsetsofedges.

Wefixan Ω -packing(L1 ,L2 ,L3 ,...,Lτ )havingaminimumnumberofedgesin theirunion.

Wecall T a transversal ofacollectionofsetsif T picksexactlyoneelement fromeachofthesets.Foranodd st-walk L,wesaythataminimalcover B isa mate of L if |B L| = τ 3.

Lemma20. Let L beanodd st-walksuchthat (G,Σ ) \ L containsatleast τ 3 pairwisedisjointodd st-walkscollectedin L.Then L hasamate B ,and B L isatransversalof L

Proof. Thesignedgraph(G,Σ )\L packsasitis st-eulerianand τ ((G,Σ )\L) <τ Let B beoneofitsminimumcovers.Byourassumption, τ ((G,Σ ) \ L) ≥ τ 3. Sinceboth(G,Σ )and(G,Σ ) \ L are st-eulerian,itfollowsthat τ ((G,Σ ) \ L) and τ havedifferentparities,andso τ ((G,Σ ) \ L)iseither τ 3or τ 1. However,observethatthelatterisnotpossibleduetoRemark19andthefact that(G,Σ ) \ L packs.Asaresult |B | = τ ((G,Σ ) \ L)= τ 3.Itisnowclear that B ∪ L containsamatefor L,andthat B isatransversalof L.

Observethatif L ⊆ L1 ∪ L2 ∪ L3 or L ∈{L4 ,...,Lτ },then(G,Σ ) \ L does containatleast τ 3pairwisedisjointodd st-walks.Thus,theprecedinglemma guaranteestheexistenceofamateforanysuchodd st-walk.Vaguelyspeaking, matesareusedasmeanstobuildconnectivity,withappropriatesigning,between theodd st-walks.

Letuscallanodd st-walk L simple ifitisanodd st-path P ;otherwisewhen L istheunionofanoddcircuit C andaneven st-path P ,wecall L a nonsimple odd st-walk.Byourdefinitionthen,when s = t alltheodd st-walksare non-simple.Foreach1 ≤ i ≤ τ ,either Li isasimpleodd st-walk Pi ,oritisa non-simpleodd st-walk Ci ∪ Pi ,where Ci isanoddcircuitand Pi isaneven st-path.

Lemma21. Oneofthefollowingholds:

(i) L1 ,L2 and L3 aresimple, (ii)atleastoneof L1 ,L2 ,L3 isnon-simple,andwhenever Lk isnon-simplefor some 1 ≤ k ≤ 3,then Ω ∈ Ck , (iii)atleasttwoof L1 ,L2 ,L3 arenon-simple,and Ω ∈ P1 ∩ P2 ∩ P3

Weanalyzeeachofthethreecasesseparately,andthetechniquesusedtotackle eachcasearedifferent.Amajordifferencebetweenourproofandtheonesfor Corollaries8,9and10(see[6,5,11,3])isinwhereanobsructionislookedfor. Inanyoftheaforementionedproofs,onlythefirstthreesetsofthe Ω -packing assistedinfindinganobstruction.Forourproof,however,thisisnolongerthe case;someoftheodd st-walksin L4 ,...,Lτ ,aswellastheirmates,helpusin findingeitheroftheobstructions.Thisconcludesouroverviewoftheproofof Theorem5.

References

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OnSimplexPivotingRules andComplexityTheory

IlanAdler,ChristosPapadimitriou ,andAviadRubinstein

UniversityofCalifornia,BerkeleyCA94720,USA

Abstract. Weshowthattherearesimplexpivotingrulesforwhichit isPSPACE-completetotellifaparticularbasiswillappearonthealgorithm’spath.Suchrulescannotbethebasisofastronglypolynomial algorithm,unlessP=PSPACE.Weconjecturethatthesamecanbe shownformostknownvariantsofthesimplexmethod.However,wealso pointoutthatDantzig’sshadowvertexalgorithmhasapolynomialpath problem.Finally,wediscussinthesamecontextrandomizedpivoting rules.

Keywords: linearprogramming,thesimplexmethod,computational complexity.

1Introduction

Linearprogrammingwasfamouslysolvedinthelate1940sbyDantzig’ssimplexmethod[8];however,manyvariantsofthesimplexmethodwereeventually provedtohaveexponentialworst-caseperformance[21],while,aroundthesame time,Karp’s1972paperonNP-completeness[18]mentionslinearprogramming asarareprobleminNPwhichresistsclassificationaseitherNP-completeor polynomial-timesolvable.Khachiyan’sellipsoidalgorithm[20]resolvedpositively thisopenquestionin1979,butwasbroadlyperceivedasapoorcompetitorto thesimplexmethod.Notlongafterthat,Karmarkar’sinteriorpointalgorithm [19]providedapracticallyviablepolynomialalternativetothesimplexmethod. However,therewasstillasenseofdissatisfactioninthecommunity:Thenumberofiterationsofboththeellipsoidalgorithmandtheinteriorpointmethod dependnotjustonthedimensionsoftheproblem(thenumberofvariables d andthenumberofinequalities n)butalsoonthenumberofbitsneededtorepresentthenumbersintheinput;suchalgorithmsaresometimescalled“weakly polynomial”.

A stronglypolynomialalgorithm forlinearprogramming(oranyproblem whoseinputisanarrayofintegers)isonethatisapolynomial-timealgorithmin theordinarysense(alwaysstopswithinanumberofstepsthatispolynomialin thetotalnumberofbitsintheinput),butitalsotakesanumberofelementary arithmeticoperationsthatispolynomialinthedimensionoftheinputarray.

TheresearchofChristosPapadimitriouandAviadRubinsteinissupportedbyNSF GrantCCF-0964033.

J.LeeandJ.Vygen(Eds.):IPCO2014,LNCS8494,pp.13–24,2014. c SpringerInternationalPublishingSwitzerland2014

Stronglypolynomialalgorithmsexistformanynetwork-relatedspecialcasesof linearprogramming,aswasfirstshownin[11].ThiswasextendedbyTardos [30]whoestablishedtheexistenceofsuchanalgorithmfor“combinatorial”linearprograms,thatis,linearprogramswhoseconstraintmatrixis0-1(or,more generally,containsintegersthatareatmostexponentiallylargeinthedimensions).However,nostronglypolynomialalgorithmisknownforgenerallinear programming.

Thefollowingsummarizesoneofthemostimportantopenproblemsinoptimizationandthetheoryofalgorithmsandcomplexity:

Conjecture1.Thereisastronglypolynomialalgorithmforlinearprogramming.

Oneparticularlyattractivedirectionforapositiveanswerforthisconjectureis thesearchforpolynomialvariantsofthesimplexmethod.Itwouldbewonderful todiscoverapivotingruleforthesimplexmethodwhich(unlikeallknown suchmethods)alwaysfindstheoptimumafteranumberofiterationsthatis polynomialin d and n.Hencethefollowingisaninterestingspeculation:

Conjecture2.Thereisapivotingruleforthesimplexmethodthatterminates afteranumberofiterationsthatis,inexpectation,polynomialin d and n

InrelationtoConjecture2,cleverrandomizedpivotingrulesofaparticularrecursivesortwerediscoveredratherrecently,withworst-casenumberofiterations thathasasubexponentialdependenceon d [16,23].Otherrecentresultsrelated toConjecture1canbefoundin[6,34].

Inthenextsectionweformalizetheconceptofa pivotingrule: Amethodfor jumpingfromonebasicsolutiontoanadjacentonethat(1)isstronglypolynomial periteration;(2)isguaranteedtoincreaseapotentialfunctionateachstep;and (3)isguaranteedtoalwaysterminateattheoptimum(orcertifyinfeasibilityor unboundedness).Wealsogiveseveralexamplesofsuchrules.Itisimportantto notethatinourdefinitionweallowpivotingrulestojumpto infeasiblebases in ordertoincludepivotingrulesotherthanoftheprimaltype.Also,ouroriginal definitioninSection2restrictspivotingrulestobedeterministic;wediscussthe importantsubjectofrandomizedrulesinSection5.

Recentlytherehasbeenaglimmerofhopethatsomestrongerformsofthe twoconjecturescouldbedisproved,afterthedisproofoftheHirschConjecture [27].TheHirschconjecture[9]positedthatthediameterofa d-dimensional polytopewith n facetsisatmost n d,thelargestknownlowerbound.The bestknownupperboundforthisdiameteristhequasi-polynomialboundsof[17]. Butevenasuper-polynomiallowerboundwouldonlyfalsifytheconjecturesfor primal pivotingrules(onesgoingthroughonlyfeasiblebases,i.e.,verticesof thepolytope),but not forthemanyotherkindsofpivotingrules(seethenext section).Furthermore,itisnowclearthatthetechniquesinvolvedinthedisproof oftheHirschconjectureareincapableofestablishinganonlinearlowerboundon thediameterofpolytopes,anditiswidelybelievedthatthereisapolynomial upperboundonthediameterofpolytopes.

Inthispaperwecontemplatewhethertheconceptsandmethodsofcomplexity theorycanbeappliedproductivelytoilluminatetheproblemofstronglypolynomialalgorithmsforlinearprogrammingandConjecture1. Weshowaresult suggestingthatPSPACE-completenessmayberelevant.

Inparticular,weproposetoclassifydeterministicpivotingrulesbythecomplexityofthefollowingproblem,whichwecall thepathproblem ofapivoting rule:Givenalinearprogramandabasicsolution,willthislatteroneappearon thepivotrule’spath?RecallthatPSPACEistheclassofproblemssolvablein polynomial memory.ThisclasscontainsNP,anditisstronglybelievedtocontainitstrictly.The pathproblem ofapivotingruleisclearlyinPSPACE, becauseitcanbesolvedbyfollowingthe(possiblyexponentiallylong)pathof therule,reusingspace;ifitisPSPACE-complete,thenthepivotingrulecannot bepolynomial(unless,ofcourse,P=PSPACE).

Butitisnotaprioriclearthattherearepivotingrulesforwhichthepath problemisPSPACE-complete.Weshow(Theorem1)thattheydoexist;unfortunately,weprovethisnotforoneofthemanyclassicalpivotingrules,butfora new,explicitlyconstructed—andfairlyunnatural—one.Weconjecturethat thesameresultholdsforessentiallyallknowndeterministicpivotingrules;such aprooflooksquitechallenging;obviously,insuchaproofmuchmorewillneed tobeencodedinthelinearprogram(which,inthepresentproof,isoflogarithmiccomplexityandisomorphicto {0, 1}n ).However,wedoexhibit(Theorem 2)apivotingrulewhosepathproblemisinP:ItisDantzig’swell-known selfdualsimplex [9](alsoknownas shadowvertexalgorithm),whichisknowntobe exponentialintheworstcase[24],buthasbeenusedinseveralsophisticated algorithmicupperboundsforlinearprogramming,suchasaverage-caseanalysis andsmoothness[5,28,2,1,33,29].Webrieflydiscusstheapparentconnectionbetweentheaverage-caseperformanceofapivotingruleandthecomplexityofits pathproblem.

Themotivationforourapproachcamefromrecentresultsestablishingthatit isPSPACE-completetocomputethefinalresultofcertainwellknownalgorithms forfindingfixpointsandequilibria[13].However,theprooftechniquesusedhere arecompletelydifferentfromthosein[13].

2Definitions

Consideranalgorithmwhoseinputisanarrayof n integers.Thealgorithmis called stronglypolynomial if – itispolynomial-timeasaTuringmachine,and – ifoneassumesthatallelementaryarithmeticoperationshavecostone,the worst-casecomplexityofthealgorithmisboundedbyapolynomialin n,and isthereforeindependentofthesizeoftheinputintegers.

Inlinearprogrammingoneseekstomaximize cT x subjectto Ax = b,x ≥ 0, where A is m × n.An m × m nonsingularsubmatrix B of A isa basis.A feasible basis B isoneforwhichthesystem BxB = b (whereby xB wedenotevector x

restrictedtothecoordinatesthatcorrespondto B )hasanonnegativesolution; inthiscase, xB iscalleda basicfeasiblesolution. Basicfeasiblesolutionsare importantbecausetheyrenderlinearprogrammingacombinatorialproblem,in thattheoptimum,ifitexists,occursatoneofthem.Wesaythattwobasesare adjacent iftheydifferinonlyonecolumn.

Therearemanyversionsoflinearprogramming(withinequalityconstraints, minimization,unrestrictedinsignvariables,etc.)buttheyareallknowntobe easilyinterreducible.Weshallfeelfreetoexpresslinearprogramsinthemost convenientofthese.

Weshallassumethatthelinearprogramsunderconsiderationarenondegenerate(notwobasesresultinthesamebasicsolution).Detectingthisconditionisnontrivial(NP-hard,asitturnsout).However,thereareseveralreasons whythisveryconvenientassumptionisinconsequential.First,arandomperturbationofalinearprogram(obtained,say,byaddingarandomsmallvectorto b)isnon-degeneratewithprobabilityone.Andsecond,simplex-likealgorithms cantypicallybemodifiedtoessentiallyperform(deterministicversionsof)this perturbationon-line,thusdealingwithdegeneracy.

Wenextdefineaclassofalgorithmsforlinearprogrammingthatarevariants ofthesimplexmethod,whatwecall pivotingrules.Tostart,werecallfromlinear programmingtheorythreeimportantkindsofbases B ,called terminalbases: – optimality: B 1 b ≥ 0,cT cT B B 1 A ≤ 0. B istheoptimalfeasiblebasisof thelinearprogram. – unboundedness: B 1 Aj ≤ 0,cj cT B B 1 Aj > 0forsomecolumn Aj of A Thisimpliesthatthelinearprogramisunboundediffeasible. – infeasibility:(B 1 )i A ≥ 0, (B 1 )i b< 0forsomerow(B 1 )i of B 1 .This meansthelinearprogramisinfeasible.

Noticethat,givenabasis,itcanbedecidedinstronglypolynomialtimewhether itisterminal(andofwhichkind).

Definition1. A pivotingrule R isastronglypolynomialalgorithmwhich,given alinearprogram (A,b,c): – producesaninitialbasis B0 ; – giveninadditionabasis B thatisnotterminal,itproducesanadjacentbasis nR (B ) suchthat φR (nR (B )) >φR (B ),where φR isapotentialfunction.

The path ofpivotingrule R forthelinearprogram (A,b,c) isthesequenceof bases (B0 ,nR (B0 ),n2 R (B0 ),...,,nk R (B0 )),endingataterminalbasis,produced by R

Obviously,anypivotingruleconstitutesacorrectalgorithmforlinearprogramming,sinceitwillterminate(bymonotonicityandfiniteness),andcanonly terminateataterminalbasis.Noticethatpivotingrulesmaypassthroughinfeasiblebasicsolutions(forexample,theycanstartwithone).Incidentally,the inclusionofinfeasiblebasesimpliesthatsuchrulesoperatenotonthelinear program’spolytope,butonits lineararrangement.Sincethelatterhasdiameter

O (mn),eventheexistenceofpolytopeswithsuper-polynomialdiameterwillnot ruleoutstronglypolynomialpivotingrules.

Therearemanyknowndeterministicpivotingrules(tiesarebrokenlexicographically,say):

1. Dantzig’srule(steepestdescent). Inthisrule(aswellasinallother primalrulesthatfollow),givenafeasiblebasis B wefirstcalculate,foreach index j notinthebasistheobjectiveincreasegradient cB j = cj cT B B 1 Aj

Define J (B )= {j : cB j > 0}.Dantzig’sruleselectsthe j ∈ J (B )withlargest cB j andbringsitinthebasis.Bynon-degeneracy(ifnotaterminalbasis), thiscompletelydeterminesthenextbasis.Aswithallprimalpivotingrules, thepotentialfunction φR istheobjective.

2. Steepestedgerule. Insteadofthemaximum cB j ,selectthelargest c B j ||B 1 Aj || .

3. Greatestimprovementrule. Webringintheindexthatresultsinthe largestincrementoftheobjective.

4. Bland’srule. Selectthesmallest j ∈ J (B ).

Foralltheserules,however,wehavenotspecifiedtheoriginalbasis B0 .This isobviouslyaproblem,sincealltheserulesareprimalandneedfeasible bases,andafeasiblebasismaynotbeaprioriavailable.Primalpivoting rulessuchasthesearebestappliednotontheoriginal m × n linearprogram (A,b,c),buttoasimple m × 2n variantcalled“thebig M version,”defined as(A|− A),b, (c|− M,..., M ),where M isalargenumber(M caneitherbe handledsymbolically,orbegivenanappropriatevaluecomputedinstrongly polynomialtime).Itistrivialnowtofindaninitialfeasiblebasis.Infact, thepivotingrulerunningonthenewlinearprogramcanbethoughtofasa slightlymodifiedpivotingruleactingontheoriginallinearprogram(when j ∈ J (B ), Aj isnegated,and cj isreplacedby M ).

5. Shadowvertexrule. Here B0 isanybasis.Given B0 ,weconstructtwo vectors c0 and b0 suchthat B0 isafeasiblebasis,andalsoadualfeasible basis,oftherelaxedlinearprogrammax cT 0 x subjectto Ax = b0 ,x ≥ 0.Now considerthelinesegmentbetweenthesetwolinearprograms,withright-hand sideandobjective λb +(1 λ)b0 and λc +(1 λ)c0 ,respectively.Moving onthislinesegmentfrom λ =0,wehavebothprimal-feasibleanddualfeasible(andhenceoptimal)solutions.Atsomepoint,oneofthetwowill becomeinfeasible(andonlyone,bynon-degeneracy).Wefindanewbasic solutionbyexchangingvariablesasdictatedbytheviolation,andcontinue. Thepotentialfunctionisthecurrent λ.When λ =1weareattheoptimum.

6. Criss-crossrules. Aclassofpivotingrulesoutsideourframework,whose firstvariantappearedin[35],goesfromone(possiblyinfeasible)basistothe otherandconvergencetoaterminalbasisisprovedthroughacombinatorial argumentthatdoesnotinvolveanexplicitpotentialfunction.However,certainsuchrules(suchasthecriss-crosspivotingrulesuggestedin[32])have beenshown([12])topossessamonotonepotentialfunction,andsotheycan beexpressedwithinourframework.

7. Dualpivotingrules. Naturally,anyoftheprimalpivotingrulescanwork onthedual.

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“You should, indeed,” struck in Olivia, with great animation. “You can’t imagine how nervous I feel. You see, you are to be the mouthpiece of all of us. If you don’t do your best, and show that we have some patriotism, as well as the North, I believe there will be a general collapse among all the Southern people here.”

Pembroke could not help laughing.

“Your anxiety, Colonel, and Miss Berkeley’s doesn’t bespeak great confidence in me.”

Olivia blushed and protested more earnestly.

“Not so, not so, sir,” cried the Colonel. “We have every confidence in you, but my boy, you had better take a look at Cicero’s orations against Catiline—and read over to-night Sheridan’s speeches—and Hayne against Webster.”

Pembroke threw himself back in his chair, and his laugh was so boyish and hearty, that Olivia was startled into joining in it.

“This is fearful,” said Olivia, bringing her pretty brows together sternly. “This is unpardonable levity. At a time like this, it is dreadful for us to stand so in awe of your self-love. Really now, we know that you are eloquence and cleverness itself, but it isn’t safe,” she continued, with an air of infinite experience, “to trust anything to chance.”

“Come down to the House to-morrow and encourage me,” replied Pembroke good humoredly, “and keep up Miles’ spirits when I begin to flounder.”

The evening was very jolly, like those old ones in Paris and in Virginia. Pembroke at last rose to go, and in parting, the Colonel clapped him on the back, while Olivia held his hand and pressed it so warmly that Pembroke’s dark face colored with pleasure, as she said:

“Now, I know I am offending you—but you can’t imagine how frightened I am. You may come out all right—but the suspense will be dreadful—” She was laughing, too, but Pembroke saw under her badinage a powerful interest in his success. He went away elated.

“At least she will see that I was worthy of more consideration than she gave me,” he thought—a common reflection to men who have been refused.

Next day the floor of the House was crowded and the galleries packed. Administration and anti-administration people were interested. Society turned out in force to hear the revelations about the late Russian Minister—the private and diplomatic galleries were filled. The Senate was not in session, and many Senators were on the floor.

After the morning hour, and the droning through of some unimportant business, the leader of the majority rose, and demanded the consideration of the resolution of inquiry relating to the recall of the Russian Minister from this country. At that a hush fell upon the crowd. The leader of the opposition rose to reply. He stated briefly that it was a matter concerning the Foreign Affairs Committee, and a member of one of the sub-committees had sole charge of it owing to the illness of the chairman. Another member then rose, and sarcastically referring to the fact that the gentleman referred to could scarcely be supposed to entertain friendly feelings toward the representative of the only foreign government which showed the slightest sympathy toward the Union in the Civil War, demanded to know by what right had the Russian Minister’s position in Washington been made untenable—and that too, at the time of the visit of a member of the Czar’s family—and was this the return the United States Government made for the Czar’s extreme friendliness? Then Pembroke stood up in his place, at a considerable distance from the Speaker. This gave him a great advantage, for it showed the fine resonant quality of his voice, clear and quite free from rant and harshness. Olivia Berkeley, who watched him from the front row in the gallery, saw that he was pale, but perfectly self-possessed. As he caught her eye, in rising, he smiled at her.

“Mr. Speaker.”

The Speaker fixed his piercing eyes upon him, and with a light tap of the gavel, said “The gentleman from Virginia has the floor.”

Pembroke used no notes. He began in a clear and dignified manner to recite the part taken by him in Volkonsky’s case—his suspicions, his demand for documents from the State Department, Volkonsky’s compromising letters, of which he read copies—the dilemma of the Department, anxious not to offend Russia but indignant at the baseness of Volkonsky—the further complication of the Grand Duke’s visit, and all which followed. He then read his statement of what had occurred at his interviews with Volkonsky, and which he had filed at the State Department.

“And here let me say,” he remarked, pausing from the reading of his minutes of his last conversation with Volkonsky, “that in some of my language and stipulations I had no authority from either the President or Secretary of State—but with the impetuosity of all honest men, I felt a profound indignation at a man of the late Minister’s character, daring to present himself as an accredited agent to this Government. In many of these instances, as for example, when I stipulated that the late Minister should not presume to shake hands with the President at his parting interview, or address him in any way, no doubt the late Minister supposed that I was instructed to make that stipulation. Sir, I was not. It was an outburst of feeling. I felt so clearly that no man of Volkonsky’s character should be permitted to touch the hand of the President of the United States, that I said so—and said so in such a way that the late Minister supposed I had the President’s authority for it.”

At this, there was an outburst of applause. The Speaker made no move to check it. Pembroke bowed slightly, and resumed in his calm and piercing voice.

Members of the House and Senate had settled themselves to hear a speech. In five minutes the old stagers had found out that there was the making of a great parliamentary speaker in this stalwart dark young man. Members leaned back and touched each other. Pens refrained from scratching. The pages, finding nothing to do, crept toward the Speaker’s desk and sat down on the carpeted steps. One little black-eyed fellow fixed his gaze on Pembroke’s face, and at the next point he made, the page, without waiting for his elders, suddenly clapped furiously. A roar of laughter and applause

followed. Pembroke smiled, and did not break silence again until the Speaker gave him a slight inclination of the head. In that pause he had glanced at Olivia in the gallery. Her face was crimson with pride and pleasure.

Outside in the corridors, the word had gone round that there was something worth listening to going on inside. The aisles became packed. A slight disturbance behind him showed Pembroke that a contingent of women was being admitted to the floor—and before him, in the reporters’ gallery, where men were usually moving to and fro, every man was at his post, and there was no passing in and out.

Pembroke began to feel a sense of triumph. His easy, but forcible delivery was not far from eloquence. He felt the pulse of his audience, as it were. At first, when he began, it was entirely cold and critical, while his blood leaped like fire through his veins, and it took all his will-power to maintain his appearance of coolness. But as his listeners warmed up, he cooled off. The more subtly he wrought them up, the more was he master of himself. His nerve did not once desert him.

Gradually he began to lead up to where he hoped to make his point—that, although of the party in opposition, he felt as deeply, and resented as instantly, any infringement of the dignity of the Government as any citizen of the republic—and that such was the feeling in his party. His own people saw his lead and applauded tremendously. Just then the Speaker’s gavel fell. Loud cries of “Go on! Go on! Give him half an hour more! Give him an hour!” rang out. Pembroke had ceased in the middle of a sentence, and had sat down.

“Is there objection to the gentleman from Virginia continuing?” asked the Speaker, in an animated voice. “The Chair hears none. The gentleman will proceed.”

The applause now turned into cheers and shouts. One very deaf old gentleman moved forward to Pembroke and, deliberately motioning a younger man out of his seat, quietly took possession of it, to the amusement of the House. The little page, who was evidently a pet of the old gentleman, stole up to him and managed to crowd in

the same chair Shouts of laughter followed this, followed by renewed applause for Pembroke, in which his opponents goodnaturedly joined. Then Pembroke felt that the time had come. He had the House with him.

He spoke for an hour. He merely took the Volkonsky incident for a text. He spoke of the regard for the common weal exhibited by his party, and he vigorously denounced his opponents for their attempt to make party capital out of that which was near and dear to all Americans. He spoke with temper and judgment, but his party realized that they had gained a powerful aid in their fight with the majority. At the last he artfully indulged in one burst of eloquence—in which he seemed carried away by his theme, but in which, like a genuine orator, he played upon his audience, and while they imagined that he had forgotten himself he was watching them. Truly they had forgotten everything but the ringing words of the speaker. He had touched the chord of true Americanism which sweeps away all parties, all prejudices. Then, amidst prolonged and vociferous cheering, he sat down. Senators and Representatives closed around him, congratulating him and shaking hands. The House was in no mood for anything after that, and a motion to adjourn was carried, nobody knew how. When at last, to escape being made to appear as if he remained to be congratulated, Pembroke was going toward the cloak room the Speaker passed near him and advanced and offered his hand. “Ah,” he cried, in his pleasant, jovial way, “right well have you acquitted yourself this day. You’ll find much better company on our side of the House, however, my young friend.”

“Thank you,” said Pembroke, smiling and bowing to the great man. “It’s not bad on my own side.”

The Speaker laughed and passed on.

Pembroke slipped out. It was a pleasant spring afternoon. The world took on for him a glorious hue just then, as it does to every man who finds his place in life, and that place an honorable one. But one thing was wanting—a tender heart to sympathize with him at that moment. Instead of turning toward his lodgings, he walked away into the country—away where he could see the blue line of the

Virginia hills. It gave him a kind of malicious satisfaction, and was yet pain to him, that Olivia would be expecting him, and that she should be disappointed. As the hero of the hour she would naturally want to greet him.

“Well,” he thought, as he struck out more vigorously still, “let us see if my lady will not peak and pine a little at being forgotten.” And yet her hurt gave him hurt, too. Love and perversity are natural allies.

It was quite dark when he returned to his lodgings. Miles was not there—gone to dinner with the Berkeleys.

About ten o’clock Miles turned up, the proudest younger brother in all America. He had all that he had heard to tell his brother. But presently he asked:

“Why didn’t you come to the Berkeleys’? The Colonel kept the carriage waiting at the Capitol for you. Olivia listened at dinner for your step, and jumped up once, thinking you had come.”

“I needed a walk in the country,” answered Pembroke, sententiously.

Miles sighed. A look came into his poor face that Pembroke had seen there before—a look that made the elder brother’s strong heart ache. Any disappointment to Olivia was a stab to this unfortunate young soul. Men, as nature made them, are not magnanimous in love. Only some frightful misfortune like this poor boy’s can make them so.

Presently Miles continued, hesitatingly:

“You must go to see her very early to-morrow. You know they return to Virginia early in the week.”

“I can’t go,” answered Pembroke, wounding himself, and the brother that he loved better than himself, in order to wound Olivia. “I must go to New York early to-morrow morning, on business. I was notified ten days ago.”

Miles said no more.

Early the next morning Pembroke was off, leaving a note for Olivia, which that young lady showed her father, and then, running up to her own room, tore into bits—and then she burst into tears. And yet it was a most kind, cordial, friendly note. When Pembroke returned, the Berkeleys had left town for the season.

CHAPTER XIX.

T quaint old house, and the straggling, half-kept grounds at Isleham were never lovelier than that spring. Sometimes the extreme quiet and repose had weighed upon Olivia’s spirits as it would upon any other young and vigorous nature. But now she had a good deal of a certain sort of excitement. She was country-bred, and naturally turned to the country for any home feeling she might have. The Colonel and Petrarch were a little bored at first. Both missed the social life at Washington. Pete had been a success in his own circle. His ruffled shirt-front, copied from his master’s, had won infinite respect among his own color. As for the natty white footmen and coachmen, their opinion and treatment, even their jeers, he regarded with lofty indifference, and classed them as among the poorest of poor white trash.

His religion, too, had struck terror to those of the Washington darkies to whom he had had a chance to expound it. His liberal promises of eternal damnation, “an’ sizzlin’ an’ fryin’ in perdition, wid de devil bastin’ ’em wid de own gravy,” had not lost force even through much repetition. “Ole marse,” Petrarch informed Olivia, “he cuss ’bout dem dam towns, an’ say he aint had nuttin’ fittin’ ter eat sence he lef’ Verginny. Ole marse, he jis’ maraudin’ an’ cussin’ ’cause he aint got nuttin’ ter do. I lay he gwi’ back naix’ year. Ef he does, I got some preachments ter make ter dem wuffless niggers d’yar, totin’ de sins ’roun’ like twuz’ gol’ an’ silver.”

It seemed as if Olivia were destined to suffer a good deal of secret mortification on Pembroke’s account. That last neglect of his had cut her to the soul. She had waked up to the fact, however, that Pembroke had taken his first rebuff in good earnest, and that nothing was left for her but that hollow pretense of friendship which men and women who have been, or have desired to be, more to each other, must affect. It was rather a painful and uncomfortable feeling to take around with her, when listening to Mrs. Peyton’s vigorous talk, or the

Rev Mr Cole’s harmless sermons, and still more harmless conversation. But it was there, and it was unconquerable, and she must simply adjust the burden that she might bear it.

The county was full of talk about Pembroke’s speech. The older people were sure that some information of his father’s great speeches in their court-house about 1849 must have reached Washington, and that Pembroke’s future was predicated upon them. Then there was a good deal in the newspapers about it. The Richmond papers printed the speech in full, together with a genealogical sketch of his family since the first Pembroke came over, with a grant of land from Charles the Second in his pocket. Likewise, Pembroke’s success was attributed almost wholly to his ancestry, and he himself was considered to have had a merely nominal share in it.

It was the long session of Congress, and there was no talk of Pembroke’s returning to the county. Whenever he did come, though, it was determined to give him a public dinner.

One afternoon in May, about the same time of year that Pembroke and Olivia had had their pointed conversation in the garden, Olivia was trimming her rose-bushes. She was a famous gardener, and a part of every morning and afternoon she might have been found looking after her shrubs and flowers. Sometimes, with a small garden hoe, she might have been seen hoeing vigorously, much to Petrarch’s disgust, who remonstrated vainly.

“Miss ’Livy, yo’ mar never did no sech a thing. When she want hoein’ done, she sen’ fur Susan’s Torm, an’ Simon Peter an’ Unc’ Silas’ Jake. She didn’t never demean herself wid no hoe in her han’.”

“But I haven’t got Susan’s Tom, nor Simon Peter nor Uncle Silas’ Jake. And besides, I am doing it because I like it.”

“Fur Gord A’mighty’s sake, Miss ’Livy, doan’ lemme hear dat none o’ de Berkeleys likes fur ter wuk. De Berkeleys allus wuz de gentlefolks o’ de county. Didn’t none on ’em like ter wuk. Ketch ole marse wukkin! Gord warn’t conjurin’ ’bout de fust families when He say, ‘By de sweat o’ de brow dey shall scuffle fer de vittals.’ He mos’

p’intedly warn’t studyin’ ’bout de Berkeleys, ’kase dey got dat high an’ mighty sperrit dey lay down an’ starve ’fo’ dey disqualify deyselfs by wukkin’.”

But Olivia stuck bravely to her plebeian amusement. On this particular afternoon she was not hoeing. She was merely snipping off straggling wisps from the great rose-trees—old-fashioned “maiden’s blush,” and damasks. She was thinking, as, indeed, she generally did when she found herself employed in that way, of Pembroke and that unlucky afternoon six years ago.

Before she knew it Pembroke was advancing up the garden walk. In a moment they were shaking hands with a great assumption of friendliness. Olivia could not but wonder if he remembered the similarity between that and just such another spring afternoon in the same place. Pembroke looked remarkably well and seemed in high spirits.

“The Colonel was out riding—and I did not need Pete’s directions to know that you were very likely pottering among your flowers at this time.”

“Pottering is such a senile kind of a word—you make me feel I am in my dotage. Doddering is the next step to pottering. And this, I remember, is the first chance I have had to congratulate you in person on your speech. Papa gives your father and your grandfather the whole credit. I asked him, however, when he wrote you to give my congratulations.”

“Which he did. It was a very cold and clammy way of felicitating a friend.”

Olivia said nothing, but she could not restrain an almost imperceptible lifting of the brows.

“The result of that speech has been,” continued Pembroke, after a little pause, “that I am in public life to stay as long as I can. That means that I shall never be a rich man. Honest men, in these times, don’t get rich on politics.”

A brilliant blush came into Olivia’s face at that. In the midst of suggestive circumstances Pembroke seemed determined to add

suggestive remarks.

“But I hardly think you could take that into consideration,” she answered, after a moment. “A man’s destiny is generally fixed by his talents. You will probably not make a great fortune, but you may make a great reputation—and to my way of thinking the great reputation is the more to be coveted.”

“Did you always think so?”

“Always.”

Then there came an awkward pause. Olivia was angry with him for asking the first question, but Pembroke seemed determined to pursue it.

“Even when I asked you to marry me on this very spot, six years ago? Then I understood that you could not marry a poor man.”

“Then,” said Olivia, calmly, and facing him, “you very much misunderstood me. I did think, as I think now, that poverty is a weight about the neck of a public man. But I can say truthfully, that it was your ability to cope with it, rather than mine, that I feared.”

“And it seems to me,” said Pembroke, calmly, “on looking back, that I was a little too aggressive—that I put rather a forced construction on what you said—and that I was very angry.”

“I was angry, too—and it has angered me every time I have thought of it in these six years, that I was made to appear mercenary, when I am far from it—that a mere want of tact and judgment should have marked me in your esteem—or anybody else’s, for that matter—as a perfectly cold and calculating woman.”

She was certainly very angry now.

“But if I was wrong,” said Pembroke, in a low, clear voice—for he used the resources of his delightful voice on poor Olivia as he had done on many men and some women before—“I have paid the price. The humiliation and the pangs of six years ago were much—and then, the feeling that, after all, there was but one woman in the world for me—ah, Olivia, sometimes I think you do not know how deep is

the hold you took upon me. You would have seen in all these years, that however I might try, I could not forget you.”

Olivia was not implacable.

When they came in the house, the Colonel was come, and in a gale of good humor. He had, however, great fault to find with Pembroke’s course. He was too conciliatory—too willing to forget the blood shed upon the battlefields of Virginia—and then and there they entered upon a political discussion which made the old-fashioned mirrors on the drawing-room wall ring again. The Colonel brought down his fist and raved. “By Jove, sir, this is intolerable. My black boy, Petrarch (Petrarch continued to be the Colonel’s boy), knows more about the subject than you do; and he’s the biggest fool I ever saw. I’ll be hanged, sir, if your statements are worth refuting.” Pembroke withstood the sortie gallantly, and at intervals charged the enemy in splendid style, reducing the Colonel to oaths and splutterings and despair.

Olivia sat in a low chair by the round mahogany table, on which the old-fashioned lamp burned softly, casting mellow lights and shades upon her graceful figure. Occasionally a faint smile played about her eyes—whereat Pembroke seemed to gain inspiration, and attacked the Colonel’s theories with renewed vigor.

Upon the Colonel’s invitation he remained all night—the common mode of social intercourse in Virginia. Next morning, the Colonel was ripe for argument. Pembroke, however, to his immense disgust, refused to enter the lists and spent the morning dawdling with Olivia in the garden. About noon, the Colonel, in a rage sent Petrarch after the renegades. Three times did he return without them. The fourth time Petrarch’s patience was exhausted.

“Marse French, fur de Lord’s sake come ter ole marse. He done got de sugar in de glasses, an’ de ice cracked up, an’ he fyarly stan’nin’ on he hade. He got out all dem ole yaller Richmun

Exameters, printed fo’ de wah, an’ he say he gwi’ bust yo’ argifyins’ all ter pieces. He mighty obstroporous, an’ you better come along.”

To this pathetic appeal Pembroke at last responded. Olivia, with downcast face, walked by his side. The Colonel was very much worked up and “mighty discontemptuous,” as Petrarch expressed it.

“This is the third time, sir—” he began to roar.

“Never mind, Colonel,” replied Pembroke, laughing. “We will have a plenty of time to quarrel. Olivia has promised to marry me in the summer.”

“By Gad, sir—”

“Have a cigar. Now, where did we leave off last night? Oh, the Virginia Resolutions of 1798.”

D. APPLETON & CO.’S PUBLICATIONS.

BRER RABBIT PREACHES

ON THE PLANTATIO

N. By J C H, author of “Uncle Remus.” With 23 Illustrations by E. W. K, and Portrait of the Author. 12mo. Cloth, $1.50.

The most personal and in some respects the most important work which Mr. Harris has published since “Uncle Remus.” Many will read between the lines and see the autobiography of the author. In addition to the stirring incidents which appear in the story, the author presents a graphic picture of certain phases of Southern life which have not appeared in his books before. There are also new examples of the folk-lore of the negroes, which became classic when presented to the public in the pages of “Uncle Remus.”

“The book is in the characteristic vein which has made the author so famous and popular as an interpreter of plantation character.”—Rochester Union and Advertiser.

“Those who never tire of Uncle Remus and his stories—with whom we would be accounted—will delight in Joe Maxwell and his exploits.”—London Saturday Review.

“Altogether a most charming book.”— Chicago Times.

“Really a valuable, if modest, contribution to the history of the civil war within the Confederate lines, particularly on the eve of the catastrophe. While Mr. Harris, in his preface, professes to have lost the power to distinguish between what is true and what is imaginative in his episodical narrative, the reader readily finds the clew. Two or three new animal fables are introduced with effect; but the history of the plantation, the printing-office, the black runaways, and white deserters, of whom the impending break-up made the community tolerant, the coon and fox hunting, forms the serious purpose of the book, and holds the reader’s interest from beginning to end. Like ‘Daddy Jake,’ this is a good anti slavery tract in disguise, and does credit to Mr. Harris’s humanity. There are amusing illustrations by E. W Kemble.”—New York Evening Post.

“A charming little book, tastefully gotten up.... Its simplicity, humor, and individuality would be very welcome to any one who was weary of the pretentiousness and the dull obviousness of the average threevolume novel.”—London Chronicle.

“The mirage of war vanishes and reappears like an ominous shadow on the horizon, but the stay-at-home whites of the Southern Confederacy were likewise threatened by fears of a servile insurrection. This dark dread exerts its influence on a narration which is otherwise cheery with boyhood’s fortunate freedom from anxiety, and sublime disregard for what the morrow may bring forth. The simple chronicle of old times ‘on the plantation’ concludes all too soon; the fire burns low and the tale is ended just as the reader becomes acclimated to the mid-Georgian village, and feels thoroughly at home with Joe and Mink. The ‘Owl and the Birds,’ ‘Old Zip Coon,’ the ‘Big Injun and the Buzzard,’ are joyous echoes of the plantation-lore that first delighted us in ‘Uncle Remus.’ Kemble’s illustrations, evidently studied from life, are interspersed in these pages of a book of consummate charm.”—Philadelphia Ledger.

FROM DUSK TO DAWN. By K P W, author of “Metzerott, Shoemaker.” 12mo. Cloth, $1.25.

This book is an original one, like its predecessor, in that it follows none of the beaten paths of fiction, and it raises questions of vital interest, and addresses itself to the reader’s thought instead of merely tickling his fancy. The influence of

one human being over another is a subject of curious analysis, as well as the relation of the individual to the community, a subject, with its varied amplifications, which is of the first moment to-day. There is a story, a romance, which will interest novelreaders, but the book will hold the attention of those for whom the average novel has little charm.

GOD’S FOOL. A Koopstad Story. By M M, author of “Joost Avelingh.” 12mo. Cloth, $1.25.

In the opinion of competent critics this new novel by Maarten Maartens represents the finest development thus far of the author’s powers, and its appearance in book form promises to cause what is termed in popular parlance the “literary sensation” of the season. At least, there can be no question regarding the high appreciation of Maarten Maartens’s work by American and English readers.

“Maarten Maartens is a capital storyteller.”—Pall Mall Gazette.

“Maarten Maartens is a man who, in addition to mere talent, has in him a vein of genuine genius.”—London Academy.

CAPT’N DAVY’S HONEYMOON. A Manx Yarn. By H C, author of

“The Deemster,” “The Scape-Goat,” etc. 12mo. Cloth, $1.00.

“A new departure by this author Unlike his previous works, this little tale is almost wholly humorous, with, however, a current of pathos underneath. It is not always that an author can succeed equally well in tragedy and in comedy, but it looks as though Mr. Hall Caine would be one of the exceptions.”— London Literary World.

“Constructed with great ingenuity. The story is full of delight.”—Boston Advertiser.

“A rollicking story of Manx life, well told.... Mr. Caine has really written no book superior in character-drawing and dramatic force to this little comedy.”— Boston Beacon.

FOOTSTEPS OF FATE. By L C, author of “Eline Vere.”

Translated from the Dutch by Clara Bell. With an Introduction by Edmund Gosse. Holland Fiction Series. 12mo. Cloth, $1.00.

“It is a very remarkable book, and can not fail to make a profound impression by its strength and originality.... Its interest is intense, and the tragedy with which it closes is depicted with remarkable grace and passion.”—Boston Saturday Evening Gazette.

“The dramatic development up to a tragical climax is in the manner of a true artist.”—Philadelphia Bulletin.

THE FAITH DOCTOR. By E

E, author of “The Hoosier Schoolmaster,” “The Circuit Rider,” etc., 12mo. Cloth, $1.50.

“One of the novels of the decade.”— Rochester Union and Advertiser.

“It is extremely fortunate that the fine subject indicated in the title should have fallen into such competent hands.”— Pittsburgh Chronicle-Telegraph.

“The author of ‘The Hoosier Schoolmaster’ has enhanced his reputation by this beautiful and touching study of the character of a girl to love whom proved a liberal education to both of her admirers.”—London Athenæum.

“‘The Faith Doctor’ is worth reading for its style, its wit, and its humor, and not less, we may add, for its pathos.”— London Spectator.

“Much skill is shown by the author in making these ‘fads’ the basis of a novel of great interest.... One who tries to keep in the current of good novel-reading must certainly find time to read ‘The Faith Doctor.’”—Buffalo Commercial.

AN UTTER FAILURE.

By M C H, author of “Rutledge.”

12mo. Cloth, $1.25.

“A story with an elaborate plot, worked out with great cleverness and with the skill of an experienced artist in fiction. The interest is strong and at times very dramatic.... Those who were attracted by ‘Rutledge’ will give hearty welcome to this story, and find it fully as enjoyable as that once immensely popular novel.”—Boston Saturday Evening Gazette.

“In this new story the author has done some of the best work that she has ever given to the public, and it will easily class among the most meritorious and most original novels of the year.”—Boston Home Journal.

“The author of ‘Rutledge’ does not often send out a new volume, but when she does it is always a literary event.... Her previous books were sketchy and slight when compared with the finished and trained power evidenced in ‘An Utter Failure.’”—New Haven Palladium.

APURITAN PAGAN. By J

G, author of “A Diplomat’s Diary,” etc. 12mo. Cloth, $1.00.

“Mrs. Van Rensselaer Cruger grows stronger as she writes.... The lines in her story are boldly and vigorously etched.”—New York Times.

“The author’s recent books have made for her a secure place in current