JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.1(1-12)Journalof - PDF document

JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.1(1-12)JournalofCombinatorialTheory,SeriesB ContentslistsavailableatScienceDirect ThedisjointpathsprobleminquadratictimeKen-ichiKawarabayashi,YusukeKobayashi,BruceReedNationalInstituteofInformatics,2-1-2,Hitotsubashi,Chiyoda-ku,Tokyo,JapanDepartmentofMathematicalInformatics,GraduateSchoolofInformationScienceandTechnology,UniversityofTokyo,Tokyo113-8656,CanadaResearchChairinGraphTheory,McGillUniversity,Montreal,CanadaProjectMascotte,INRIA,LaboratoireI3S,CNRS,Sophia-Antipolis,France articleinfoabstract Articlehistory:Received6July2009Availableonlinexxxx Keywords:DisjointpathsQuadratictimeGraphminorTree-width Weconsiderthefollowingwell-knownproblem,whichiscalleddisjointpathsproblem.Foragivengraphandasetofofterminalsin,theobjectiveisto“ndvertex-disjointpathsconnectinggivenpairsofterminalsortoconcludethatsuchpathsdonotexist.Wepresentantimealgorithmforthisproblemfor“xed.ThisimprovesthetimecomplexityoftheseminalresultbyRobertsonandSeymour,whogaveantimealgorithmforthedisjointpathsproblemfor“xed.NotethatPerkovicandReed(2000)announcedin[24](withoutproofs)thatthisproblemcanbesolvedintime.Ouralgorithmimpliesthatthereisantimealgorithmfortheedge-disjointpathsproblem,theminorcontainmentproblem,andthelabeledminorcontainmentproblem.Infact,thetimecomplexityofallthealgorithmswiththemostexpensivepartdependingonRobertsonandSeymoursalgorithmcanbeimprovedto,forexample,themembershiptestingforminor-closedclassofgraphs.2011ElsevierInc.Allrightsreserved. 1.Introduction1.1.BackgroundofthedisjointpathsproblemInthevertex-(edge-)disjointpathsproblem,wearegivenagraphandasetofpairsofvertices),...,((whicharesometimescalledterminals),andwehavetodecidewhetherornotvertex-(edge-)disjointpathssuchthatfor E-mailaddresses:k_keniti@nii.ac.jp(K.Kawarabayashi),kobayashi@mist.i.u-tokyo.ac.jp(Y.Kobayashi),breed@cs.mcgill.ca(B.Reed).0095-8956/$…seefrontmatter2011ElsevierInc.Allrightsreserved.doi:10.1016/j.jctb.2011.07.004 JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.2(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesBFurthermore,we“ndsuchpathsiftheyexist.Thisiscertainlyacentralprobleminalgorithmicgraphtheoryandcombinatorialoptimization.Seethesurveys[9,32].Ithasattractedattentioninthecon-textsoftransportationnetworks,VLSIlayoutandvirtualcircuitroutinginhigh-speednetworksorInternet.AbasictechnicalproblemhereistointerconnectcertainprescribedchannelsŽonthechipsuchthatwiresbelongingtodifferentpinsdonottoucheachother.Inthissimplestform,theprob-lemmathematicallyamountsto“ndingdisjointtreesinagraphordisjointpathsinagraph,eachconnectingagivensetofvertices.Letusgivepreviousknownresultsonthedisjointpathsproblem.Ifisapartoftheinputoftheproblem,thenthisisoneofKarpsoriginalNP-completeproblems[12],anditremainsNP-completeevenifisconstrainedtobeplanar(Lynch[21]).TheseminalworkofRobertsonandSeymoursaysthatthereisapolynomialtimealgorithm(actuallyantimealgorithm)forthedisjointpathsproblemwhenthenumberofterminals,,is“xed(inthispaper,weshallrefertothisproblemasdisjointpathsproblemŽ).Actually,thisalgorithmisoneofthespin-offsoftheirgroundbreakingworkonthegraphminorprojectspanning23papers,andgivingseveraldeepandprofoundresultsandtechniquesindiscretemathematics.Thedisjointpathsproblemisaspecialcaseof“ndingamulti-commodity”ow.Inthemulti-commodity”owquestion,thecommoditiesatthesourcesaredifferentandthedemandateachisforaspeci“ccommodity.Thisisthetypeofquestionweneedtoresolvewhensend-inginformationthroughtheinformationhighwaynetworkandsohasbecomeincreasinglyofinteresttocomputerscientists(see,forexampletheworkofChekurietal.[3…6]andofTardosandKlein-berg[15…18]).Onespecialcasewhichisofgreatinterestisthatalldemandsareatmost12.Thisproblemsettingbehavesverydifferentfromthedisjointpathsproblem.Indeedtherearemanysuch”owtypeproblemsforwhichthehalfintegralversioncanbeatleastapproximatelysolvedalthoughtheintegralversionisintractable(see[20,22]).Asimilarsituationholdswithrespecttothepathproblem.TheproofofcorrectnessofRobertsonandSeymoursalgorithmrequiresalmostallofthegraphminorsprojectpapersandmorethan500pages.Ontheotherhand,KawarabayashiandReed[13]gaveanearlylineartimealgorithmforthehalfintegralversion,whichimprovesthepre-viousknownresultbyKleinberg[16]whogaveantimealgorithm.Inaddition,thecorrectnessofthisalgorithmismuchsimplerthanthatofRobertsonandSeymours.1.2.MotivationandmainresultsOurmotivationisthatitseemsthatthetimecomplexityofRobertsonandSeymoursalgo-rithmistooexpensive.Reedannouncedin[27](seealso[24])thatheprovedthatthisproblemcanbesolvedintime.Howeveradetaileddescriptionofthealgorithmwasnotfullywrittendown.Wenowcometoknowtanglesandbrambles,andtheircorrespondinggridminorsmoreclosely,sowethinkthatweshouldbeabletoimprovethetimecomplexity.In[13],twoofusgaveanearlyŽlineartimealgorithmforthehalfintegralversionofthedisjointpathsproblem.Namely,thetimecomplexityis.ThisgreatlyimprovestheresultbyKleinberg[16].Wenowtrytoim-provethetimecomplexityofRobertsonandSeymoursalgorithm.Letusremarkthattheresultsin[26,29]showthatthereisalineartimealgorithmforthedisjointpathsproblemwhenaninputgraphisplanar.Also,thereisalineartimealgorithmforthedisjointpathsproblemwhenanin-putgraphisaboundedgenusgraph[8,19].So,itwouldbeconceivablethatthetimecomplexityofRobertsonandSeymoursalgorithmcanbeimprovedtolinearornearlylinear.However,therearealotoftechnicaldiculties;aswementioned,thehalfintegralversionofthedisjointpathsproblemismucheasier.Inthispaper,wemanagedtoprovethefollowingtheorem,whichimprovesthetimecomplexityofRobertsonandSeymoursalgorithmtoquadratic.Theorem1.1.ThereisanOtimealgorithmforthekdisjointpathsproblemfor“xedk. Recently,amuchshorterproofforthecorrectnessofthegraphminoralgorithmisobtainedin[14].Theproofhingesuponasigni“cantlyshorterproofoftheuniquelinkagetheoremŽ[35]. JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.3(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesB1.3.OverviewOuralgorithmfollowsRobertson…Seymoursalgorithm[33].So,letus“rstsketchtheRobertson…Seymoursalgorithmonthedisjointpathsproblem.Atahighlevel,itisbasedonthefollowingtwocases:eitheragivengraphhasboundedtree-widthorelseithaslargetree-width.Inthe“rstcase,onecanapplydynamicprogrammingtoatree-decompositionofboundedtree-width,see[1,2,33].Thesecondcaseagainbreaksintotwocasesdependingonwhetherhasalargecliqueminorornot.Supposethathasalargecliqueminor.Ifwecanlinkuptheterminalstotheminor,thenwecanusethiscliqueminortolinkuptheterminalsinanydesiredway.Otherwise,thereisasmallseparationsuchthatthelargecliqueminoriscutofffromtheterminalsbythisseparation.Inthiscase,wecanprovethatthereisavertexinthecliqueminorwhichisirrelevant,i.e.,thedisjointpathsproblemisfeasibleinifandonlyifitisinSo,supposedoesnothavealargecliqueminor.Thenonecanprovethat,afterdeletingboundednumberofvertices,thereisalargewallwhichisessentiallyplanar.Thismakesitpossibletoprovethatthemiddlevertexofthiswallisirrelevant.Thisrequiresthewholegraphminorpapers,andthestructuretheoremofgraphminors[34].RobertsonandSeymourcouldonlygiveantimealgorithmto“ndsuchanirrelevantver-tex.Thenthealgorithmrecursesinthegraph.Thusifwecouldgiveanalgorithmto“ndsuchanirrelevantvertex,wecouldproveTheorem1.1.Thisisourmaintask.Weneedtoconsiderthetwocases,namely,agraphwithorwithoutalargecliqueminor.Inbothcases,weneedto“ndanirrelevantvertexintime.ThiswillbeprovedinSections4and5.Thetechnicaldicultieswehavetoovercomearethefollowingtwopoints:1.Whenthereisalargecliqueminor,wehaveto“ndanirrelevantvertexintime,ratherthantime.Thismeansthatwecannotusethestandard”owalgorithm(sincetheinputgraphcouldhaveedges).WeshallusethealgorithmbyNagamochiandIbaraki[23]to“ndsuchavertex.Roughly,atthebeginningofouralgorithm,wehavetoconstructaspannerŽoftheinputgraphwithatmostedges,whichmaintainsconnectivitybetweeneachvertexinandtheterminalsin.Thisallowsusto“ndasmallseparationintime,andhenceweareableto“ndanirrelevantvertexinthelargecliqueminorinInaddition,afterdeletingtheirrelevantvertex,wehavetoupdatethisspannerŽin2.Whenthereisnolargecliqueminor,intime,wehaveto“ndalargewallwhichisclosesttoaleafŽintheseminalgraphminordecompositiontheorem.Thenwehaveto“ndanearlyŽplanarembeddinginducedbythiswallintime.Notethatatthemoment,thereareedgesin2.PreliminaryInthispaper,alwaysmeanthenumberofverticesofagivengraphandthenumberofedgesofagivengraph,respectively.SometimeswesaydisjointpathsŽ,whichmeanvertex-disjointpathsŽ.Letbeagraph.Forasubgraph,thevertexsetandtheedgesetofaredenotedby,respectively.Aseparationisapairofedge-disjointsubgraphssuchthat.Theorderoftheseparation.For,letdenotethesetofverticesinthatareadjacentto,andlettX]bethesubgraphinducedby.Wenowlookatde“nitionsofthetree-widthandwalls.2.1.Tree-widthTree-widthwasintroducedbyHalinin[10],butitwentunnoticeduntilitwasrediscoveredbyRobertsonandSeymour[30]and,independently,byArnborgandProskurowski[1].tree-decompositionofagraphconsistsofatreeandasubtreeforeachvertexsuchthatifisanedgeofintersect.Foreachnodeofthetree,weletbethesetofverticessuchthat.Theofatree-decompositionisthemaximumof JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.4(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesB Fig.1.Anelementarywallofheight8. Fig.2.Awallofheight3.1overthenodes.Thetree-widthofagraphistheminimumwidthamongallpossibletree-decompositionsofthegraph.Wecanapplydynamicprogrammingtosolveproblemsongraphsofboundedtree-width,inthesamewaythatweapplyittotrees(seee.g.[1]),providedthatwearegivenaboundedwidthtree-decomposition.RobertsonandSeymourdevelopedthe“rstpolynomialtimealgorithmforconstructingatree-decompositionofagraphofboundedwidth[33],andeventuallycameupwithanalgorithmwhichrunsintime,forthisproblem.Reed[25]developedanalgorithmfortheproblemwhichrunstime,andthenBodlaender[2]developedalineartimealgorithm.Thisalgorithmwasfurtherimprovedin[24].Theorem2.1.(SeeBodlaender[2].)Forany“xedintegerw,thereexistsanOtimealgorithmthat,givenagraphG,either“ndsatree-decompositionofGofwidthworconcludesthatthetree-widthofGismorethanw.2.2.WallAnelementarywallofheighteightisdepictedinFig.1.Anelementarywallofheightforissimilar.Itconsistsoflevelseachcontainingbricks,whereabrickisacycleoflengthsix.wallofheightisobtainedfromanelementarywallofheightbysubdividingsomeoftheedges,i.e.replacingtheedgeswithinternallyvertex-disjointpathswiththesameendpoints(seeFig.2).Theofawallaretheverticesofdegreethreewithinit.Anywallhasauniqueplanarembedding.Theperimeterofawall,denotedperistheboundaryoftheuniquefaceinthisembeddingwhichcontains4Oneofthemostimportantresultsconcerningthetree-widthisthemainresultofGraphMinorsV[31]whichsaysthefollowing:Theorem2.2.(SeeRobertsonandSeymour[31].)Foranyr,thereexistsaconstantfsuchthatifGhastree-widthatleastf,thenGcontainsawallWofheightr.Thebestknownupperboundforisgivenin[7,27,36].Itis20.Thebestknownlowerboundis,see[36].Furthermore,suchawallcanbefoundinlineartime. JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.5(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesBTheorem2.3.(Followsfrom[2,24,31].)For“xedintegerr,inagraphGwithtree-widthatleastf,wecan“ndawallWofheightrinlineartime.Herewegiveanoutlineofthelineartimealgorithm.Bythealgorithmin[24],wecan“ndinlineartimeasubgraphoftree-widthatleastandatree-decompositionofofwidthatmost2.Then,sincehasawallofheightbyTheorem2.2,itcanbefoundinlineartimebythedynamicprogrammingmethod[2].3.Findinga”atlargewallin-minor-freegraphsInthissection,weapplyastructuralresultofRobertsonandSeymourconcerninggraphswhichhavealargetree-widthbutnolargecliqueminor.Tostatethisresultwewillneedafewde“nitions.Recallthattheofawallaretheverticesofdegreethreewithinit.Anywallhasauniqueplanarembedding.Foranywallinagivengraph,thereisauniquecomponentcontaining.Thecompass,denotedcomp,isthesubgraphofinducedbysubwallofawallisawallwhichisasubgraphof.Asubwallofproperifitconsistsofconsecutivebricksfromeachofconsecutiverowsof.Theexteriorofapropersubwallofawall.WesayapropersubwallifthecompassofisdisjointfromAwallisifitscompassdoesnotcontaintwovertex-disjointpathsconnectingthediago-nallyoppositecorners.Notethatifthecompassofhasaplanarembeddingwhosein“nitefaceisboundedbytheperimeterofisclearly”at.Inordertocharacterize”atwalls,weusetheresultofSeymour[37],Thomassen[41],andothersonthe2disjointpathsproblem.Wenowmentionthecharacterizationofa”atwall.Bythecharacterizationofgraphscontaining2disjointpaths(see[37,Theorem4.1]forexample),awallis”atifandonlyiftherearepairwisedisjointsetscompcontainingnocornersofsuchthat(1)for1(2)for13,and(3)ifisthegraphobtainedfromcompbydeletingandaddingnewedgesjoiningeverypairofdistinctverticesinforeach,thencanbedrawninaplanesothatallcornersareontheouterfaceboundary.Wecallwiththeseconditionsattachedvertexsets.Ifsuchexist,wesaythatcompcanbeembeddedintoaplaneupto3-separations,andanembeddingasin(3)iscalleda.Itiseasytoseethatanypropersubwallofa”atwallmustbeboth”atanddividing.Furthermore,ifaretwoverticesofa”atwallandthereisapathbetweenthemwhichisinternallydisjointfromtheneitherarebothonperorsomebrickcontainsbothofFinally,westatethemainresultinthissection.RobertsonandSeymour(Theorem(9.6)in[33])provedthefollowingalgorithmicresult:Theorem3.1.(SeeRobertsonandSeymour[33]andKapadiaetal.[11].)Forany“xedintegerst,thereisacomputableconstantfsuchthatthefollowingcanbedoneinOtime,wheremisthenumberofedgesofagivengraphG.InputAgraphG,awallHofheightatleastf-minor,orasubsetXoforderatmostandtdisjointpropersubwallsHofheighthsuchthateachHisdividingand”atinGXfori.Inaddition,a”atembeddingofHisalsogivenfor JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.6(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesBThealgorithmin[33]takestimebecauseRobertsonandSeymourusedantimeal-gorithmtotestthe”atness(i.e.,givea”atembeddinginthecompassofinthesecondconclusion),andthispartisthemostexpensive.Moreprecisely,RobertsonandSeymour“rstprovedthatthereistimealgorithmtogettheconclusionofTheorem3.1,butwithouttheconclusion”atŽ(thisstepcorrespondsto(9.4)in[33]thatrunsintime).Thenforeachofdividingsubwalls,testwhetherornotitis”at(thisstepcorrespondsto(8.1)in[33]thatrunsintime).Thereisnowtimealgorithmtotestthe”atnessbyKapadia,LiandReed[11],whichimprovestheprevi-ousbestknownresultbyTholey[39,40]whichgivesantimealgorithm,wheretheistheinverseoftheAckermannfunction(seetextbyTarjan[38]).Notethatthealgorithmsin[11,39,40]also“nda”atembeddingofawall(ifitis”at)inthesamerunningtime.Actually,in(nearly)lineartime,thesealgorithms“ndattachedvertexsetsandreduceagiveninstanceofthe2disjointpathsproblemtoaninstanceinagraphthathasnoseparationoforderatmostthreesuchthatcontainsalltheterminals(seeProperty(P)in[40]).Afterthisreductiona”atembeddingisequivalenttoaplanarembeddingwhichcanbefoundinlineartime.Thusifweusethealgorithmin[11]fortestingthe”atness,wecangetantimealgorithm,asclaimedinTheorem3.1.Notethat[11]isapaperundersubmission.Ifweusetheresultin[39,40]insteadofthatin[11],therunningtimeinTheorem3.1becomes,andconsequentlytherunningtimeinTheorem1.1becomes,whichisslightlylargerthanIneithercase,withthehelpofTheorem3.1,we“nddesireddisjointpathsoranirrelevantvertexintime.Wedescribetheprocedureinthenextsection.4.IrrelevantverticesLetusrecallthatavertexisirrelevantifthedisjointpathsproblemisfeasibleinifandonlyifitisin.Inthissection,wegivesometheoremsconcerningirrelevantvertices.4.1.WhenthegraphhasalargecliqueminorLetus“rstgivetheoremsconcerningagraphwithalargecliqueminor.Theorem4.1.(SeeRobertsonandSeymour[33,Theorem(5.4)].)LetsbeterminalsinagivengraphG.IfthereisacliqueminoroforderatleastkinG,andthereisnoseparationoforderatmostinGsuchthatAcontainsalltheterminalsandBAcontainsatleastonenodeofthecliqueminor,thentherearedisjointpathsPwithtwoendsinsforik.Furthermore,suchpathscanbefoundinOIfthereisaseparationoforderatmost21suchthatcontainsalltheterminalsandcontainsatleastonevertexofthecliqueminor,wecan“ndavertexofthecliqueminorwhichisirrelevantintime(seeargumentsin[33,Theorem(6.2)]).Inordertoimprovetherunningtimefrom,weneedtheresultbyNagamochiandIbaraki[23].Theygaveanalgorithmtoreducethenumberofedgesfromkeepingtheconnectivityofthegraph.Moreprecisely,foragraphandaninteger,theiralgorithm“ndsasubgraphsuchthat,wheredenotesthevertexconnectivitybetween.Suchagraphissaidtobeat-certi“cate.Thisproceduretakestime,soifwenewlyconstructa-certi“cateeachtimeafterdeletinganirrelevantnode,therunningtimebecomesexpensive.Inordertoavoidthis,weshowthefollowing:Theorem4.2.Letkbea“xedinteger.SupposewearegivenagraphG,terminalssacliqueminorKoforderk,andak-certi“cateGofGwith.Wecan“ndeitherdesiredkdisjointpathsPwithtwoendsinsforikinOtimeor JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.7(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesBagraphGandak-certi“cateGsuchthatcontainsallterminals,andthekdisjointpathsprobleminGisequivalenttotheoriginalprobleminProof.First,we“ndinasmallestseparationoforderatmost21suchthatcontainsalltheterminals,containsatleastonenodeofthecliqueminor.Notethatifthesmallestorderofsuchaseparationisatleast2,wecan“nddesireddisjointpathsinbyTheorem4.1,andtherunningtimeis.Ifthesmallestorderisatmost21,wecan“ndsuchaseparationintimebyusingasimple”owalgorithmin.Then,weobtainbydeletingallverticesandaddingnewedgesjoiningeverypairofdistinctverticesin.OnecanseethatthisproceduredoesnotaffectthesolutionbyTheorem4.1.Furthermore,byexecutingthesameprocedure,weobtainagraphforany.Sinceatmostedges,wecanapplythealgorithmofNagamochiandIbarakitotime.Thenweobtaina2-certi“cateInTheorem4.2,weassumedthatacliqueminoroforder3isgiven.Buthowdowe“ndsuchacliqueminorinlineartime?Hereisoneway.Theorem4.3.(SeeReedandWood[28,Theorem2.1].)Forany“xedintegert,ifGhasatleastedges,thenthereisanOtimealgorithmto“ndaK-minor.ThistheoremimprovesAlgorithm(6.4)in[33].Notethatalthoughthenumberoftheedgesinnotnecessarily,weonlyneedtofocuson2edgesto“nda-minor.Thus,therunningtimeisnot.OuralgorithmwillkeepapplyingTheorem4.3togetacliqueminoroforder3.AfterapplyingTheorem4.3,wemayassumethathasatmost2edges.Hence,inwhatfollowsinthissection,weassumethatanddescribehowwe“ndanirrelevantvertexinagraphwithno-minor.4.2.WhenthegraphhasnolargecliqueminorNext,wediscussthecasewhenthegraphhasnolargecliqueminor.Inordertostatethetheorem,weneedtode“nerealizablepartitions.Letbeagraphandbeavertexset.Apartitionrealizableiftherearedisjointtreessuchthatfor.RobertsonandSeymour[33]showedthefollowingtheorem.Theorem4.4.(SeeRobertsonandSeymour[33,Theorem(4.1)].)LetGbeagraphwithatree-decompositionofwidthatmostwfor“xedw,andletZbeavertexsetoforderatmostfor“xedk.ThenthereisantimealgorithmtoenumerateallrealizablepartitionsofZinGforsomefunctionfofkNotethattheproblemofenumeratingallrealizablepartitionscorrespondstotheproblemcalled-folioin[33].Itisgeneralizedtotheproblemcalledl-folioin[33],whichplaysacentralroleintheproofoftheseminalresultofRobertsonandSeymour.Wealsonotethatalthoughtheresultin[33]isstatedintermsofbranch-width,Žitdoesnotcauseanyproblemsbecausebranch-widthdiffersonlybyaconstantfactorfromtree-width.Nowwearereadytostatetheresultconcerningagraphwithoutalargecliqueminor.Theorem4.5.(SeeRobertsonandSeymour[33,Theorem(10.3)].)Forany“xedintegerk,thereisacomputableconstantfsatisfyingthefollowingifthereisasubsetXoforderatmostsuchthatthereisa”atwallWofheightfinGX,thenthereisavertexvinWsuchthatvisirrelevant.Furthermore,ifwehavea”atembeddingofcompwithattachedvertexsetsA JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.8(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesBallrealizablepartitionsofZinAforeachi,whereAcompandZcompthenwecan“ndanirrelevantvertexvinlineartime.WenowgivesomeremarksondifferencesbetweenthistheoremandtheoriginalstatementofTheorem(10.3)in[33].TheoriginalstatementofTheorem(10.3)in[33]isacombinationofTheorem4.5aboveandTheorem(10.1)in[33],andsothecondition(W2)doesnotappearintheoriginalstatementexplicitly.InTheorem(10.3)in[33],theyusearuraldivisioninsteadoftheattachedvertexsetsFor,thecorrespondingruraldivisioncanbeobtainedasfollows.Foreachpairofsetscompcomp,replacethembytheirunion.Fortheresultingsets,letbethesubgraphofinducedbyfor.Then,theruraldivisionconsistsofthegraphsandthegraphseachconsistingofasingleedge(togetherwithitsendvertices)notcontainedinoneofAlthoughtheoriginalstatementofTheorem(10.3)in[33]saysthatitrequiresquadratictimeintheabovestatement,thepreciserunningtimeislinear.Thisisbecausethenumberofiterations(denotedby)intheproofofTheorem(10.3)of[33]isboundedbyaconstantwhichonlydependsWealsonotethattheoriginalstatementrequiresaofeach,whichconsistsof(W2)andthefollowinginformation:(W3)theclockwiseorderingofinthe”atembeddingofcomp(W4)foreachvertexcomp,avertexthatcanbeanendpointofapathfromsuchthatdoesnothit,exceptforSincethe”atembeddingofcompgivesus(W3)and(W4)inlineartime,weomittheminthestatement.Intheoriginalstatementof[33],theyconsider”atwallsbutdonotusetheir”atembeddings,andsotheymentiontheinformation(W3)and(W4)explicitly.Theorem(10.1)in[33]showsthat(W2)canbecomputedinquadratictimeifallthecomponentsponentsA1],...,G[Al]havetree-widthboundedbya“xedconstant.Byimprovingtherunningtimeofthisstatementtolineartime,weshowthefollowingtheorem.Theorem4.6.LetGX,andWbeasinthe“rstpartofTheorem.Ifwehavea”atembeddingofcompwithattachedvertexsetsAsuchthatallthecomponentsGGA1],...,G[Al]havetree-widthboundedbya“xedconstantg,wecan“ndinOtimeanirrelevantvertexv.Proof.ByTheorem4.5,itsucestoshowthat(W2)canbecomputedinlineartimewhenallthecomponentsponentsA1],...,G[Al]havetree-widthboundedbya“xedconstant.Letbethecom-passofForeach,byTheorems2.1and4.4,intime,wecanenumerateallrealizablepartitions.Ontheotherhand,since3andThus,itcanbedoneintime,andTheorem4.6follows.5.KeytheoremsTheprevioustheorem,Theorem4.6,isourmaintool,buthowdowe“ndasubsetawallsuchthat JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.9(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesBisa”atwallofheight,and(C3)allthecomponentsponentsA1],...,G[Al]havetree-widthatmost,whereareattachedvertexsetsasinthe”atembeddingofcompinlineartime?RobertsonandSeymour[33]describedthefollowingalgorithm,whichtakesTheorem5.1.(SeeRobertsonandSeymour[33,Theorem(9.8)].)Forany“xedintegerk,therearecomputableconstantshandgsuchthat,ifagivengraphGhastree-widthatleasth,thenthereisanOalgorithmto“ndeitheraK-minororapairsatisfyingtheconditionsOurmaincontributioninthispaperisthefollowing:Theorem5.2.Forany“xedintegerk,therearecomputableconstantshandgsuchthat,ifagivengraphGhastree-widthatleasth,thenthereisanOtimealgorithmto“ndeitheraK-minororapairsatisfyingtheconditionsProof.Weset,whereisasinthecondition(C2),isasinTheorem2.3,isasinTheorem3.1.SettheconstantsuchthatForagivengraphwithtree-widthatleast,we“rstapplyTheorem4.3.Ifthereisaminor,wearedone.Thuswemayassumethathasatmost2edges.ByapplyingTheorem2.3,wenowhaveawallofheightinhand.WeapplyThe-orem3.1withforthewall.Sincehasatmostedges,thetimecomplexityofTheorem3.1isnow.Ifwegeta-minor,wearedone.Thusbythede“nitionof,wemayassumethatthereexistasetdividingand”atpropersubwallsofheight.Wetakesuchtwosubwalls,andletbevertexsetsasinthe”atembeddingofthecompassofthewallfor2.Foreachturn,wetestwhetherornothastree-widthatleast.ThiscanbedoneinlineartimebyTheorem2.1.IfallofofAi,1],G[Ai,2],...,G[Ai,li]havetree-widthatmost,thensatis“es(C1)…(C3).Otherwise,wemayassumethatthatAi,1]hastree-widthatleastfor2.Furthermore,sincearedisjoint,wemayassumealsothat2.Letbeatmostthreeverticesinthecompassthatareadjacenttoavertexin.Then,isacutsetinsuchthatthatA1,1]isoneofthecomponentsof.WerepeatthisalgorithmwiththeinputgraphaphA1,1XX].Weonlytakecareofhowwechoosetwosubwallsofheight.InthegraphaphA1,1XX],byapplyingTheorems2.3and3.1,wecan“nddividingand”atsubwallsofheight.Since2,thereexisttwodividingand”atsubwallsofheightthatdonotcontainanyvertexof.Thus,whenweexecutethesameprocedureforinthenextiteration,weonlyneedtolookatthegraphphA1,1].Alltheaboveoperationscanbedoneintime,asdescribed.Letustakeasthetimecomplexityintheaboveoperationswiththeinputgraphvertices.Whenwerepeatthealgo-rithm,wethrowawayhalfoftheverticesofthecurrentgraph.Thus,thetimecomplexitywillbe 2)+T(n +···,whichisstill.ThiscompletestheproofofTheorem5.2.6.AlgorithmLetbeapositiveinteger.Inthissection,wedescribeouralgorithmforthedisjointpathsproblem.AlgorithmforthedisjointpathsproblemInput:Agraphvertices,andterminals JID:YJCTBAID:2713/FLA[m1G;v1.58;Prn:27/07/2011;9:20]P.10(1-12)K.Kawarabayashietal./JournalofCombinatorialTheory,SeriesBdisjointpathssuchthatfor,iftheyexist.DescriptionStep1.Computea2-certi“catetimebythealgorithmof[23],andgotoStep2.Step2.Whilethegraphhasatleast2edges,wedetecta-minorbyTheorem4.3,andremovesomeverticesasinTheorem4.2.Ifthegraphhaslessthan2edges,thengotoStep3.Step3.Testwhetherornotacurrentgraphhastree-widthatmost,whereisasinTheo-rem5.2.Ifithas,thenTheorem2.1givesrisetoatree-decompositionofwidthatmost.WethenuseTheorem4.4tosolvetheproblem.Ifthegraphhastree-widthmorethan,gotoStep4.Step4.ApplyTheorem5.2tothegraph.Ifwegeta-minor,thenapplyTheorem4.2toconstructasmallerinstanceintime.IfwegetawallasinTheorem5.2,thenapplyTheorem4.6to“ndanirrelevantvertexintime.Then,deletetheirrelevantvertexandgotoStep3.First,thetotalrunningtimeofStep2isbyTheorem4.2.WhenwegotoStep3afterexecutingStep2,thenumberofedgesis,andhenceittakestimeinStep4to“ndoneirrelevantvertexinthegraph.Sincewedeleteatmostvertices,thetotalrunningtimeofStep4.ThiscompletestheproofofTheorem1.1.7.ConcludingremarksWeconcludethispaperbyobservingthatotherproblemscanalsobesolvedinquadratictime.Corollary7.1(Thekedge-disjointpathsproblem).ThereisanOtimealgorithmfortheedge-disjointpathsproblemwhenkisa“xedconstant,i.e.,fortheproblemofconstructingpathsPthatarenotnecessarilyvertex-disjointbutedge-disjoint.edge-disjointpathsproblemcanbereducedtothevertex-disjointpathsproblembycon-structingthelinegraph.However,sincethenumberofverticesofthelinegraphis,byusingthisnaivereduction,therunningtimebecomes.Inordertoimprovetherunningtimeto,we“rstreducethenumberofedgestobyexecutingthesameproceduresasTheorems4.2and4.3.Withthispreprocessing,wecanreducetheedge-disjointpathsproblemtothevertex-disjointpathsprobleminagraphwithvertices,whichyieldsCorollary7.1.ThesimilarprooftoTheorem1.1alsoworksforthefollowingproblems:Corollary7.2(Thelabeledminortesting).ThereisanOtimealgorithmforthelabeledminorcontainmentproblem.Thatis,givenagraphGandnon-nullsubsetsZ,withk,wherekisa“xedconstant,wecantestinOtimewhetherornotthereexistmutuallyvertex-disjointconnectedsubgraphsofG,withZCorollary7.3(Theminortesting).ThereisanOtimealgorithmfortheminorcontainmentproblem.Thatis,fora“xedgraphHandagivengraphG,wecantestwhetherornotGhasaminorisomorphictoHinOCorollaries7.2and7.3areobtainedbyfollowingtheproofofTheorem1.1inthesamewayasin[33].Wehavetousetheconceptof-folioŽinsteadofdisjointpathsŽandrealizablepartitions,Žbutthesameargumentalsoworks.Finally,wenotethatthetimecomplexityofallthealgorithmswiththemostexpensivepartde-pendingonRobertsonandSeymoursalgorithmcanalsobeimprovedto,forexample,themembershiptestingforaminor-closedclassofgraphs.Therearemany,soweomitthem. 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