Organization
ProgramCommittee
JemalAbawajy DeakinUniversity,Australia
HideoBannai KyushuUniversity,Japan
LjiljanaBrankovic UniversityofNewcastle,Australia
StoltingBrodal AarhusUniversity,Denmark
PinoCaballero-GilUniversityofLaLaguna,Spain
CharlieColbourn ArizonaStateUniversity,USA
MaximeCrochemoreKing’sCollegeLondon,UK
PinarDundar EgeUniversity,Turkey
JiriFiala CharlesUniversityinPrague,CzechRepublic
DaliborFroncek UniversityofMinnesotaDuluth,USA
RobertoGrossi Università diPisa,Italy
JanHolub CzechTechnicalUniversityinPrague,CzechRepublic CostasIliopoulos King’sCollegeLondon,UK
RalfKlasing LaBRIandCNRS,France
ChristianKomusiewiczTUBerlin,Germany
JanKratochvíl CharlesUniversityinPrague,CzechRepublic
DieterKratsch UniversitedeLorraine-Metz,France
GregoryKucherov UniversityParis-EstMarne-la-ValleeandCNRS, France
ThierryLecroq Université deRouen,France
ZsuzsannaLipták Università diVerona,Italy
PaulManuel KuwaitUniversity,Kuwait
MirkaMiller UniversityofNewcastle,Australia
IanMunro UniversityofWaterloo,Canada
KunsooPark SeoulNationalUniversity,SouthKorea
SolonPissis King’sCollegeLondon,UK
HebertPérez-Ros ésUniversityofLleida,Spain
SohelRahman BangladeshUniversityofEngineeringandTechnology, Bangladesh
VojtaRodl EmoryUniversity,USA
FrankRuskey UniversityofVictoria,Canada
BillSmyth McMasterUniversity,Canada
LynetteVanZijl StellenboschUniversity,SouthAfrica
ProgramCommitteeCo-chairs
JanKratochvíl CharlesUniversityinPrague,CzechRepublic MirkaMiller UniversityofNewcastle,Australia
ProblemSessionCo-chairs
UweLeck UniversityofWisconsin-Superior,USA ZsuzsannaLipták Università diVerona,Italy
ProceedingsTechnicalEditor
XiaofengGu UniversityofWisconsin-Superior,USA
SteeringCommittee
CostasIliopoulos King’sCollegeLondon,UK MirkaMiller UniversityofNewcastle,Australia BillSmyth McMasterUniversity,Canada
OrganizingCommittee
SergeiBezrukov UniversityofWisconsin-Superior,USA DaliborFroncek(Chair)UniversityofMinnesotaDuluth,USA XiaofengGu UniversityofWisconsin-Superior,USA StevenRosenberg UniversityofWisconsin-Superior,USA UweLeck UniversityofWisconsin-Superior,USA
AdditionalReviewers
Avrachenkov,Konstantin Baca,Martin Baisya,Dipankar Bari,Md.Faizul Bevern,René Van Boeckenhauer,Hans-Joachim Burcsi,Péter Caballero-Gil,Candido CaceresCruz,Jose Cechlarova,Katarina Chen,Jiehua Cheng,Eddie Cicalese,Ferdinando D’Arco,Paolo Dev,Himel
Dogan,Derya Escoffer,Bruno Feria-Puron,Ramiro Fernau,Henning Fertin,Guillaume Fici,Gabriele Firoz,Jesun Foucaud,Florent Gerbner,Daniel Grigorious,Cyriac Gu,Xiaofeng Harju,Tero Hartung,Sepp Hocquard,Hervé Hoksza,David
Hossain,Md.Iqbal Hsieh,Sun-Yuan Hüffner,Falk I.,Tomohiro Inenaga,Shunsuke Irvine,Veronika Islam,A.S.M.Sohidull Keil,Mark Lefebvre,Arnaud Li,Rao Liu,Daphne Lukovszki,Tamas Martin-Fernandez,Francisco Mary,Arnaud Medvedev,Paul Miltzow,Tillmann Molina-Gil,Jezabel Mondal,Debajyoti Nisse,Nicolas Oner,Tahsin Ordin,Burak Peterlongo,Pierre Phanalasy,Oudone Pineda-Villavicencio, Guillermo Prieur-Gaston,Elise
Rajan,Bharati Rios-Solis,Yasmin Rivals,Eric Rosenberg,Steve Ryan,Joe Ryjacek,Zdenek Saba,Sahand Salikhov,Kamil Scholtzova,Jirina Sebe,Francesc Sgall,Jiri Sgall,Jiří Shaw,DipanLal Sorge,Manuel Stephen,Sudeep Suchy,Ondrej Tantau,Till Tiwary,HansRaj Valla,Tomáš Valtr,Pavel Vialette,Stéphane Walen,Tomasz Weimann,Oren Xu,Min Zhu,Xuding Zohora,FatemaTuz
Contents
OntheComplexityofVariousParameterizationsofCommonInduced SubgraphIsomorphism......................................1 FaisalN.Abu-Khzam, ÉdouardBonnet,andFlorianSikora
ApproximationandHardnessResultsfortheMaximumEdges inTransitiveClosureProblem..................................13 AnnaAdamaszek,GuillaumeBlin,andAlexandruPopa
QuantifyingPrivacy:ANovelEntropy-BasedMeasureofDisclosureRisk...24 MousaAlfalaylehandLjiljanaBrankovic
OntheGaloisLatticeofBipartiteDistanceHereditaryGraphs...........37 NicolaApollonio,MassimilianoCaramia,andPaoloGiulioFranciosa
FastandSimpleComputationsUsingPrefixTablesUnderHamming andEditDistance..........................................49 CarlBarton,CostasS.Iliopoulos,SolonP.Pissis,andWilliamF.Smyth BorderCorrelations,Lattices,andtheSubgraphComponentPolynomial....62 FrancineBlanchet-Sadri,MichelleCordier,andRachelKirsch
ComputingMinimumLengthRepresentationsofSetsofWords ofUniformLength.........................................74 FrancineBlanchet-SadriandAndrewLohr
ComputingPrimitively-RootedSquaresandRunsinPartialWords........86 FrancineBlanchet-Sadri,JordanNikkel,J.D.Quigley,andXufanZhang
3-ColoringTriangle-FreePlanarGraphswithaPrecolored9-Cycle........98 IlkyooChoi,JanEkstein,PřemyslHolub,andBernardLidický
ComputingHeatKernelPagerankandaLocalClusteringAlgorithm.......110 FanChungandOliviaSimpson
A C-magicRectangleSetandGroupDistanceMagicLabeling...........122 SylwiaCichacz
SolvingMatchingProblemsEfficientlyinBipartiteGraphs.............128 SelmaDjelloul
A3-ApproximationAlgorithmforGuardingOrthogonalArtGalleries withSlidingCameras.......................................140 StephaneDurocherandSaeedMehrabi
OnDecomposingtheCompleteGraphintotheUnion ofTwoDisjointCycles......................................153
SaadI.El-Zanati,UthoompornJongthawonwuth,HeatherJordon, andCharlesVandenEynden
ReconfigurationofVertexCoversinaGraph.......................164 TakehiroIto,HiroyukiNooka,andXiaoZhou
SpaceEfficientDataStructuresforNearestLargerNeighbor............176 VarunkumarJayapaul,SeungbumJo,VenkateshRaman, andSrinivasaRaoSatti
PlayingSeveralVariantsofMastermindwithConstant-SizeMemory isnotHarderthanwithUnboundedMemory.......................188 GeroldJägerandMarcinPeczarski
OnMaximumCommonSubgraphProblemsinSeries-ParallelGraphs......200 NilsKriege,FlorianKurpicz,andPetraMutzel
Profile-BasedOptimalMatchingsintheStudent/ProjectAllocationProblem....213 AugustineKwanashie,RobertW.Irving,DavidF.Manlove, andColinT.S.Sng
TheMin-maxEdge q-ColoringProblem...........................226 TommiLarjomaaandAlexandruPopa
SpeedingupGraphAlgorithmsviaSwitchingClasses.................238
NathanLindzey
Studyof jðDÞ for D ¼f2; 3; x; yg ...............................250 DanielCollisterandDaphneDer-FenLiu
SomeHamiltonianPropertiesofOne-Con flictGraphs.................262 ChristianLaforestandBenjaminMomège
SequenceCoveringArraysandLinearExtensions....................274 PatrickC.MurrayandCharlesJ.Colbourn
Minimum r -StarCoverofClass-3OrthogonalPolygons...............286 LeonidasPaliosandPetrosTzimas
EmbeddingCirculantNetworksintoButterflyandBenesNetworks........298 R.SundaraRajan,IndraRajasingh,PaulManuel,T.M.Rajalaxmi, andN.Parthiban
KineticReverse k -NearestNeighborProblem.......................307 ZahedRahmati,ValerieKing,andSueWhitesides
EfficientlyListingBoundedLength st -Paths........................318
RomeoRizzi,GustavoSacomoto,andMarie-FranceSagot
MetricDimensionforAmalgamationsofGraphs.....................330
RinoviaSimanjuntak,SaladinUttunggadewa,andSuhadiWidoSaputro
ASuffixTreeOrNotaSuffixTree?.............................338
TatianaStarikovskayaandHjalteWedelVildhøj
DeterministicAlgorithmsfortheIndependentFeedbackVertex SetProblem..............................................351
YumaTamura,TakehiroIto,andXiaoZhou
LosslessSeedsforSearchingShortPatternswithHighErrorRates........364
ChristopheVroland,MikaëlSalson,andHélèneTouzet
AuthorIndex
OntheComplexityofVariousParameterizations ofCommonInducedSubgraphIsomorphism
FaisalN.Abu-Khzam1 , ´ EdouardBonnet2 ,andFlorianSikora2(B)
1 LebaneseAmericanUniversity,Beirut,Lebanon faisal.abukhzam@lau.edu.lb
2 PSL,Universit´eParis-Dauphine,LAMSADE,UMRCNRS7243,Paris,France {florian.sikora,edouard.bonnet}@dauphine.fr
Abstract. MaximumCommonInducedSubgraph (henceforth MCIS)isamongthemoststudiedclassical NP-hardproblems.MCIS remains NP-hardonmanygraphclassesincludingbipartitegraphs,planargraphsand k -trees.Littleisknown,however,abouttheparameterizedcomplexityoftheproblem.Whenparameterizedbythevertex covernumberoftheinputgraphs,theproblemwasrecentlyshowntobe fixed-parametertractable.Capitalizingonthisresult,weshowthatthe problemdoesnothaveapolynomialkernelwhenparameterizedbyvertex coverunless NP ⊆ coNP/poly .Wealsoshowthat MaximumCommon ConnectedInducedSubgraph (MCCIS),whichisavariantwhere thesolutionmustbeconnected,isalsofixed-parametertractablewhen parameterizedbythevertexcovernumberofinputgraphs.Bothproblemsareshowntobe W [1]-completeonbipartitegraphsandgraphsof girthfiveand,unless P = NP,theydonotbelongtotheclass XP when parameterizedbyaboundonthesizeoftheminimumfeedbackvertex setsoftheinputgraphs,thatissolvingtheminpolynomialtimeisvery unlikelywhenthisparameterisaconstant.
1Introduction
Acommoninducedsubgraphoftwographs G1 and G2 isagraphthatisisomorphictoinducedsubgraphsofeach.Theproblemoffindingacommoninduced subgraphofmaximumnumberofvertices(oredges)hasmanyapplicationsin anumberofdomainsincludingbioinformaticsandchemistry[11–13, 16, 17].In thedecisionversionoftheproblem,wearegivenaninteger k andthequestion istodecideifthereisasolutionwithatleast k vertices.Wesaythat k isthe naturalparameteroftheproblem,thatisthesolutionsize.
Concerningitsclassicalcomplexity, MaximumCommonInducedSubgraph is NP-complete,andremainssoonbipartitegraphsandgraphswith boundedtreewidth.However,theproblemisin P fortrees[10]andgraphsof (both)boundedtreewidthandboundeddegree[3].
Adecisionsubproblemof MaximumCommonInducedSubgraph isthe wellknown InducedSubgraphIsomorphism (ISI)problem,whichconsistsof decidingwhether G1 isisomorphictoaninducedsubgraphof G2 .Inotherwords, c SpringerInternationalPublishingSwitzerland2015 J.Kratochv´ıletal.(Eds.):IWOCA2014,LNCS8986,pp.1–12,2015. DOI:10.1007/978-3-319-19315-1 1
2F.N.Abu-Khzametal.
itisequivalentto MaximumCommonInducedSubgraph where k = |G1 |. Inthiscase G1 iscalledthepatterngraphwhile G2 isthehostgraph.ISIis W [1]-hardingeneral,byastraightforwardreductionfrom k -Clique.Therefore MCISisalso W [1]-hard.Ontheotherhand,ifISIisin FPT onacertaingraph class,thensoisMCIS.Toseethis,notethatanarbitraryinstance(G1 ,G2 ,k ) ofMCIScanbereducedinfpt-timetotwoinstancesofISIbyenumeratingall possiblegraphson k verticesandcheckingwhethereachisaninducedsubgraph ofeachofthetwoinputgraphs.ThisimpliesthatISIandMCIShavethesame parameterizedcomplexitywhenparameterizedbythesolutionsize,whichwe refertoasthenaturalparameterinthispaper.Ofcourse,thelatterreduction takestime O (2k 2 )(multipliedbythetimeneededtosolveISIonthegiven graphclass),whichmakesitprohibitivelyimpractical.Weshallprovideasimpler reductionthattakes O (ck )-timeonaclassofgraphsthatincludes H -minorfree graphsandgraphsofboundeddegree.
Anotherwaytodealwiththehardnessofaproblemistostudyitscomplexity withrespecttoauxiliary(orstructural)parameters,tobetterunderstandthe behavioroftheproblem(seeforexample[8]).MCISisalreadyhardongraphs withboundedtreewidth,being NP-hardonforests,asweshallobservebased onaclassicalresultfromGareyandJohnson[10].Accordingly,theproblemis W [1]-hardwhenparameterizedbythetreewidthoftheinputgraphs.Therefore weneedtolookforbiggerparameters.Weshallstudytheproblemwithrespect tothesizeofa(minimum)feedbackvertexsetthatofa(minimum)vertexcover ofinputgraphs.WeobservethatMCISisnotin XP whenparameterizedby thefeedbackvertexsetnumberoftheinputgraphs.Thisalsoimpliesthatthe problemisnotin XP whenparameterizedbytreewidth.
WeobservethatISIremains W [1]-hardongraphswhere G1 hasa k -vertex coverbyareductionfromthe W [1]-hard InducedBipartiteMatching problem[14]:ifthepatternconsistsof k disjointedges,itsvertexcoveris k .Therefore, MCISis W [1]-hardwhentheparameteristhevertexcoverofoneoftheinput graphs,eveniftheothergraphisbipartite.However,iftheparameter k isthe combinationofthevertexcoverofbothinputgraphs,thentheproblemisin FPT,witharunningtimeof O ((24k )k )[1].WeshallproveinSect. 3 thatMCIS doesnothaveapolynomial-sizekernelinthiscaseunless NP ⊆ coNP/poly
Wealsoconsiderthe MaximumCommonConnectedInducedSubgraph problem.Weobservethattheproblemisin FPT ongraphsofboundeddegree andshowittobe W [1]-completeontheclassofbipartitegraphs,eveniftheinput graphis C4 -free.Consequently,MCCISis W [1]-completeongraphsofgirthfive. Finally,weshowthatMCCISisfixed-parametertractablewhenparameterized byaboundontheminimumvertexcoversoftheinputgraphs.
2Preliminaries
Twofinitegraphs G1 =(V1 ,E1 )and G2 =(V2 ,E2 )are isomorphic ifthereis abijection π : V1 → V2 suchthat ∀u,v ∈ V1 : uv ∈ E1 ⇔ π (u)π (v ) ∈ E2 . Givenagraph G =(V,E ),agraph G =(V ,E )isan inducedsubgraph of G
if V ⊆ V and E = E (V )= {uv ∈ E | u,v ∈ V },i.e. E istheedgesetwith bothextremitiesin V .Wealsosaythat G isthesubgraphof G inducedby V .
The girth ofagraph G isthelengthoftheshortestcyclecontainedin G. Contractinganedge uv consistsofdeleting uv andreplacingthevertices u and v byasinglevertex w intheincidencerelation(edgesincidenton u or v become incidenton w ).Agraph H isa topologicalminor ofgraph G if H isobtained fromasubgraphof G byapplyingzeroormoreedgecontractions.Givenafixed graph H ,afamily F ofgraphsissaidtobe H -minorfree if H isnotaminorof anyelementof F
The MaximumCommonInducedSubgraph problemisdefinedformally asfollows.
MaximumCommonInducedSubgraph(MCIS):
• Input:Twographs G1 =(V1 ,E1 )and G2 =(V2 ,E2 ).
• Output:Aninducedsubgraph G1 of G1 isomorphictoaninduced subgraph G2 of G2 withamaximumnumberofvertices.
MaximumCommonConnectedInducedSubgraph (MCCIS)isdefined asMCISwiththeadditionalrestrictionthatthesolutionmustbeconnected. InducedSubgraphIsomorphism isdefinedsimilarly:
InducedSubgraphIsomorphism(ISI):
• Input:Twographs G1 =(V1 ,E1 )and G2 =(V2 ,E2 ).
• Output:Aninducedsubgraph G1 of G1 isomorphicto G2
Parameterizedcomplexity. Aparameterizedproblem(I,k )issaid fixed-parameter tractable (orintheclass FPT)w.r.t.(withrespectto)parameter k ifitcanbe solvedin f (k ) ·|I |c time(i.e.infpt-time),where f isanycomputablefunctionand c isaconstant(see[7, 15]formoredetailsaboutfixed-parameter tractability).Theparameterizedcomplexityhierarchyiscomposedoftheclasses FPT ⊆ W [1] ⊆ W [2] ⊆···⊆ XP.Theclass XP containsproblemssolvablein time f (k ) ·|I |g (k ) ,where f and g areunrestrictedfunctions.A W [1]-hardproblemisnotfixed-parametertractable(unless FPT = W [1])andonecanprove W [1]-hardnessbymeansofa parameterizedreduction froma W [1]-hardproblem.Thisisamappingofaninstance(I,k )ofaproblem A1 in g (k ) ·|I |O (1) time(foranycomputablefunction g )intoaninstance(I ,k )for A2 suchthat (I,k ) ∈ A1 ⇔ (I ,k ) ∈ A2 and k ≤ h(k )forsomefunction h.
Apowerfultechniquetodesignparameterizedalgorithmsis kernelization.In short,kernelizationisapolynomial-timeself-reductionalgorithmthattakesan instance(I,k )ofaparameterizedproblem P asinputandcomputesanequivalentinstance(I ,k )of P suchthat |I | h(k )forsomecomputablefunction h and k k .Theinstance(I ,k )iscalleda kernel inthiscase.Ifthefunction h ispolynomial,wesaythat(I ,k )isapolynomialkernel.Itiswellknown thataproblemisin FPT iffithasakernel,butthisequivalenceyieldssuperpolynomialkernels(ingeneral).Todesignefficientparameterizedalgorithms,a
4F.N.Abu-Khzametal.
kernelofpolynomial(orevenlinear)sizein k isimportant.However,somelower boundsonthesizeofthekernelcanbeshownunlesssomepolynomialhierarchycollapses.Toshowthisresult,wewillusethecrosscompositiontechnique developedbyBodlaenderetal.[4].
Definition1(PolynomialEquivalenceRelation[4]). Anequivalencerelation R on Σ∗ issaidtobe polynomial ifthefollowingtwoconditionshold:(i) Thereisanalgorithmthatgiventwostrings x,y ∈ Σ∗ decideswhether x and y belongtothesameequivalenceclassintime (|x| + |y |)O (1) .(ii)Foranyfinite set S ⊆ Σ∗ theequivalencerelation R partitionstheelementsof S intoatmost (maxx∈S |x|)O (1) classes.
Definition2(OR-Cross-Composition[4]). Let L ⊆ Σ∗ beasetandlet Q ⊆ Σ∗ × N beaparameterizedproblem.Wesaythat L cross-composes into Q ifthereisapolynomialequivalencerelation R andanalgorithmwhich,given t strings x1 ,x2 ,...,xt belongingtothesameequivalenceclassof R,computes aninstance (x∗ ,k ∗ ) ∈ Σ∗ × N intimepolynomialin t i=1 |xi | suchthat:(i) (x∗ ,k ∗ ) ∈ Q ⇔ xi ∈ L forsome 1 i t.(ii) k ∗ isboundedbyapolynomialin maxt i=1 |xi | +log t.
Proposition1([4]). Let L ⊆ Σ∗ beasetwhichis NP-hardunderKarpreductions.If L cross-composesintotheparameterizedproblem Q,then Q hasno polynomialkernelunless NP ⊆ coNP/poly
Aparameterizedproblemissaidtobe fixed-parameterenumerable ifallfeasible solutionscanbeenumeratedin O (f (k )|I |c )where f isacomputablefunctionof theparameter k only,and c isaconstant.
3StructuralParameterizationofMaximumCommon InducedSubgraph
Letusfirstrecallthat tw (G) fvs(G) vc(G),where tw (G)(resp. fvs(G), vc(G))representsthetreewidth(resp.thefeedbackvertexsetnumber,thevertexcovernumber)of G [8].Asnotedbefore,iftheparameteristhecombination of tw (G1 )and tw (G2 )thenMCISisknowntobe W [1]-hard.Evenmore,ifthe parameteristhecombinationof fvs(G1 )and fvs(G2 )(whichisbiggerthanthe combinationofthetreewidth),thentheproblemisnotevenin XP since MaximumCommonInducedSubgraph and InducedSubgraphIsomorphism are NP-hardonforests,acasewheretheparameterisequalto0.Indeed,onecan modifythereductionfrom 3-partition donebyGareyandJohnsonin[10]for SubforestIsomorphism toourproblem,bybuildingchainsof B +3vertices insteadof B +1in G2 suchthateachchainof G1 isseparatedbyavertex.The followingtheoremfollows.
Theorem1. Unless P = NP, MaximumCommonInducedSubgraph isnot in XP whenparameterizedbyaboundontheminimumfeedbackvertexsetsof thepairofinputgraphs.
ThehardnessofMCISonforestalsoimpliesthefollowing.
Corollary1. Unless P = NP, MaximumCommonInducedSubgraph is notin XP whenparameterizedbythetreewidthoftheinputgraphs.
Itwasshownin[1]thatMCISisin FPT iftheparameteristhecombinationof vc(G1 )and vc(G2 ).Accordingly,theproblemhasakernel,butnopolynomial boundisknownonitssize.Weshowthat,inthiscase,thekernelcannotbe polynomialunless NP ⊆ coNP/poly
Theorem2. Unless NP ⊆ coNP/poly , MaximumCommonInducedSubgraph hasnopolynomialkernelwhenparameterizedbythesumofthesizesof vertexcoversinthetwoinputgraphs.
Proof. WewilldefineanOR-cross-compositionfromthe NP-complete Clique, problem,wherethegiveninstanceisatuple(Gc ,l )andthequestioniswhether thegraph Gc containsacliqueon l vertices.
Given t instances,(Gc 1 ,l1 ), (Gc 2 ,l2 ),..., (Gc t ,lt ),of Clique,where Gc i isa graphand li ∈ N, ∀1 i t,wedefineourequivalencerelation R suchthatany stringsthatarenotencodingvalidinstancesareequivalent,and(Gc i ,li ), (Gc j ,lj ) areequivalentiff |V (Gc i )| = |V (Gc j )|,and li = lj .Hereafter,weassumethat V (Gc i )= {1,...,n} and li = l ,forany1 i t.Wewillbuildaninstanceof MaximumCommonInducedSubgraph parameterizedbythevertexcover (G1 ,G2 ,l ,Z )where G1 and G2 aretwographs, l ∈ N and Z ⊆ V (G2 )isa vertexcoverof G2 computedinfpt-time,suchthatthereisasolutionofsize l for MaximumCommonInducedSubgraph iffthereisan i, 1 i t such thatthereisasolutionofsize l in Gc i .Wewillnowdescribehowtobuild G1 and G2
Tobuild G2 (seealsoFig. 1):
–
V (G2 )= {p,q,r }∪{ai | 1 i t}∪{euv | 1 u<v n}∪{vi | 1 i n}, –
E (G2 )1 = {pq,pr,qr }, –
E (G2 )2 = {rai | 1 i t}, – E (G2 )3 = {ai euv | uv ∈ E (Gc i )}, –
E (G2 )4 = {euv vu ,euv vv |∀1 u<v n}, –
E (G2 )= E (G2 )1 ∪ E (G2 )2 ∪ E (G2 )3 ∪ E (G2 )4 .
Tobuild G1 (seealsoFig. 2):
– V (G1 )= {p,q,r,a}∪{ei | 1 i l 2 }∪{vi | 1 i l }, – E (G1 )1 = {pq,pr,qr,ra}, – E (G1 )2 = {aei | 1 i l 2 }, –
E (G1 )3 = {ei vu ,ei vv |∀1 i l 2 ,ei = uv }, –
E (G1 )= E (G1 )1 ∪ E (G1 )2 ∪ E (G1 )3 .
Weset l = |V (G1 )|,and Z = {p,r }∪{euv |1 u<v n}.Itiseasytoseethat Z isindeedavertexcoverfor G2 andthatitssizeisequalto n(n 1) 2 +2,which
Illustrationoftheconstructionof G2
Fig.2. Illustrationoftheconstructionof G1 .
ispolynomialin n andhenceinthesizeofthelargestinstance.Notethatthe sizeofthegraph G1 doesnotdependon t andispolynomialin n,sothesizeof itsvertexcoverisalsopolynomialin n andindependentof t.
Letusshowthat G1 isaninducedsubgraphof G2 iffatleastoneofthe Gc i ’s hasacliqueofsize l . (⇐)Supposethat Gc i hasacliqueofsize l .Wedenoteby S ⊆ V (Gc i )a cliqueofsizeexactly l in Gc i .Weshowthatthereisaninducedsubgraph S of G2 ofsize l ,isomorphicto G1 .Weset V (S )= {p,q,r }∪{ai }∪{euv | ∀uv ∈ E (S )}∪{vu |u ∈ S }.Onecaneasilycheckthatthissubgraphisisomorphic to G1 .
(⇒)Assumenowthat G1 isaninducedsubgraphof G2 .Denoteby S the subgraphof G2 isomorphicto G1 .Notethattheonlytrianglein G2 is pqr .
Fig.1.
Indeed, T (V (G2 ) \{p})isbipartite.Thetriangle pqr in G1 hasthereforeto match pqr in G2 .Moreover, r in G1 hastomatch r in G2 since p and q haveno edgesbesidestheclique pqr .Thevertex a in G1 canonlymatchavertex ai for some i ∈{1,...,t}.Then, e1 upto e( l 2) in G1 hastomatch l 2 verticesin {euv | 1 u<v n} of G2 whichcorrespondtoactualedgesin Gc i .Finally, v1 upto vl in G1 hastomatch l verticesamongthe vj ’sin G2 .Notethatthenumberof edgesin E (G1 )3 betweenthe ej ’sandthe vj ’sisexactly2 l 2 = l (l 1).More precisely,each ej touches2edgesin E (G1 )3 andeach vj touches l 1edges in E (G1 )3 .Inordertogetamatchin G2 ,oneshouldfindasetof l 2 edges inducingexactly l vertices.So,thissetof l verticesisacliquein Gc i
NotethattheparameterofMCISinthisreductionisexactlythesizeof G1 Therefore,thisnegativeresultholdsforISItoo.
DespitethefactthatISIandMCIShavethesameparameterizedcomplexity whenparameterizedbythenaturalparameter,theyexhibitdifferentcomplexitieswithrespecttostructuralparameters.Infact,thelatterisnotevenin XP whenparameterizedbythevertexcoverofonlyoneofthetwographswhileISI is FPT whenparameterizedbythevertexcoverofthesecond(host)graph.To seethis,notethatwhenthehostgraphhasa k -vertexcover,theminimumsize ofavertexcoverinthepatterngraphmustbeboundedbytheparameter k , otherwisewehaveanoinstance.Theclaimfollowsfromthefixed-parameter tractabilityofMCISinthiscase[1].
AlthoughMCISisnotin XP w.r.t.somestructuralparameterssuchas treewidthandfeedbackvertexsetnumber,itis,togetherwithMCCISandISI, in W [1]w.r.t.thenaturalparameter.
Theorem3. MCIS,MCCISandISIare W [1]-completew.r.t.thenaturalparameter.
Proof. SinceISI,MCISandMCCISare W [1]-hardbyastraightforwardreduction from k -Clique,itsufficestoshowmembershipin W [1].In[5],itisshownthatif aproblemcanbereducedinFPTtimetosimulatinganon-deterministicsingletapedTuringMachinehaltinginatmost f (k )steps,forsomefunction f ,then itisin W [1].TheTuringMachinecanhaveanalphabetandasetofstatesof sizedependingonthesizeoftheinputoftheinitialproblem.Inourcase,we candesignaTuringMachinethatguessesin2k stepsthecorrespondingright k verticesin G1 (forISIthispartisnotnecessary)andtheright k verticesin G2 (ouralphabetbeingisomorphictoanindexingof V (G1 ) ∪ V (G2 ))andthen checkintime O (k 2 )whetherthetwoinducedsubgraphsareisomorphic(and thattheyareconnectedforMCCIS).
WenowturnourattentiontothecasewheretheMCISisparameterizedby acombinationofthenaturalparameterandsomestructuralparameter.For example,considerthecasewheretheparameteristhesumofsomebound t on thefeedbackvertexsetoftheinputgraphsandthenaturalparameter k .The problemis FPT inthiscasesincegraphsof t-feedbackvertexsetare H -minor free(let H bethe“fixed”graphconsistingofadisjointunionof t +1triangles).
8F.N.Abu-Khzametal.
Moreover,weknowISIis FPT inthiscasedueto[9].However,andasstated inSect. 1,asolutiontoaninstanceofMCIS(inthiscase)isobtainedviaan exhaustiveenumerationof O (2k 2 )instancesofISI.Thiscanbeimprovedon classesofgraphsthataregivenwithsomefixedcoloring t,suchas H -minorfree graphsandgraphsofboundedmaximumdegree.Infact,iftheinputtoMCISis apairof t-coloredgraphs,thenareductionalgorithmwouldfirstcheckwhether eachofthetwographshasan(independent)colorclassofsize k .Ifso,thenboth haveanedgelesscommonsubgraphofsize k .Otherwise,theorderofatleast oneofthetwographs,say G1 ,issmallerthan tk .Insuchcase,thealgorithm proceedsbyrunninga(fixed-parameter)algorithmforISIoneachofthe O (2tk ) inducedsubgraphsof G1
In[14]itwasshownthat InducedMatching is W [1]-hardonbipartite graphs.Asmentionedearlier,thisprovesthatMCISis W [1]-hardinthiscase. WeshowthatMCISremains W [1]-hardon C4 -freebipartitegraphs,whichproves its W [1]-hardnessongraphsofgirthfive.
Theorem4. MaximumCommonInducedSubgraph is W [1]-completew.r.t. sizeofthesolution,asparameter,evenon C4 -freebipartitegraphs.
Proof. Membershipin W [1]comesfromTheorem 3.Forthehardness,consider thefollowingreductionfromthe W [1]-hardproblem Clique.Givenaninstance (G =(V,E ),k )of Clique,webuildaninstance(G1 ,G2 ,k )ofourproblemas follows.Thegraph G2 isthebipartiteincidencegraphof G (thebipartitionis betweenverticesrepresenting V andverticesrepresenting E ),thegraph G1 is thebipartiteincidencegraphof Kk ,and k = k + k 2 = |V (G1 )|.
Notethatabipartiteincidencegraphis C4 -freesince,inasimplegraph,no twoedgesareincidentonthesamepairofvertices.
Itisclearthat G1 occursasaconnectedinducedsubgraphof G2 iffthereis acliqueofsize k in G,becausew.l.o.g. k> 2andtheverticesrepresentingedges in G1 and G2 areofdegree2.
Corollary2. MaximumCommonInducedSubgraph is W [1]-complete w.r.t.sizeofthesolutionongraphsofgirthfive.
4MaximumCommonConnectedInducedSubgraph
MaximumCommonConnectedInducedSubgraph istrivially FPT whenever InducedSubgraphIsomorphism is FPT,including H -minorfreegraphs, sincetheenumerationofall O (2k 2 )possibleinducedconnectedsubgraphscanbe usedasdescribedbefore.Theconverseisalsotrue.Infact,aninstance(G1 ,G2 ,k ) ofISIcanbereducedtoanequivalentinstance(G1 ,G2 ,k +1)ofMCCISbyletting Gi bethegraphobtainedbyaddingasingle(universal)vertexto Gi that ismadeadjacenttoallotherverticesof Gi .ItfollowsthatMCISandMCCIS havethesameparameterizedcomplexitywithrespecttothenaturalparameter (i.e.,solutionsize).
NotethatMCISis NP-hardonforestswhileMCCISissolvableinpolynomialtimeinthiscase:giventwoforests G1 and G2 ,runthepolynomial-timeMCCIS algorithmof[2]oneverypairoftreesfrom G1 and G2 . InthefollowingofthissectionwestudythecomplexityofMCCISwith respecttostructuralparameters.
Lemma1. InducedconnectedSubgraphIsomorphism is NP-hardeven whenbothgraphshavefeedbackvertexsetnumberequaltoone.
Proof. Givenaninstanceof InducedSubgraphIsomorphism onforests G1 and G2 (eachwithatleast2trees),webuildaninstanceof InducedconnectedSubgraphIsomorphism byaddingauniversalvertex(connectedto everynode)in G1 andin G2 .Onecanseethatthesetwouniversalverticesmust bematchedtogethersincetheyaretheonlyoneswithsufficientlyhighdegree. Then,thereisasolutionfor InducedSubgraphIsomorphism iffthereis asolutionfor InducedconnectedSubgraphIsomorphism.Theresultof courseholdsforMCCIStoo.
Corollary3. Unless P = NP, MaximumCommonConnectedInduced Subgraph isnotin XP whenparameterizedbyaboundoftheminimumfeedback vertexsetnumberoftheinputgraphs(andhencethenwhenparameterizedbya boundonthetreewidthofeachofthetwoinputgraphs).
Giventheabovenegativeresult,thenextquestioniswhetherMCCISisin FPT w.r.t.theparametervertexcover.In[1],aparameterizedalgorithmispresented forMCISwhentheparameterisaboundontheminimumvertexcovernumber oftheinputgraphs.However,thatalgorithmcannothelpusmuchforsolving MCCISsinceitreliesontheexistenceofafeasiblesolutionofsizeatleast ≈ n k whichconsistsofmappingthetwo big independentsetsofthetwo graphsontoeachother.Ofcourse,thisisnotafeasiblesolutionforMCCIS. InthefollowingweprovethatMCCISisfixed-parametertractablew.r.t. k = max(vc(G1 ),vc(G2 )).
Theorem5. MaximumCommonConnectedInducedSubgraph parameterizedbyaboundonthevertexcoversoftheinputgraphsisfixed-parameter tractable.
Proof. Intime O ∗ (2k )(even O ∗ (1.2738k )[6]),wecanfindminimumvertexcovers C1 and C2 in G1 and G2 respectively.Let I (j ) betheindependentset V (Gj ) \ Cj for j ∈{1, 2}.Byassumption,ourparameter k ismax(C1 ,C2 ),sowecan enumeratealltripartitionsof C1 and C2 intime O ∗ (9k ).Wedenoteby C1,m , C1,u and C1,i (respectively C2,m , C2,u and C2,i )thethreesetsofatripartitionof C1 (respectively C2 ).For j ∈{1, 2}, Cj,u correspondstotheverticesof Cj that arenotmatched,sotheymaybedeleted. Cj,m comprisestheverticesmatchedto C3 j,m (thatis,tothevertexcoveroftheothergraph),and Cj,i arethevertices matchedto I (3 j ) ,theindependentsetoftheothergraph.SeeFig. 3
Weobservethatfor j ∈{1, 2}, I (j ) canbepartitionedintoatmost2k classes oftwins: I (j ) 1 ,I (j ) 2 ,...I (j ) 2k .Aclassoftwinsinthiscontextisasetofverticeswith
10F.N.Abu-Khzametal.
anidenticalneighborhoodinthevertexcoverandthereareatmost2k subsets of Cj .Potentially,someclassescanbeempty:theycorrespondtoasubsetofthe vertexcover Cj thatisnotthe(exact)neighborhoodofanyvertexin I (j ) . Atthispoint,wecanenumeratethemappingsbetween C1,m and C2,m intime O ∗ (k k )andthemappingsbetween Cj,i and I (3 j ) intime O ∗ ((2k )k )= O ∗ (2k 2 ). Indeed,tomatchavertex u withavertex v oratwinof v isequivalent.Thus, intime O ∗ ((9k )k 2k 2 )wecanenumerateallthesolutionsofMCISwhereonly verticesof I (1) couldstillbematchedtoverticesof I (2) .Theoptimalmapofthe independentsetscanbedoneinlineartimebymatchingthegreatestnumberof verticesineach equivalent twinclass(whichisthesizeofthesmallerofthetwo equivalenttwinclasses),whereatwinclass I (j ) r in I (j ) isequivalenttoatwin class I (3 j ) s in I (3 j ) iftheverticesof N (I (j ) r ) \ Cj,u and N (I (3 j ) s ) \ C3 j,u are inone-to-onecorrespondence.
Fig.3. IllustrationoftheproofofTheorem 5.Dashedboxesrepresenttheclassesinside theindependentset.Arrowsrepresentthematchingbetweensetsofvertices.
TofindasolutionforMCCIS,thealgorithmdescribedintheaboveproofenumeratesallpossiblemaximalcommoninducedsubgraphsintime O ∗ ((9k )k 2k 2 ). Assuch,itcanbeusedasanenumerationalgorithmforMCIS.
Theorem6. MaximumCommonInducedSubgraph parameterizedbyvertexcover,isfixed-parameterenumerable.
Finally,thefollowingcorollariesfolloweasilyfromtheproofsofTheorems 2 and 4 sincethegraphsusedinbothproofsareconnected.
Corollary4. MaximumCommonConnectedInducedSubgraph,parameterizedbyaboundontheminimumvertexcoversofinputgraphs,doesnot haveapolynomial-sizekernelunless NP ⊆ coNP/poly .
Corollary5. MaximumCommonConnectedInducedSubgraph is W [1]completeonbipartitegraphsandgraphsofgirthfive.
Table1. SummaryofdifferentparameterizedcomplexityresultsofISI,MCISand MCCISfordifferentstructuralparameters.
vc + vc vc + fvs fvs + fvs vc
ISI FPT;noPolyKernel (Theorem 2)
M(C)CIS FPT ([1],Theorem 5); noPolyKernel
Open / ∈ XP
FPT for vc(G2 ), / ∈ XP for vc(G1 )
Open / ∈ XP (Corollary 3) / ∈ XP
Inthefollowingtablewegiveasummaryofsomeresultsobtainedinthis paperalongwithopenquestions.NotethatforISI, vc + fvs isnotthesame parameteras fvs + vc.Inthelatter,theparameterisaboundonthevertex coverof G2 (aswellasthefeedbackvertexsetof G1 )whichmakesISIin FPT, whileitremainsopenfor vc + fvs.WealsonotethatISIisnotin XP w.r.t. vc(G1 )byasimplereductionfrom IndependentSet (let G2 beanedgeless graphon k vertices,thenitsvertexcovernumberis0).
5Conclusion
Westudiedthe MaximumCommonInducedSubgraph and MaximumCommonConnectedInducedSubgraph problemswithrespecttothesolution sizeasnaturalparameteronspecialgraphsclasses,suchasforests,bipartite graphsandgraphsofgirthfive.Thetwoproblemsarefixed-parametertractable on H -minorfreegraphs,whichincludeforests,buttheyare W [1]-completeon bipartitegraphsandgraphsofgirthfive.
Wealsoconsideredtheuseofauxiliaryparameters,suchasaboundonthe minimumvertexcoversoftheinputgraphs.AlthoughbothMCISandMCCIS arein FPT inthiscase,weprovedthatnokernelofpolynomialboundcan beobtainedunless NP ⊆ coNP/poly .WenotedthatMCISisnotevenin XP withrespecttoother(smaller)auxiliaryparameters,suchastreewidthandfeedbackvertexset(seeTable 1).Afewcorrespondingopenproblemsremaintobe addressed.Forexample,areMCIS/MCCISin FPT whenparameterizedbythe combinationofthevertexcovernumberandthefeedbackvertexsetnumber,or bythevertexcovernumberandthetreewidth?
Acknowledgements. Workpartiallysupportedbythebilateralresearchcooperation CEDREbetweenFranceandLebanon(grantnumber30885TM).
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ApproximationandHardnessResults fortheMaximumEdgesinTransitive ClosureProblem
AnnaAdamaszek
1 ,GuillaumeBlin2,3 ,andAlexandruPopa4(B)
1 Max-Planck-InstitutF¨urInformatik,Saarbr¨ucken,Germany anna@mpi-inf.mpg.de
2 LaBRI,UMR5800,UniversityofBordeaux,33400Talence,France
3 CNRS,LaBRI,UMR5800,33400Talence,France
guillaume.blin@labri.fr
4 FacultyofInformatics,MasarykUniversity,Brno,CzechRepublic popa@fi.muni.cz
Abstract. Inthispaperwestudythefollowingproblem,namedMaximumEdgesinTransitiveClosure,whichhasapplicationsincomputationalbiology.Givenasimple,undirectedgraph G =(V,E )anda coloringofthevertices,removeacollectionofedgesfromthegraphsuch thateachconnectedcomponentis colorful (i.e.,itdoesnotcontaintwo identicallycoloredvertices)andthenumberofedgesinthetransitive closureofthegraphismaximized.
TheproblemisknowntobeAPX-hard,andnoapproximationalgorithmsareknownforit.Weimprovethehardnessresultbyshowingthat theproblemisNP-hardtoapproximatewithinafactorof |V |1/3 ε ,for anyconstant ε> 0.Additionally,weshowthattheproblemisAPXhardalreadyforthecasewhenthenumberofvertexcolorsequals3.We complementtheseresultsbyshowingthefirstapproximationalgorithm fortheproblem,withapproximationfactor √2 OPT.
1Introduction
TheMaximumEdgesinTransitiveClosureproblemweconsiderinthispaper belongstotheframeworkofcolorfulcomponentsproblems.
Colorfulcomponentsframework: Givenasimple,undirectedgraph G = (V,E )andacoloring σ : V → C oftheverticeswithcolorsfromagivenset C , removeacollectionofedges E ⊆ E from G suchthateachconnectedcomponent intheresultinggraph G =(V,E \E )isa colorfulcomponent (i.e.,itdoesnot containtwoidenticallycoloredvertices).Wewantthegraph G tobeoptimal accordingtosomefixed optimizationmeasure
Inourproblem,theoptimizationmeasureisthenumberofedgesinthetransitiveclosure.Foragraphconsistingof k connectedcomponents,eachcontaining respectively a1 ,a2 ,...,ak vertices,thenumberofedgesinthetransitiveclosure ofthegraphis
c SpringerInternationalPublishingSwitzerland2015 J.Kratochv´ıletal.(Eds.):IWOCA2014,LNCS8986,pp.13–23,2015. DOI:10.1007/978-3-319-19315-1 2
MaximumEdgesinTransitiveClosure(MEC): Givenasimple,undirectedgraph G =(V,E )andacoloring σ : V → C ofthevertices,remove acollectionofedges E ⊆ E from G suchthateachconnectedcomponentin theresultinggraph G =(V,E \E )iscolorful,andthenumberofedgesinthe transitiveclosureof G ismaximum.
Motivation. Thecolorfulcomponentsframeworkismotivatedbyapplicationsin comparativegenomics[8, 10],whichisafundamentalbranchofbioinformatics studyingtherelationshipofthegenomestructurebetweendifferentbiological species.Oneofthekeyproblemsinthisarea,themultiplealignmentofgene orders,canbecapturedasagraphtheoreticalproblem,usingthecolorfulcomponentsframework,wherethecolorfulgraphsrepresentsimilarityrelationships betweengenesfromdifferenthomologousgenefamilies.Apartitionintocolorful componentscorrespondsthentoapartitionofgenesintoorthologysets,where anytwogenesfromthesamegenomebelongtodifferentorthologysets.Werefer thereaderto[10]foramoredetaileddescriptionoftheconnectionbetweenthe multiplealignmentofgeneordersandthegraphtheoreticframeworkconsidered.
Theunderstandingoforthologousgenesoftwodifferentgenomesasoriginatingfromasinglegeneinthemostrecentcommonancestorofthetwospecies leadstotransitivityasapropertyoftheorthologyrelation.Thismotivatesthe studyofMEC(see[10]formoredetails,andforadiscussionwhyMECyields goodresultsinpractice).
RelatedWork. TheMaximumEdgesinTransitiveClosureproblemhasbeen introducedbyZhengetal.[10].Theypresentheuristicalgorithmsfortheproblem,withoutgivinganyworst-caseapproximationguarantee.TheyalsoconjecturetheproblemtobeNP-hard.AdamaszekandPopa[1]provethatMECis APX-hard,eveninthecaseof4vertexcolors.
ThecolorfulcomponentsframeworkappearedfirstinthepaperbyZheng etal.[10]andhasbeenformallydefinedbyAdamaszekandPopa[1],although problemswhichfitintothisframeworkhavealreadybeenstudiedearlier.We nowsummarizeknownresultsfortheseproblems.
IntheproblemnamedeitherColorfulComponents[3, 4]orMinimumOrthogonalPartition[5, 10],theobjectivefunctionistominimizethenumberofedges removedfrom G toobtainagraphinwhichallconnectedcomponentsarecolorful.Bruckneretal.[4]showthattheproblemis NP -hardforthreeormorecolors andtheystudyfixed-parameteralgorithmsfortheproblem.Their NP -hardness reductioncanbemodifiedslightlytoshowtheAPX-hardnessoftheproblem (see[1]).Zhengetal.[10]andBruckneretal.[3]studyheuristicapproaches fortheproblem,andHeetal.[5]presentanapproximationalgorithmforsome specialcaseoftheproblem.Asthegeneralproblemisaspecialcaseofthe MinimumMulti-MultiwayCutproblem,itadmitsa O (log |C |)approximation algorithm[2].
Zhengetal.[10]introducetheMinimumSingletonVerticesproblem(MSV), wherethegoalistominimizethenumberofisolatedverticesintheresulting graph.Zhengetal.[10]presentheuristicalgorithmsfortheproblem,without givinganyworst-caseapproximationguarantee.Theyalsoconjecturedthatthe problemisNP-hard.Tremblay-SavardandSwenson[9]consideraMaximum OrthogonalEdgeCoverproblem(MAX-OREC),whichisadualproblemto MSV.There,thegoalistocoveramaximumnumberofverticesofagraph usingvertex-disjoint,non-singletonconnectedcolorfulsubgraphs.In[9],a2/3approximationalgorithmforMAX-ORECispresented.AdamaszekandPopa[1] provethatMSV(andthereforealsoMAX-OREC)canbesolvedexactlyinpolynomialtime,thusdisprovingtheconjecturein[10].
AdamaszekandPopa[1]introduceanotherproblem,termedMinimumColorfulComponents,inwhichthegoalistodeleteasubsetofedgessuchthat theresultinggraphhasonlycolorfulcomponentsandthenumberofconnected componentsisminimized.Theyshowthatthisproblemcannotbeapproximated withinafactorof |V |1/14 ε unless P = NP ,andwithinafactor |V |1/2 ε unless ZPP = NP .
OurResults. InthispaperweimprovethehardnessresultsfortheMECproblem, andwepresentthefirstapproximationalgorithm.
First,weshowthatMECisAPX-hardevenforthecasewhen |C | =3.This settlesthecomplexityoftheproblemwhenthenumberofcolorsisaconstant, asfor |C | =2theMECproblemcanbesolvedexactlyinpolynomialtimeby usingamaximummatchingalgorithm.Ourproofisviaareductionfromthe MaximumBounded3-DimensionalMatchingproblem(Max3-DM-3).
Forthegeneralcase,whenthenumberofcolorsisarbitrary,weshowthat MECisNP-hardtoapproximatewithinafactorof |V |1/3 ε foranyconstant ε> 0.Thisresultholdseveniftheinputgraphisatreeandeachcolorappears atmosttwiceinthegraph.WeusethesamereductionfromtheIndependentSet asRizziandSikoraforprovinghardnessofapproximationoftheGraphMotif problem[7].
Wealsoshowthefirstpolynomial-timeapproximationalgorithmforMEC, whichhasaratioof √2 · OPT.Weusetheexactpolynomialtimealgorithmfor theMinimumSingletonVerticesproblem[1]toobtainapartitionintocolorful componentsandthenweshowthatthispartitionhasabigenoughnumberof edgesinthetransitiveclosure.
2APX-hardnessofMECfor |C | =3
Inthissection,weprovethattheMECproblemrestrictedtoinstancesusing only3colorsisAPX-hard.TheproofisviaareductionfromtheMaximum Bounded3-DimensionalMatchingproblem.Thisresultstrengthenstheonepresentedin[1],whichholdsforprobleminstancesusing4colors.
Beforewegivethereduction,wefirststatethedefinitionofMax3-DM-3and theknownhardnessresultforit.
MaximumBounded3-DimensionalMatching(Max3-DM-3): Theinput consistsofpairwisedisjointsets X , Y , Z andacollection T ⊆ X × Y × Z of
16A.Adamaszeketal.
triplessuchthateachelementfrom X , Y and Z occursinatleastoneandat mostthreetriplesin T .Theaimistofindafeasiblesubsetoftriples T ⊆ T (i.e.,notwoelementsof T agreeonanycoordinate)ofmaximumcardinality.
Theorem1(Theorem4.4in[6],Rephrased). Thereexistsaconstant ε> 0 suchthatitisNP-hardtodistinguishbetweentheinstancesofMax3-DM-3with thefollowingproperties:
1.Thereisafeasiblecollectionoftriples T ⊆ T suchthateveryelementof X , Y and Z belongstosometriplein T
2.Foreveryfeasiblecollectionoftriples T ⊆ T lessthan (1 ε) fractionof elementsfrom X ∪ Y ∪ Z belongtosometripleof T .
Withoutlossofgeneralitywecanassumethat |X | = |Y | = |Z | = n,sinceif |X |, |Y | and |Z | aredifferent,thenthecase1ofTheorem 1 cannothold.Also,define N = |T |.Itholdsthat N ≤ 3n,sinceeachelementof X ∪ Y ∪ Z appearsin atmostthreetriples.Intherestofthesection,weuseOPT3DM todenotethe sizeofanoptimalsolutionofaMax3-DM-3instance(theinstancewereferto willalwaysbeclearfromthecontext),andOPTMEC todenotethevalueofan optimalsolution(i.e.,thenumberofedgesinthetransitiveclosureofthegraph) oftheMECinstanceobtainedviathereduction.
Reduction. Givenaninstance(X,Y,Z,T )ofMax3-DM-3,wecreatean instance(G =(V,E ),σ )oftheMECprobleminthefollowingway.SeeFig. 1 for apartialillustration.Wecreatethesetofvertices V asfollows.
1.Foreachtriple tj ∈ T ,weaddsixvertices {tX j ,tY j ,tZ j ,tXY j ,tXZ j ,tYZ j }.
2.Foreachelement xi ∈ X (resp. yi ∈ Y and zi ∈ Z ),weaddacorresponding vertex xi (resp. yi and zi ).
Wehavethat |V | =6 ·|T | + |X | + |Y | + |Z | =6N +3n.Letusnowdefinethe coloring σ : V → C oftheverticesusingthesetofcolors C = {1, 2, 3}.
1.Forany1 ≤ i ≤ n and1 ≤ j ≤ N , σ (xi )= σ (tXY j )= σ (tZ j )=1.
2.Forany1 ≤ i ≤ n and1 ≤ j ≤ N , σ (yi )= σ (tYZ j )= σ (tX j )=2.
3.Forany1 ≤ i ≤ n and1 ≤ j ≤ N , σ (zi )= σ (tXZ j )= σ (tY j )=3. Finally,letusdefinethecollectionofedges E
1.Foreach1 ≤ j ≤ N ,eachof {tX j ,tXY j ,tXZ j }, {tY j ,tXY j ,tYZ j }, {tZ j ,tXZ j ,tYZ j } formsacliqueofsizethree.
2.Foreach1 ≤ i ≤ n and1 ≤ j ≤ N ,if xi (resp. yi and zi )appearsin tj , connect xi (resp. yi and zi )to tX j (resp. tY j and tZ j ).
Analysis. Informally,weshowthataninstanceofMax3-DM-3whereallthe vertices X ∪ Y ∪ Z canbecoveredbyafeasiblecollectionoftriples T correspondstoaninstanceofMECwithalargeoptimalvalue,i.e.,thegraphcan bepartitionedintocolorfulcomponentsinducingalargetransitiveclosure.On theotherhand,weshowthataninstanceofMax3-DM-3wherenomorethan (1 ε)fractionofthevertices X ∪ Y ∪ Z canbecoveredbyanyfeasiblesetof triplescorrespondstoaninstanceofMECwithamuchsmalleroptimalvalue. Wenowanalyzebothcases.
Fig.1. Asubgraphcorrespondingtoatriple tj =(xi ,yk ,zl ).Colorsoftheverticesare denotedusingthelinestyles:solid,dottedanddashedlinesrespectivelycorresponds tocolors1,2and3.
Lemma1. Let (X,Y,Z,T ) beaninstanceofMax3-DM-3whereOPT3DM = n, i.e.,wherealltheverticesof X ∪ Y ∪ Z canbecoveredbyafeasiblecollection oftriples.ThenforthecorrespondinginstanceofMEC,wehaveOPTMEC ≥ 6N +3n
Proof. ThecolorfulcomponentsoftheMECinstanceareconstructedasfollows. Foreachtriple tj ∈ T (thereare n ofthem),weaddthreecolorfulcomponents, eachcomponentconsistingofthreevertices.Givenatriple tj =(xi ,yk ,zl ),the colorfulcomponentsare {xi ,tX j ,tXZ j }, {yk ,tY j ,tXY j } and {zl ,tZ j ,tYZ j }.Foreach triple tj ∈ T \ T (thereare N n ofthem),wecreatetwocolorfulcomponents, eachconsistingofthreevertices: {tX j ,tXZ j ,tZ j } and {tXY j ,tY j ,tYZ j }.SeeFig. 2 for anillustration.
As T isafeasiblecollectionoftriples,thatisasetoftriplessuchthatnotwo elementsagreeonanycoordinate,weobtainafeasiblepartitionofthegraphinto colorfulcomponents.Clearly,thetotalnumberofedgesinthetransitiveclosure equals9n +6(N n)=6N +3n,sinceeachofthe n triplesin T inducesthree colorfulcomponentsofsizethreeandeachofthe N n othertriplesinduces twocolorfulcomponentsofsizethree.
Lemma2. Let (X,Y,Z,T ) beaninstanceofMax3-DM-3whereOPT3DM < (1 ε)n,i.e.,whereeveryfeasiblecollectionoftriplescoverslessthana (1 ε) wofvertices X ∪ Y ∪ Z .Then,forthecorrespondinginstanceofMEC,wehave OPTMEC < 6N +3n(1 ε/2).
Proof. Let(G =(V,E ),σ )betheinstanceoftheMECproblemcorresponding toaninstanceofMax3-DM-3asdefinedinthelemmastatement.Foranytriple tj =(xi ,yk ,zl ) ∈ T ,let Gtj beasubgraphof G inducedbythefollowingsetof vertices {xi ,yk ,zl ,tX j ,tY j ,tZ j ,tXY j ,tXZ j ,tYZ j },asshowninFig. 1
Letusfixanoptimalsolution S fortheMECproblemfor(G,σ ).Thissolutiondefinesapartition Γ of G intocolorfulcomponents.First,noticethateach colorfulcomponentiscontainedwithinsomesubgraph Gt .Indeed,byconstruction,theonlyverticeswhichbelongtomultiplesubgraphs Gtj arethevertices
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Boyer, W. S. Johnnie Kelly. (N ’20)
Butler, E. P. Swatty. (Ap ’20)
Croy, H. Turkey Bowman. (N ’20)
France, A., pseud. Little Pierre. (F ’21)
Galsworthy, J: Awakening. (Ja ’21)
Gatlin, D. Missy. (D ’20)
McKishnie, A. P. Son of courage. (D ’20)
Masters, E. L. Mitch Miller. (D ’20)
Porter, E. Mary Marie. (Je ’20)
Vorse, M. M. Growing up. (S ’20)
Christmas stories
Rinehart, M. Truce of God. (F ’21)
Court-room scenes
Hill, F: T. Tales out of court. (D ’20)
Crime and criminals
Gregory, J. Ladyfingers. (Je ’20)
Leverage, H: Where dead men walk. (Ag ’20)
Packard, F. L. From now on. (Mr ’20)
Packard, F. L. White Moll. (Jl ’20)
Roche, A. S. Uneasy street. (Mr ’20)
Rowland, H: C. Duds. (Mr ’20)
Williamson, C: N. and A. M. Second latchkey. (Je ’20)
Detective stories
See Mystery stories
Divorce
Maxwell, W: B. For better, for worse. (N ’20)
Economic problems
Cadmus and Harmonia, pseuds. Island of sheep. (Ap ’20)
Enterprise
Sullivan, A. Rapids. (S ’20)
European war
Abdullah, A. Man on horseback. (Ap ’20)
Battersby, H: F. P. Edge of doom. (Je ’20)
Bazin, R. F. N. M. Pierre and Joseph. (Je ’20)
Blasco Ibáñez, V. Enemies of women. (F ’21)
Cook, W. V: Grey fish. (S ’20)
Gibbs, G: F. Splendid outcast. (Ap ’20)
Gibbs, P. H. Wounded souls. (N ’20)
Herbert, A. P. Secret battle. (Mr ’20)
Jeffery, J. E. Side issues. (N ’20)
Kelly, T: H. What outfit, Buddy? (My ’20)
Latzko, A. Judgment of peace. (Ap ’20)
Mackay, H. G. Chill hours. (Ap ’20)
Mason, A. E: W. Summons. (D ’20)
Maurois, A. Silence of Colonel Bramble. (Jl ’20)
Mayran, C. Story of Gotton Connixloo. (Ja ’21)
Montague, M. P. England to America. (Je ’20)
Oppenheim, E: P. Devil’s paw. (D ’20)
Robinson, E. H. Maid of Mirabelle. (O ’20)
Sawyer, R. Leerie. (S ’20)
Sinclair, M. Romantic. (D ’20)
Zanella, N. By the waters of Fiume. (Jl ’20)
Belgium
Vane, G: Waters of strife. (S ’20)
Great Britain
Ayres, R. M. Richard Chatterton, V. C. (Jl ’20)
Baxter, A. B. Parts men play. (Ja ’21)
Benson, E: F: Robin Linnet. (Mr ’20)
Benson, S. Living alone. (My ’20)
Copplestone, B. Last of the Grenvilles. (Ap ’20)
Dawson, C. W: Little house. (N ’20)
Frankau, G. Peter Jameson. (My ’20)
Galsworthy, J: Tatterdemalion. (My ’20)
Hamilton, C. M. William—an Englishman. (Jl ’20)
Jenkin, A. M. N. End of a dream. (Jl ’20)
Jepson, E. Pollyooly dances. (Ag ’20)
Locke, W: J: House of Baltazar. (Mr ’20)
Maxwell, W: B. Glamour. (Ap ’20)
Onions, B. Bridge of kisses. (D ’20)
United States
Austin, M. No. 26 Jayne street. (Je ’20)
Montague, M. P. Uncle Sam of Freedom Ridge. (S ’20)
Poole, E. Blind. (D ’20)
Schem, L. C. Hyphen. (Ja ’21)
Sherwood, M. P. World to mend. (N ’20)
Fairy tales
Tarn, W: W. Treasure of the isle of mist. (Jl ’20)
High cost of living
Dodge, H: I. Skinner makes it fashionable. (Je ’20)
Historical novels
Canada
Wilson, M. Forging of the pikes. (My ’20)
Cape Colony
Juta, R. Cape Currey. (S ’20)
Discovery & exploration (17th century)
Forbes, G: Adventures in southern seas. (Ja ’21)
Egypt
Couperus, L: M. A. Tour. (Jl ’20)
Haggard, H: R. Ancient Allan. (Je ’20)
England
Hudson, W: H: Dead Man’s Plack, and An old thorn. (F ’21)
Locke, G. E. Ronald o ’ the moors. (O ’20)
McCarthy, J. H. Henry Elizabeth. (S ’20)
MacDonald, G. North door. (O ’20)
McFadden, G. V. Preventive man. (Jl ’20)
Orczy, E. His Majesty’s well-beloved. (Ap ’20)
France
Bailey, H: C. Barry Leroy. (Je ’20)
Brooks, C: S. Luca Sarto. (Ap ’20)
Iceland
Hewlett, M. H: Light heart. (Jl ’20)
Hewlett, M. H: Outlaw. (Ap ’20)
Italy
Drummond, H. Maker of saints. (S ’20)
Vorse, M. M. Ninth man. (O ’20)
Jerusalem
Mapu, A. Sorrows of Noma. (Ap ’20)
Middle ages
Ross, R. Revels of Orsera. (F ’21)
United States
Daviess, M. T. Matrix. (Ap ’20)
Fox, J:, jr. Erskine Dale, pioneer. (D ’20)
Gregg, F. M. Founding of a nation. (O ’20)
Lynn, M. Free soil. (Ja ’21)
Maule, M. K. Prairie-schooner princess. (N ’20)
Peadexter, H. Red belts. (Mr ’20)
Shafer, D. C. Barent Creighton. (S ’20)
White, S. E: Rose dawn. (D ’20)
Horse racing
Richards, G. Double life. (F ’21)
Yates, L. B. Autobiography of a race horse. (Ag ’20)
Humor and satire
Ashford, D. Daisy Ashford: her book. (S ’20)
Beerbohm, M. Seven men. (D ’20)
Cannan, G. Windmills. (Ag ’20)
Darlington, W. A. Alf’s button. (S ’20)
Dickson, H. Old Reliable in Africa. (D ’20)
Dodge, H: I. Skinner makes it fashionable. (Je ’20)
Huxley, A. L. Limbo. (Ag ’20)
Irwin, W. A. Trimmed with red. (Ag ’20)
Kelland, C. B. Efficiency Edgar. (Jl ’20)
Leacock, S. B. Winsome Winnie. (Ja ’21)
Mackall, L. Scrambled eggs. (D ’20)
Mackenzie, C. Poor relations. (Mr ’20)
Marshall, R. Enchanted golf clubs. (Mr ’20)
Putnam, N. It pays to smile. (F ’21)
Street, J. L. Sunbeams, Inc. (D ’20)
Ullman, A. E: “Line’s busy.” (N ’20)
Immigrants in America
Cournos, J: Mask. (Mr ’20)
Yezierska, A. Hungry hearts. (D ’20)
Industrial conditions
Hall, H. S. Steel preferred. (N ’20)
Jewish life
Pearl, B. Sarah and her daughter. (Jl ’20)
Journalism
Dodge, L: Whispers. (Je ’20)
George, W. L. Caliban. (O ’20)
Macaulay, R. Potterism. (D ’20)
Williams, S. C. Unconscious crusader. (Je ’20)
Williams, W. W. Goshen street. (N ’20)
Law and lawyers
Train, A. C. Tutt and Mr Tutt. (Je ’20)
Legends and folktales
Shedlock, M. L. Eastern stories and legends. (D ’20)
Flemish
Coster, C: T. H. de. Flemish legends. (N ’20)
Irish
MacManus, S. Top o ’ the mornin’. (N ’20)
Japanese
Ozaki, Y. T. Romances of old Japan. (Mr ’20)
Letters (stories in letter form)
Lucas, E: V. Verena in the midst. (N ’20)
Ridsdale, K. Gate of fulfillment. (Je ’20)
Locality, Novels of
Adirondacks
Longstreth, T: M. Mac of Placid. (O ’20)
Africa
Battersby, H: F. P. Edge of doom. (Je ’20)
Benoit, P. Atlantida. (S ’20)
Burroughs, E. R. Tarzan the untamed. (O ’20)
Dickson, H. Old Reliable in Africa. (D ’20)
Stockley, C. Pink gods and blue demons. (Ag ’20)
Alaska
Rutzebeck, H. Alaska man ’ s luck. (Ja ’21)
Arizona
Noyes, A. Beyond the desert. (D ’20)
Armenia
Inchbold, A. C. Love and the crescent. (S ’20)
Boston
Brown, A. Wind between the worlds. (S ’20)
Brazil
Elliott, L. W. Black gold. (F ’21)
Graça Aranha, J. P. da. Canaan. (Ap ’20)
California
Chase, J. S. Penance of Magdalena. (Ag ’20)
Porter, R. N. Girl from Four Corners. (My ’20)
White, S. E: Rose dawn. (D ’20)
Canada
Binns, O. Mating in the wilds. (S ’20)
Cody, H. A. Glen of the high north. (O ’20)
Cullum, R. Heart of Unaga. (N ’20)
Curwood, J. O. Valley of silent men. (O ’20)
Durkin, D. Heart of Cherry McBain. (N ’20)
Footner, H. Fur bringers. (D ’20)
Gibbon, J: M. Conquering hero. (N ’20)
Kendall, R. S. Luck of the mounted. (F ’21)
McKowan, E. Graydon of the Windermere. (F ’21)
Pinkerton, K. S. and R. E. Long traverse. (S ’20)
Pinkerton, K. S. and R. E. Penitentiary Post. (Ag ’20)
Sidgwick, C., and Garstin, C. Black knight. (O ’20)
Stringer, A. J: A. Prairie mother. (N ’20)
White, S: A. Ambush. (N ’20)
Cape Cod
Cooper, J. A. Tobias o ’ the light. (Ag ’20)
Lincoln, J. C. Portygee. (Je ’20)
Chicago
Borden, T. M. Romantic woman. (My ’20)
Webster, H: K. Mary Wollaston. (D ’20)
China
Lamb, H. Marching sands. (My ’20)
Merwin, S: Hills of Han. (D ’20)
Miln, L. J. Mr Wu. (Ap ’20)
Weale, B. L. P. Wang the ninth. (Ja ’21)
Connecticut
Minnigerode, M. Laughing house. (D ’20)
Egypt
Bradley, M. Fortieth door. (Ap ’20)
Weigall, A. E: P. B. Madeline of the desert. (D ’20)
England
Anstruther, E. H. A. Husband. (Jl ’20)
Baxter, A. B. Parts men play. (Ja ’21)
Johnston, H. H. Mrs Warren’s daughter. (Jl ’20)
O’Sullivan, Mrs D. Mr Dimock. (F ’21)
England (London)
Brunner, E. Celia and her friends. (S ’20)
Brunner, E. Celia once again. (S ’20)
Cannan, G. Time and eternity. (My ’20)
Desmond, S. Passion. (Jl ’20)
Ervine, S. G. Foolish lovers. (Jl ’20)
Pryde, A. Marqueray’s duel. (Jl ’20)
Woolf, V. Night and day. (N ’20)
England (Oxford)
Morley, C. D. Kathleen. (Ap ’20)
England (provincial and rural)
Ayscough, J:, pseud. Abbotscourt. (Je ’20)
Benson, E: F: “Queen Lucia.” (S ’20)
Blundell, M. E. Beck of Beckford. (F ’21)
Brighouse, H. Marbeck inn. (Ap ’20)
Buckrose, J. E., pseud. Young hearts. (S ’20)
Easton, D. Golden bird. (S ’20)
Galsworthy, J: In chancery. (N ’20)
Gambier, K. Girl on the hilltop. (Ag ’20)
Hicks Beach, S. E. Shuttered doors. (Je ’20)
Hocking, J. Passion for life. (Jl ’20)
Kaye-Smith, S. Tamarisk town. (Jl ’20)
Lorimer, N. O. With other eyes. (O ’20)
Marshall, A. Many Junes. (Je ’20)
Merrick, H. Mary-girl. (S ’20)
Oldmeadow, E. J. Coggin. (Mr ’20)
Pedler, M. House of dreams-come-true. (S ’20)
Phillpotts, E. Miser’s money. (Je ’20)
Rainsford, W. H. That girl March. (F ’21)
Stevenson, G: Benjy. (Je ’20)
Swinnerton, F. A. September. (Mr ’20)
Thurston, E. T. Sheepskins and grey russet. (Je ’20)
Turner, J: H. Place in the world. (My ’20)
Vachell, H. A. Whitewash. (Jl ’20)
Ward, M. A. Harvest. (My ’20)
Whitham, G. I. St John of Honeylea. (Je ’20)
Far East
Rideout, H: M. Foot-path way. (Je ’20)
Florida
Oyen, H: Plunderer. (Je ’20)
Germany
Henry, S. O. Villa Elsa. (Je ’20)
India
Comfort, W. L., and Dost, Z. K. Son of power. (D ’20)
Lowis, C. C. Four blind mice. (D ’20)
Mundy, T. Told in the East. (F ’21)
Savi, E. W. When the blood burns. (D ’20)
Tracy, L: Sirdar’s sabre. (D ’20)
Ireland
Carleton, W: Stories of Irish life. (Mr ’20)
Ervine, S. G. Foolish lovers. (Jl ’20)
Galway, C. Towards the dawn. (D ’20)
Hannay, J. O. Up, the rebels! (Mr ’20)
Hinkson, K. Love of brothers. (D ’20)
MacFarlan, A. Inscrutable lovers. (Ap ’20)
MacGill, P. Maureen. (Je ’20)
MacNamara, B. Clanking of chains. (My ’20)
O’Duffy, E. Wasted island. (N ’20)
O’Kelly, S. Golden barque; and The weaver ’ s grave. (D ’20)
O’Riordan, C. O. Adam of Dublin. (Ja ’21)
Page, G. Paddy-the-next-best-thing. (F ’21)
Poore, I. M. Rachel Fitzpatrick. (S ’20)
Somerville, E. A. O., and Martin, V. F.
Mount Music. (Je ’20)
Italy
Forster, E: M. Where angels fear to tread. (Ap ’20)
Hudson, S. Richard Kurt. (Jl ’20)
Kennard, J. S. Memmo. (Ja ’21)
Lombardi, C. Cry of youth. (Jl ’20)
Kansas
Lynn, M. Free soil. (Ja ’21)
Kentucky
Buck, C: N. Tempering. (Je ’20)
Louisiana
Perry, S. G: Palmetto. (N ’20)
Mexico
Grogan, G. William Pollok, and other tales. (Je ’20)
Minnesota
Lewis, S. Main street. (N ’20)
Mississippi river
Spears, R. S. River prophet. (Ag ’20)
Nebraska
Maule, M. K. Prairie-schooner princess. (N ’20)
New England
Bassett, S. W. Wall between. (D ’20)
Brown, A. Homespun and gold. (Ja ’21)
Gerould, G. H. Youth in Harley. (O ’20)
Howells, W: D. Vacation of the Kelwyns. (N ’20)
Spofford, H. E. Elder’s people. (My ’20)
Williams, W. W. Goshen street. (N ’20)
New York (city)
Austin, M. 26 Jayne street. (Je ’20)
Bullard, A. Stranger. (Jl ’20)
Chamberlain, G: A. Taxi. (Ag ’20)
Corbett, E. F. Puritan and pagan. (Ja ’21)
Duganne, P. Prologue. (N ’20)
Forman, H: J. Fire of youth. (Ap ’20)
Frank, W. D: Dark mother. (D ’20)
Hudson, H:, 2d, pseud. Spendthrift town. (D ’20)
Hughes, R. What’s the world coming to? (Jl ’20)
Pearl, B. Sarah and her daughter. (Jl ’20)
Raine, W: M. Big-town round-up. (F ’21)
Spadoni, A. Swing of the pendulum. (Ap ’20)
Wharton, E. N. Age of innocence. (N ’20)
New York (state)
White, G. Storm country Polly. (Jl ’20)
New Zealand
Mander, J. Story of a New Zealand river. (My ’20)
Ohio
Anderson, S. Poor white. (D ’20)
Anderton, D. Cousin Sadie. (F ’21)
Paris
Audoux, M. Marie Claire’s workshop. (D ’20)
Pennsylvania
Deland, M. W. Old Chester secret. (D ’20)
Martin, H. R. Schoolmaster of Hessville. (N ’20)
Morris, H. S. Hannah Bye. (Ja ’20)
Myers, A. B. Patchwork. (My ’20)
Philippine islands
Martin, M. W. Green god’s pavilion. (O ’20)
Rome
Richardson, N. Pagan fire. (Ja ’21)
San Francisco
Dobie, C: C. Blood red dawn. (Jl ’20)
Spadoni, A. Swing of the pendulum. (Ap ’20)
Scotland
Douglas, O. E. Penny plain. (D ’20)
Niven, F: J: Tale that is told. (D ’20)
Watson, R. Stronger than his sea. (F ’21)
South Africa
Dell, E. M. Top of the world. (N ’20)
Millin, S. G. Dark river. (D ’20)
Young, F. E. M. Almonds of life. (O ’20)
South Carolina
Oemler, M. Purple heights. (N ’20)
South seas
Barbour, R. H:, and Holt, H. P. Joan of the island. (Ag ’20)
Conrad, J. Rescue. (Je ’20)
Grimshaw, B. Terrible island. (D ’20)
Switzerland
In the mountains. (N ’20)
United States (middlewestern)
Dell, F. Moon-calf. (D ’20)
Howe, E. W. Anthology of another town. (Ja ’21)
Watts, M. S. Noonmark. (F ’21)
United States (northwestern)
Sinclair, B. W: Poor man ’ s rock. (D ’20)
United States (southern)
Griffiths, G. Lure of the manor. (Ag ’20)
Hoffman, M. E. Lindy Loyd. (O ’20)
Olmstead, F. Stafford’s Island. (Je ’20)
Ragsdale, L. Next-besters. (S ’20)
Sampson, E. S. Mammy’s white folks. (S ’20)
Turner, G: K. Hagar’s hoard. (N ’20)
United States (southwestern)
Barry, R: H. Fruit of the desert. (S ’20)
Bennet, R. A. Bloom of cactus. (Mr ’20)
Dunn, J. A. E. Dead man ’ s gold. (S ’20)
Gregory, J. Man to man. (D ’20)
Hooker, F. C. Long dim trail. (N ’20)
Shedd, G: C. Iron furrow. (Ap ’20)
White, S. E: Killer. (Jl ’20)
United States (western)
Abbott, K. Wine o ’ the winds. (Ag ’20)
Bower, B. M., pseud. Quirt. (Jl ’20)
Brand, M. Trailin’! (Jl ’20)
Burt, K. N. Hidden Creek. (O ’20)
Coolidge, D. Wunpost. (N ’20)
Croy, H. Turkey Bowman. (N ’20)
Dorrance, E. A. and J. F. Glory rides the range. (Jl ’20)
Dunn, J. A. E. Turquoise Cañon. (Ap ’20)
Grey, Z. Man of the forest. (Mr ’20)
Hendryx, J. B. Gold girl. (Ag ’20)
Lynde, F. Girl, a horse and a dog. (S ’20)
Marshall, E. Voice of the pack. (Je ’20)
Raine, W: M. Oh, you Tex! (Ag ’20)
Richards, C. E. Tenderfoot bride. (F ’21)
Ritchie, R. W. Trails to Two Moons. (D ’20)
Titus, H. Last straw. (Jl ’20)
White, W: P. Hidden trails. (O ’20)
White, W: P. Lynch lawyers. (Jl ’20)
White, W: P. Paradise Bend. (F ’21)
Virginia
Bailey, T. Trumpeter swan. (D ’20)
Johnston, M. Sweet Rocket. (D ’20)
Wales
Evans, C. My neighbors. (My ’20)
Young, F. B. and E. B. Undergrowth. (Ja ’21)
Washington (state)
Kyne, P. B. Kindred of the dust. (Ag ’20)
West Virginia
Dillon, M. C. Farmer of Roaring Run. (Mr ’20)
Marriage
Borden, T. M. Romantic woman. (My ’20)
Byrne, D. Foolish matrons. (N ’20)
Couperus, L: M. A. Inevitable. (D ’20)
Edginton, H. M. Married life. (Ag ’20)
Hamilton, C. His friend and his wife. (Je ’20)
Harris, C. M. Happily married. (Ap ’20)
Kerr, S. Painted meadows. (Je ’20)
Widdemer, M. I’ve married Marjorie. (S ’20)
Wylie, I. A. R. Children of storm. (N ’20)
Moving picture stories
Luther, M. L. Presenting Jane McRae. (S ’20)
Witwer, H. C: Kid Scanlan. (Ag ’20)
Witwer, H. C: There’s no base like home. (Ag ’20)
Musicians
Close, E. Cherry Isle. (D ’20)
Leadbitter, E. Rain before seven. (Jl ’20)
Mix, J. I. At fame’s gateway. (My ’20)
Schauffler, R. H. Fiddler’s luck. (Jl ’20)
Mystery stories
Allison, W: Secret of the sea. (Ap ’20)
Allison, W: Turnstile of night. (D ’20)
Benoit, P. Secret spring. (Je ’20)
Biss, G. Door of the unreal. (Ja ’21)
Brebner, P. J. Ivory disc. (Ag ’20)
Brown, E. A. That affair at St Peter’s. (Jl ’20)
Burt, K. Red lady. (Je ’20)
Camp, C: W. Gray mask. (Jl ’20)
Capes, B. E: J. Skeleton key. (Ag ’20)
