Parameterized and Exact Computation: 6th International Symposium, IPEC 2011, Saarbrücken, Germany, September 6-8, 2011. Revised Selected Papers PDF
Preview Parameterized and Exact Computation: 6th International Symposium, IPEC 2011, Saarbrücken, Germany, September 6-8, 2011. Revised Selected Papers
Lecture Notes in Computer Science 7112 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 MadhuSudan MicrosoftResearch,Cambridge,MA,USA DemetriTerzopoulos UniversityofCalifornia,LosAngeles,CA,USA DougTygar UniversityofCalifornia,Berkeley,CA,USA GerhardWeikum MaxPlanckInstituteforInformatics,Saarbruecken,Germany Dániel Marx Peter Rossmanith (Eds.) Parameterized and Exact Computation 6th International Symposium, IPEC 2011 Saarbrücken, Germany, September 6-8, 2011 Revised Selected Papers 1 3 VolumeEditors DánielMarx MTASZTAKI Lágymányosiu.11 1111Budapest,Hungary E-mail:[email protected] PeterRossmanith RWTHAachen LehrgebietTheoretischeInformatik Ahornstraße55 52056Aachen,Germany E-mail:[email protected] ISSN0302-9743 e-ISSN1611-3349 ISBN978-3-642-28049-8 e-ISBN978-3-642-28050-4 DOI10.1007/978-3-642-28050-4 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2011945507 CRSubjectClassification(1998):F.2,G.1-2,I.3.5,F.1,G.4,E.1,I.2.8 LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues ©Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliable toprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readybyauthor,dataconversionbyScientificPublishingServices,Chennai,India Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface TheInternationalSymposiumonParameterizedandExactComputation(IPEC, formerly IWPEC) is an international symposium series that covers research in all aspects of parameterized and exact algorithms and complexity. Started in 2004 as a biennial workshop, it became an annual event in 2008. This volume contains the papers presented at IPEC 2011: the 6th Inter- national Symposium on Parameterized and Exact Computation held during September 6–8, 2011 in Saarbru¨cken. The symposium was part of ALGO 2011, whichalsohostedthe19thEuropeanSymposiumonAlgorithms(ESA2011),the 11thWorkshoponAlgorithms for Bioinformatics(WABI 2011),the 11thWork- shop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2011), the 9th Workshop on Approximation and Online Algorithms(WAOA2011),andthe7thInternationalSymposiumonAlgorithms forSensorSystems,WirelessAdHocNetworksandAutonomousMobileEntities (ALGOSENSORS).ThefivepreviousmeetingsoftheIPEC/IWPECserieswere held in Bergen, Norway (2004), Zu¨rich, Switzerland (2006), Victoria, Canada (2008), Copenhagen, Denmark (2009), and Chennai, India (2010). TheIPEC2011plenarykeynotetalkwasgivenbyMartinGrohe(Humboldt- Universita¨t zu Berlin) on “Excluding Topological Subgraphs.” We had two ad- ditional invited tutorial speakers: Hans L. Bodlaender (Utrecht University, The Netherlands) speaking on kernels and Fedor V. Fomin (University of Bergen, Norway) speaking on width measures. We thank the speakers for accepting our invitation. Inresponsetothecallforpapers,40papersweresubmitted.Eachsubmission wasreviewedbyatleastthree,andonaverage3.8,reviewers.Thereviewerswere eitherProgramCommitteemembersorinvitedexternalreviewers.TheProgram Committee held electronic meetings using the EasyChair system, went through extensive discussions,and selected 21 of the submissions for presentationat the symposiumandinclusioninthisLNCSvolume.TheProgramCommitteedecided to awardthe ExcellentStudent PaperAward to the paper “A Faster Algorithm for Dominating Set Analyzed by the Potential Method” by Yoichi Iwata (The University of Tokyo). We thank Frances Rosamond for sponsoring the award. We are very grateful to the ProgramCommittee, and the external reviewers theycalledon,forthehardworkandexpertisewhichtheybroughttothedifficult selectionprocess.Wealsowishtothankalltheauthorswhosubmittedtheirwork for our consideration. October 2011 Da´niel Marx Peter Rossmanith Organization Program Committee Hans L. Bodlaender Utrecht University, The Netherlands Rod Downey Victoria University of Wellington, New Zealand David Eppstein University of California, Irvine, USA Pinar Heggernes University of Bergen, Norway Thore Husfeldt IT University of Copenhagen, Denmark and Lund University, Sweden Iyad Kanj DePaul University, Chicago, USA Dieter Kratsch Universit´e Paul Verlaine, Metz, France Daniel Lokshtanov University of Bergen, Norway Da´niel Marx (Co-chair) Budapest University of Technology and Economics, Hungary Rolf Niedermeier TU Berlin, Germany Peter Rossmanith (Co-chair) RWTH Aachen University, Germany Ulrike Stege University of Victoria, Canada St´ephan Thomass´e E´quipe AlGCo, LIRMM-Universit´e Montpellier 2, France Ryan Williams IBM Almaden Research Center, USA Gerhard Woeginger TU Eindhoven, The Netherlands Additional Reviewers Abu-Khzam, Faisal Hartung, Sepp Otachi, Yota Bevern, Ren´e van Hermelin, Danny Paulusma, Dani¨el Bredereck, Robert Hlinˇeny´, Petr Pilipczuk, Michal(cid:4) Cao, Yixin Hof, Pim van ’t Reidl, Felix Chen, Jiehua Hu¨ffner, Falk Rooij, Johan M. M. van Creignou, Nadia Jansen, Bart Saurabh, Saket Cygan, Marek Kern, Walter Scott, Allan Damaschke, Peter Komusiewicz, Christian Sikdar, Somnath Dell, Holger Kowaluk, Miroslaw Sorge, Manuel Fernau, Henning Kratsch, Stefan Szeider, Stefan Fleischer, Rudolf Langer, Alexander Taslaman, Nina Flum, Jo¨rg Liedloff, Mathieu Thilikos, Dimitrios M. Fomin, Fedor V. Lonergan, Steven Wahlstro¨m, Magnus Gaspers, Serge McCartin, Catherine Watt, Nathaniel Golovach, Petr Mnich, Matthias Weller, Mathias Grandoni, Fabrizio Nichterlein, Andr´e Table of Contents On Multiway Cut Parameterizedabove Lower Bounds ................ 1 Marek Cygan, Marcin Pilipczuk, Michal(cid:2) Pilipczuk, and Jakub Onufry Wojtaszczyk ParameterizedComplexity of Firefighting Revisited .................. 13 Marek Cygan, Fedor V. Fomin, and Erik Jan van Leeuwen ParameterizedComplexity in Multiple-Interval Graphs: Domination .... 27 Minghui Jiang and Yong Zhang A Faster Algorithm for Dominating Set Analyzed by the Potential Method......................................................... 41 Yoichi Iwata Contracting Graphs to Paths and Trees............................. 55 Pinar Heggernes, Pim van ’t Hof, Benjamin L´evˆeque, Daniel Lokshtanov, and Christophe Paul Increasing the Minimum Degree of a Graph by Contractions........... 67 Petr A. Golovach, Marcin Kamin´ski, Dani¨el Paulusma, and Dimitrios M. Thilikos Planar Disjoint-Paths Completion.................................. 80 Isolde Adler, Stavros G. Kolliopoulos, and Dimitrios M. Thilikos Sparse Solutions of Sparse Linear Systems: Fixed-Parameter Tractability and an Application of Complex Group Testing............ 94 Peter Damaschke New Upper Bounds for MAX-2-SAT and MAX-2-CSP w.r.t. the Average Variable Degree .......................................... 106 Alexander Golovnev Improved Parameterized Algorithms for above Average Constraint Satisfaction ..................................................... 118 Eun Jung Kim and Ryan Williams On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal...................................................... 132 Bart M.P. Jansen and Stefan Kratsch Kernel Bounds for Path and Cycle Problems ........................ 145 Hans L. Bodlaender, Bart M.P. Jansen, and Stefan Kratsch VIII Table of Contents On the Hardness of Losing Width.................................. 159 Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal(cid:2) Pilipczuk, and Saket Saurabh Safe Approximation and Its Relation to Kernelization ................ 169 Jiong Guo, Iyad Kanj, and Stefan Kratsch Simpler Linear-Time Kernelization for Planar Dominating Set ......... 181 Torben Hagerup Linear-Time Computation of a Linear Problem Kernel for Dominating Set on Planar Graphs ............................................ 194 Ren´e van Bevern, Sepp Hartung, Frank Kammer, Rolf Niedermeier, and Mathias Weller Tight Complexity Bounds for FPT Subgraph Problems Parameterized by Clique-Width................................................. 207 Hajo Broersma, Petr A. Golovach, and Viresh Patel Finding Good Decompositions for Dynamic Programming on Dense Graphs ......................................................... 219 Eivind Magnus Hvidevold, Sadia Sharmin, Jan Arne Telle, and Martin Vatshelle ParameterizedMaximum Path Coloring ............................ 232 Michael Lampis On Cutwidth Parameterizedby Vertex Cover........................ 246 Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal(cid:2) Pilipczuk, and Saket Saurabh Twin-Cover:Beyond Vertex Cover in ParameterizedAlgorithmics ...... 259 Robert Ganian Author Index.................................................. 273 On Multiway Cut Parameterized (cid:2) above Lower Bounds Marek Cygan1, Marcin Pilipczuk1, Michal(cid:4) Pilipczuk1, and Jakub Onufry Wojtaszczyk2 1 Instituteof Informatics, University of Warsaw, Poland {cygan@,malcin@,mp248287@students.}mimuw.edu.pl 2 Google Inc., Cracow, Poland [email protected] Abstract. Inthispaperweconsidertwoabove lower boundparameter- izations of the Node Multiway Cut problem — above the maximum separating cut and abovea natural LP-relaxation — and provethem to be fixed-parameter tractable. Our results imply O∗(4k) algorithms for Vertex Cover above Maximum Matching and Almost 2-SAT as well as an O∗(2k) algorithm for Node Multiway Cut with a stan- dard parameterization by the solution size, improving previous bounds for theseproblems. 1 Introduction The study of cuts and flows is one of the most active fields in combinatorial optimization. However, while the simplest case, where we seek a cut separating two given vertices of a graph, is algorithmically tractable, the problem becomes hard as soon as one starts to deal with multiple terminals. For instance, given threeverticesinagraphitisNP-hardtodecidewhatisthesmallestsizeofacut that separates every pair of them (see [4]). The generalization of this problem — the well-studied Node Multiway Cut problem — asks for the size of the smallest set separating a given set of terminals. The formal definition is as follows: Node Multiway Cut Input: A graph G=(V,E), a set T ⊆V of terminals and an integer k. Question:Does there exista setX ⊆V \T of sizeatmostk suchthatany path between two different terminals intersects X? For various approaches to this problem we refer the reader for instance to [8,2,4,13]. Beforedescribingourresults,letusdiscussthe methodologywewillbe work- ing with. We will be studying Node Multiway Cut (and several other prob- lems) from the parameterized complexity point of view. Note that since the (cid:2) ThefirsttwoauthorswerepartiallysupportedbyNationalScienceCentregrantno. N206 567140 and Foundation for Polish Science. D.MarxandP.Rossmanith(Eds.):IPEC2011,LNCS7112,pp.1–12,2012. (cid:2)c Springer-VerlagBerlinHeidelberg2012 2 M. Cygan et al. solution to our problem is a set of k vertices and it is easy to verify whether a solution is correct, we can solve the problem by enumerating and verifying all the O(|V|k)sets ofsize k. Therefore,for every fixedvalue of k,our problemcan be solved in polynomial time. This approach, however, is not feasible even for, say, k =10. The idea of parameterized complexity is to try to split the (usually exponential)dependencyonkfromthe(hopefullyuniformlypolynomial)depen- dency on|V| — so we look for an algorithmwhere the degree ofthe polynomial does not depend on k, e.g., an O(Ck|V|O(1)) algorithm for a constant C. Formally, a parameterized problem Q is a subset of Σ∗ ×N for some finite alphabetΣ,wheretheintegeristheparameter.Wesaythattheproblemisfixed parametertractable(FPT)ifthereexistsanalgorithmsolvinganyinstance(x,k) in time f(k)poly(|x|) for some (usually exponential) computable function f. It is known that a decidable problem is FPT iff it is kernelizable: a kernelization algorithm for a problem Q takes an instance (x,k) and in time polynomial in |x|+k produces an equivalent instance (x(cid:4),k(cid:4)) (i.e., (x,k) ∈ Q iff (x(cid:4),k(cid:4)) ∈ Q) suchthat |x(cid:4)|+k(cid:4) ≤g(k) for some computable function g.The function g is the size of the kernel, and if it is polynomial, we say that Q admits a polynomial kernel. The reader is invited to refer to now classical books by Downey and Fellows [5], Flum and Grohe [7] and Niedermeier [15]. The typical parameterization takes the solution size as the parameter. For instance, Chen et al. [2] have shown an algorithm solving Node Multiway Cut in time O(4knO(1)), improving upon the previous result of Daniel Marx [13]. However, in many cases it turns out we have a natural lower bound on the solution size — for instance, in the case of the Vertex Cover problem the cardinality of the maximal matching is such a lower bound. It can happen that this lower bound is large — rendering algorithms parameterized by the solutionsizeimpractical.Forsomeproblems,betteranswershavebeenobtained by introducingthe socalledparameterization above guaranteed value,i.e.taking astheparameterthedifferencebetweentheexpectedsolutionsizeandthelower bound. The idea was first proposed in [12]. An overview of this currently active researcharea can be found in the introduction to [10]. We will consider two natural lower bounds for Node Multiway Cut — the separating cut and the LP-relaxation solution. Let I = (G,T,k) be a Node Multiway Cut instanceandlets=|T|.Byaminimum solutiontoI wemean a set X ⊆V \T of minimum cardinality that disconnects the terminals, even if |X|>k. Foraterminalt∈T asetS ⊆V\T isaseparatingcut(alsocalledanisolating cut) of t if t is disconnected from T \{t} in G[V \S] (the subgraph induced by V \S). Let m(I,t) be the size of a minimum isolating cut of t. Notice that for anytthevaluem(I,t)canbefoundinpolynomialtimeusingstandardmax-flow techniques. Moreover, the maximum of these values over all t is a lower bound for the size of the minimum solution to I — any solution X has, in particular, to separate t from all the other terminals. Now we consider a different approach to the problem, stemming from linear programming.LetP(I)denotethesetofallsimplepathsconnectingtwodifferent On Multiway Cut Parameterized above Lower Bounds 3 terminals in G. Garg et al. [8] gave a 2-approximation algorithm for Node Multiway Cut using the following natural LP-relaxation: (cid:2) minimize dv (1) v∈V\T (cid:2) subject to dv ≥1 ∀P ∈P(I) v∈P∩(V\T) dv ≥0 ∀v ∈V \T In other words, the LP-relaxation asks to assign for each vertex v ∈ V \T a non-negative weight dv, such that the distance between pair of terminals, with respect to the weights dv, is at least one. This is indeed a relaxation of the original problem — if we restrict the values dv to be integers, we obtain the original Node Multiway Cut . The above LP-relaxationhas exponential number of constraints, as P(I) can be exponentially big in the input size. However, the optimal solution for this LP-relaxation can be found in polynomial time either using separation oracle and ellipsoid method or by solving an equivalent linear program of polynomial size (see [8] for details). By LP(I) we denote the cost of the optimal solution of the LP-relaxation (1). As the LP-relaxation is less restrictive than the original Node Multiway Cut problem, LP(I) is indeed a lower bound on the size of the minimum solution. We can now define two above lower bound parameters: L(I) = k − LP(I) and C(I) = k −maxt∈T m(I,t), and denote by NMWC-a-LP (Node Mul- tiway Cut above LP-relaxation) and NMWC-a-Cut (Node Multiway CutaboveMaximum Separating Cut)theNodeMultiway Cut problem parameterized by L(I) and C(I), respectively. WesaythataparameterizedproblemQisinXP,ifthereexistsanalgorithm solving any instance (x,k) in time |x|f(k) for some computable function f, i.e., polynomialforanyconstantvalueofk.TheNMWC-a-Cutproblemwasdefined and shown to be in XP by Razgon in [17]. Ourresults. InSection2,usingtheideasofXiao[20]andbuildinguponanalysis of the LP relaxation by Guillemot [9], we prove a Node Multiway Cut in- stanceI canbe solvedinO∗(4L(I))time1,whicheasilyyields anO∗(2C(I))-time algorithm. Both algorithms run in polynomial space. Consequently we prove both NMWC-a-LP and NMWC-a-Cut problems to be FPT, solving an open problem of Razgon [17]. Observe that if C(I) > k the answer is trivially nega- tive, hence as a by-product we obtain an O∗(2k) time algorithm for the Node Multiway Cut problem, improving the previously best known O∗(4k) time algorithm by Chen et al. [2]. By considering a line graphof the input graph,it is easy to see that an edge- deletionvariantofMultiway Cutiseasierthanthenode-deletionone,andour results hold also for the edge-deletion variant. We note that the edge-deletion 1 O∗()istheO()notationwithsuppressedfactorspolynomialinthesizeoftheinput.
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