{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11340","name":"Solving Partition Problems Almost Always Requires Pushing Many Vertices Around","abstract":"A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G[A] and G[B] satisfy properties Pi_A and Pi_B, respectively. This so-called (Pi_A,Pi_B)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (Pi_A,Pi_B)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an \"almost correct\" bipartition (A',B'), and pushes appropriate vertices from A' to B' and vice versa to eventually arrive at a correct bipartition.\nIn this paper, we study whether (Pi_A,Pi_B)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where Pi_A is the set of P_3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and Pi_B is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP not subseteq coNP\/poly, (Pi_A,Pi_B)-Recognition admits a polynomial kernel if and only if H contains a graph of order at most 2. In both the kernelization and the lower bound results, we make crucial use of the pushing process.","keywords":["Fixed-parameter algorithms","Kernelization","Vertex-partition problems","Reduction rules","Cross-composition"],"author":[{"@type":"Person","name":"Kanj, Iyad","givenName":"Iyad","familyName":"Kanj","affiliation":"School of Computing, DePaul University Chicago, USA"},{"@type":"Person","name":"Komusiewicz, Christian","givenName":"Christian","familyName":"Komusiewicz","affiliation":"Fachbereich Mathematik und Informatik, Philipps-Universit\u00e4t Marburg, Germany","funding":"CK gratefully acknowledges support by the DFG, project MAGZ, KO 3669\/4-1."},{"@type":"Person","name":"Sorge, Manuel","givenName":"Manuel","familyName":"Sorge","affiliation":"Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel","funding":"MS gratefully acknowledges support by the People Programme (Marie Curie Actions) of the European Union\u2019s Seventh Framework Programme (FP7\/2007-2013) under REA grant agreement number 631163.11, by the Israel Science Foundation (grant number 551145\/14), and by the European Research Council (ERC) underthe European Union\u2019s Horizon 2020 research and innovation programme under grant agreement number 714704."},{"@type":"Person","name":"van Leeuwen, Erik Jan","givenName":"Erik Jan","familyName":"van Leeuwen","affiliation":"Department of Information and Computing Sciences, Utrecht University, The Netherlands"}],"position":51,"pageStart":"51:1","pageEnd":"51:14","dateCreated":"2018-08-14","datePublished":"2018-08-14","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kanj, Iyad","givenName":"Iyad","familyName":"Kanj","affiliation":"School of Computing, DePaul University Chicago, USA"},{"@type":"Person","name":"Komusiewicz, Christian","givenName":"Christian","familyName":"Komusiewicz","affiliation":"Fachbereich Mathematik und Informatik, Philipps-Universit\u00e4t Marburg, Germany","funding":"CK gratefully acknowledges support by the DFG, project MAGZ, KO 3669\/4-1."},{"@type":"Person","name":"Sorge, Manuel","givenName":"Manuel","familyName":"Sorge","affiliation":"Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel","funding":"MS gratefully acknowledges support by the People Programme (Marie Curie Actions) of the European Union\u2019s Seventh Framework Programme (FP7\/2007-2013) under REA grant agreement number 631163.11, by the Israel Science Foundation (grant number 551145\/14), and by the European Research Council (ERC) underthe European Union\u2019s Horizon 2020 research and innovation programme under grant agreement number 714704."},{"@type":"Person","name":"van Leeuwen, Erik Jan","givenName":"Erik Jan","familyName":"van Leeuwen","affiliation":"Department of Information and Computing Sciences, Utrecht University, The Netherlands"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ESA.2018.51","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6315","volumeNumber":112,"name":"26th Annual European Symposium on Algorithms (ESA 2018)","dateCreated":"2018-08-14","datePublished":"2018-08-14","editor":[{"@type":"Person","name":"Azar, Yossi","givenName":"Yossi","familyName":"Azar"},{"@type":"Person","name":"Bast, Hannah","givenName":"Hannah","familyName":"Bast"},{"@type":"Person","name":"Herman, Grzegorz","givenName":"Grzegorz","familyName":"Herman"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11340","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6315"}}}