Solving Partition Problems Almost Always Requires Pushing Many Vertices Around
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.
In 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.
Fixed-parameter algorithms
Kernelization
Vertex-partition problems
Reduction rules
Cross-composition
Theory of computation~Parameterized complexity and exact algorithms
Theory of computation~Algorithm design techniques
51:1-51:14
Regular Paper
Iyad
Kanj
Iyad Kanj
School of Computing, DePaul University Chicago, USA
Christian
Komusiewicz
Christian Komusiewicz
Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
CK gratefully acknowledges support by the DFG, project MAGZ, KO 3669/4-1.
Manuel
Sorge
Manuel Sorge
Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel
MS gratefully acknowledges support by the People Programme (Marie Curie Actions) of the European Union’s 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’s Horizon 2020 research and innovation programme under grant agreement number 714704.
Erik Jan
van Leeuwen
Erik Jan van Leeuwen
Department of Information and Computing Sciences, Utrecht University, The Netherlands
10.4230/LIPIcs.ESA.2018.51
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Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan van Leeuwen
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