eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-14
51:1
51:14
10.4230/LIPIcs.ESA.2018.51
article
Solving Partition Problems Almost Always Requires Pushing Many Vertices Around
Kanj, Iyad
1
Komusiewicz, Christian
2
Sorge, Manuel
3
van Leeuwen, Erik Jan
4
School of Computing, DePaul University Chicago, USA
Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel
Department of Information and Computing Sciences, Utrecht University, The Netherlands
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol112-esa2018/LIPIcs.ESA.2018.51/LIPIcs.ESA.2018.51.pdf
Fixed-parameter algorithms
Kernelization
Vertex-partition problems
Reduction rules
Cross-composition