Solving partition problems almost always requires pushing many vertices around

Iyad Kanj, Christian Komusiewicz, Manuel Sorge, Erik Jan Van Leeuwen

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations


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 ΠA and ΠB, respectively. This so-called (πA, πB)-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 (πA, πB)-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 (πA, πB)-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 πA is the set of P3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and πB is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP ⊈ coNP/poly, (πA, πB)-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.

Original languageEnglish
Title of host publication26th European Symposium on Algorithms, ESA 2018
EditorsHannah Bast, Grzegorz Herman, Yossi Azar
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770811
StatePublished - 1 Aug 2018
Event26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland
Duration: 20 Aug 201822 Aug 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference26th European Symposium on Algorithms, ESA 2018


  • Cross-composition
  • Fixed-parameter algorithms
  • Kernelization
  • Reduction rules
  • Vertex-partition problems

ASJC Scopus subject areas

  • Software


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