TY - GEN

T1 - Solving partition problems almost always requires pushing many vertices around

AU - Kanj, Iyad

AU - Komusiewicz, Christian

AU - Sorge, Manuel

AU - Van Leeuwen, Erik Jan

N1 - Publisher Copyright:
© Iyad Kanj, Christian Komusiewicz, Manuel Sorge, and Erik Jan van Leeuwen.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Cross-composition

KW - Fixed-parameter algorithms

KW - Kernelization

KW - Reduction rules

KW - Vertex-partition problems

UR - http://www.scopus.com/inward/record.url?scp=85052538573&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2018.51

DO - 10.4230/LIPIcs.ESA.2018.51

M3 - Conference contribution

AN - SCOPUS:85052538573

SN - 9783959770811

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 26th European Symposium on Algorithms, ESA 2018

A2 - Bast, Hannah

A2 - Herman, Grzegorz

A2 - Azar, Yossi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 26th European Symposium on Algorithms, ESA 2018

Y2 - 20 August 2018 through 22 August 2018

ER -