Parameterized Complexity of Independent Set in H-Free Graphs
In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously K_{1, 4}, any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning C_4-free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey's theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.
Parameterized Algorithms
Independent Set
H-Free Graphs
Theory of computation~Fixed parameter tractability
17:1-17:13
Regular Paper
É. B. is supported by the LABEX MILYON (ANR-10- LABX-0070) of Université de Lyon, within the program "Investissements d’Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). N. B. and P. C. are supported by the ANR Project DISTANCIA (ANR-17-CE40-0015) operated by the French National Research Agency (ANR).
A full version of the paper is available at [É. Bonnet et al., 2018], https://arxiv.org/abs/1810.04620.
Édouard
Bonnet
Édouard Bonnet
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Nicolas
Bousquet
Nicolas Bousquet
CNRS, G-SCOP laboratory, Grenoble-INP, France
Pierre
Charbit
Pierre Charbit
Université Paris Diderot, IRIF, France
Stéphan
Thomassé
Stéphan Thomassé
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France, Institut Universitaire de France
Rémi
Watrigant
Rémi Watrigant
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
10.4230/LIPIcs.IPEC.2018.17
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É. Bonnet, N. Bousquet, P. Charbit, S. Thomassé, and R. Watrigant
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