Quasi-polynomial time approximation schemes for the maximum weight independent set problem in h-free graphs

Maria Chudnovsky, Marcin Pilipczuk, Michał Pilipczuk, Stéphan Thomassé

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n1−ε for any ε > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction. In H-free graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever H contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of H is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomial-time solvability results for small graphs H such as P5, P6, the claw, or the fork. We prove that for every such “possibly tractable” graph H there exists an algorithm that, given an Hfree graph G and an accuracy parameter ε > 0, finds an independent set in G of cardinality within a factor of (1 − ε) of the optimum in time exponential in a polynomial of log |V (G)| and ε1. That is, we show that for every graph H for which Maximum Independent Set is not known to be APX-hard in H-free graphs, the problem admits a quasi-polynomial time approximation scheme in this graph class. Our algorithm works also in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set.

Original languageEnglish (US)
Title of host publication31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
EditorsShuchi Chawla
PublisherAssociation for Computing Machinery
Pages2260-2278
Number of pages19
ISBN (Electronic)9781611975994
StatePublished - Jan 1 2020
Event31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States
Duration: Jan 5 2020Jan 8 2020

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume2020-January

Conference

Conference31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
CountryUnited States
CitySalt Lake City
Period1/5/201/8/20

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Chudnovsky, M., Pilipczuk, M., Pilipczuk, M., & Thomassé, S. (2020). Quasi-polynomial time approximation schemes for the maximum weight independent set problem in h-free graphs. In S. Chawla (Ed.), 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 (pp. 2260-2278). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 2020-January). Association for Computing Machinery.