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An Algorithmic Weakening of the Erdős-Hajnal Conjecture

Authors: Édouard Bonnet, Stéphan Thomassé, Xuan Thang Tran, and Rémi Watrigant

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

We study the approximability of the Maximum Independent Set (MIS) problem in H-free graphs (that is, graphs which do not admit H as an induced subgraph). As one motivation we investigate the following conjecture: for every fixed graph H, there exists a constant δ > 0 such that MIS can be n^{1-δ}-approximated in H-free graphs, where n denotes the number of vertices of the input graph. We first prove that a constructive version of the celebrated Erdős-Hajnal conjecture implies ours. We then prove that the set of graphs H satisfying our conjecture is closed under the so-called graph substitution. This, together with the known polynomial-time algorithms for MIS in H-free graphs (e.g. P₆-free and fork-free graphs), implies that our conjecture holds for many graphs H for which the Erdős-Hajnal conjecture is still open. We then focus on improving the constant δ for some graph classes: we prove that the classical Local Search algorithm provides an OPT^{1-1/t}-approximation in K_{t, t}-free graphs (hence a √{OPT}-approximation in C₄-free graphs), and, while there is a simple √n-approximation in triangle-free graphs, it cannot be improved to n^{1/4-ε} for any ε > 0 unless NP ⊆ BPP. More generally, we show that there is a constant c such that MIS in graphs of girth γ cannot be n^{c/(γ)}-approximated. Up to a constant factor in the exponent, this matches the ratio of a known approximation algorithm by Monien and Speckenmeyer, and by Murphy. To the best of our knowledge, this is the first strong (i.e., Ω(n^δ) for some δ > 0) inapproximability result for Maximum Independent Set in a proper hereditary class.

Cite as

Édouard Bonnet, Stéphan Thomassé, Xuan Thang Tran, and Rémi Watrigant. An Algorithmic Weakening of the Erdős-Hajnal Conjecture. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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  author =	{Bonnet, \'{E}douard and Thomass\'{e}, St\'{e}phan and Tran, Xuan Thang and Watrigant, R\'{e}mi},
  title =	{{An Algorithmic Weakening of the Erd\H{o}s-Hajnal Conjecture}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-128894},
  doi =		{10.4230/LIPIcs.ESA.2020.23},
  annote =	{Keywords: Approximation, Maximum Independent Set, H-free Graphs, Erd\H{o}s-Hajnal conjecture}
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