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An Algorithmic Weakening of the Erdős-Hajnal Conjecture

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

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Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Stéphan Thomassé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Xuan Thang Tran
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Rémi Watrigant
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France

Funding

This work was 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).

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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) https://doi.org/10.4230/LIPIcs.ESA.2020.23

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation
  • Maximum Independent Set
  • H-free Graphs
  • Erdős-Hajnal conjecture

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