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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2020.23
URN: urn:nbn:de:0030-drops-128894
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12889/
Bonnet, Édouard ;
Thomassé, Stéphan ;
Tran, Xuan Thang ;
Watrigant, Rémi
An Algorithmic Weakening of the Erdős-Hajnal Conjecture
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.
BibTeX - Entry
@InProceedings{bonnet_et_al:LIPIcs:2020:12889,
author = {{\'E}douard Bonnet and St{\'e}phan Thomass{\'e} and Xuan Thang Tran and R{\'e}mi Watrigant},
title = {{An Algorithmic Weakening of the Erdős-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 = {Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12889},
URN = {urn:nbn:de:0030-drops-128894},
doi = {10.4230/LIPIcs.ESA.2020.23},
annote = {Keywords: Approximation, Maximum Independent Set, H-free Graphs, Erdős-Hajnal conjecture}
}
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Keywords: |
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Approximation, Maximum Independent Set, H-free Graphs, Erdős-Hajnal conjecture |
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Collection: |
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28th Annual European Symposium on Algorithms (ESA 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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26.08.2020 |
