# An algorithmic weakening of the Erdős-Hajnal conjecture

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. P_6-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 C_4-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.

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