Quasi-polynomial-time algorithm for Independent Set in P_t-free graphs via shrinking the space of induced paths
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In a recent breakthrough work, Gartland and Lokshtanov [FOCS 2020] showed a quasi-polynomial-time algorithm for Maximum Weight Independent Set in P_t-free graphs, that is, graphs excluding a fixed path as an induced subgraph. Their algorithm runs in time n^𝒪(log^3 n), where t is assumed to be a constant. Inspired by their ideas, we present an arguably simpler algorithm with an improved running time bound of n^𝒪(log^2 n). Our main insight is that a connected P_t-free graph always contains a vertex w whose neighborhood intersects, for a constant fraction of pairs {u,v}∈V(G)2, a constant fraction of induced u-v paths. Since a P_t-free graph contains 𝒪(n^t-1) induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound. We also show that the same approach can be used to obtain quasi-polynomial-time algorithms for related problems, including Maximum Weight Induced Matching and 3-Coloring.
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