Quasi-polynomial-time algorithm for Independent Set in P_t-free graphs via shrinking the space of induced paths

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 , 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 . 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 , 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.