Quasi-polynomial-time algorithm for Independent Set in Pt-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 sub-graph. Their algorithm runs in time n(O(log3 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(log2 n)). Our main insight is that a connected Pt-free graph always contains a vertex w whose neighborhood intersects, for a constant fraction of pairs {u, v} is an element of (V (G) 2), a constant fraction of induced u - v paths. Since a Pt-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.