Independent Set on -Free Graphs in Quasi-Polynomial Time

Abstract

We present an algorithm that takes as input a graph $G$ with weights on the vertices, and computes a maximum weight independent set $S$ of $G$ . If the input graph $G$ excludes a path $P_{k}$ on $k$ vertices as an induced subgraph, the algorithm runs in time $n^{O(k^{2}\log^{3}n)}$ . Hence, for every fixed $k$ our algorithm runs in quasi-polynomial time. This resolves in the affirmative an open problem of [Thomasse, SODA'20 invited presentation]. Previous to this work, polynomial time algorithms were only known for $P_{4}$ -free graphs [Corneil et al., DAM'81], $P_{5}$ -free graphs [Lokshtanov et al., SODA'14], and $P_{6}$ -free graphs [Grzesik et al., SODA'19]. For larger values of $t$ , only $2^{O(\sqrt{kn\log n})}$ time algorithms [Bacso et al., Algorithmica'19] and quasipolynomial time approximation schemes [Chudnovsky et al., SODA'20] were known. Thus, our work is the first to offer conclusive evidence that Independent Set on $P_{k}$ - free graphs is not NP-complete for any integer $k$ . Additionally we show that for every graph $H$ , if there exists a quasi-polynomial time algorithm for Independent Seton $C$ -free graphs for every connected component $C$ of $H$ , then there also exists a quasi-polynomial time algorithm for Independent Set on $H$ -free graphs. This lifts our quasi-polynomial time algorithm to $T_{k}$ -free graphs, where $T_{k}$ has one component that is a $P_{k}$ , and $k-1$ components isomorphic to a fork (the unique 5-vertex tree with a degree 3 vertex).