Independent Set on P_k-Free Graphs in Quasi-Polynomial Time

摘要

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(k2log3n). Hence, for every fixed k our algorithm runs in quasi-polynomial time. This resolves in the affirmative an open problem of [Thomassé, SODA'20 invited presentation]. Previous to this work, polynomial time algorithms were only known for P₄-free graphs [Corneil et al., DAM'81], P₅-free graphs [Lokshtanov et al., SODA'14], and P₆-free graphs [Grzesik et al., SODA'19]. For larger values of t, only 2O(√{knlogn}) time algorithms [Bacsó 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).}