Quasi-Polynomial Time Approximation Schemes for the Maximum Weight Independent Set Problem in [math]-Free Graphs

SIAM Journal on Computing, Volume 53, Issue 1, Page 47-86, February 2024.
Abstract. In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of [math] for any [math]. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction. In [math]-free graphs, that is, graphs not containing a fixed graph [math] as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever [math] contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of [math] is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomial-time solvability results for small graphs [math] such as [math], [math], the claw, or the fork. We prove that for every such “possibly tractable” graph [math] there exists an algorithm that, given an [math]-free graph [math] and an accuracy parameter [math], finds an independent set in [math] of cardinality within a factor of [math] of the optimum in time exponential in a polynomial of [math] and [math]. Furthermore, an independent set of maximum size can be found in subexponential time [math]. That is, we show that for every graph [math] for which Maximum Independent Set is not known to be APX-hard and SUBEXP-hard in [math]-free graphs, the problem admits a quasi-polynomial time approximation scheme and a subexponential-time exact algorithm in this graph class. Our algorithms also work in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set.