QUASI-POLYNOMIAL TIME APPROXIMATION SCHEMES FOR THE MAXIMUM WEIGHT INDEPENDENT SET PROBLEM IN H-FREE GRAPHS

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 n(1-epsilon) for any epsilon > 0. 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 H-free graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever H 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 H 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 H such as P-5, P-6, the claw, or the fork. We prove that for every such "possibly tractable" graph H there exists an algorithm that, given an H-free graph H and an accuracy parameter epsilon > 0, finds an independent set in G of cardinality within a factor of (1-epsilon) of the optimum in time exponential in a polynomial of log vertical bar V(G)vertical bar and epsilon(-1). Furthermore, an independent set of maximum size can be found in subexponential time 2(O(vertical bar V(G)|8/9log vertical bar|V(G vertical bar)). That is, we show that for every graph H for which MAXIMUM INDEPENDENT SET is not known to be APX-hard and SUBEXP-hard in H-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.