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 \(n1-ε\) for any \(ε > 0\). 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 \(G\) and an accuracy parameter \(ε > 0\), finds an independent set in \(G\) of cardinality within a factor of \((1-ε )\) of the optimum in time exponential in a polynomial of \(log |V(G)|\) and \(ε⁻¹\). Furthermore, an independent set of maximum size can be found in subexponential time \(2^{O(|V(G)|8/9 log |V(G)|)}\). 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.Keywordsmaximum weight independent sethereditary graph classesapproximation schemethree-in-a-treeMSC codes68R1005C6905C85