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 $n^{1-\varepsilon}$ for any $\varepsilon > 0$. 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 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 algorithm works also 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.

翻译：在最大独立集问题中，我们需要在给定图中寻找一组两两不相邻的顶点，使得其基数尽可能大。在一般图中，这一经典问题已知是NP困难的，且对于任意ε>0，无法在因子n^{1-ε}内进行近似。因此，研究各类图中的最大独立集问题复杂度，以期获得更好的可解性结果，是一个活跃的研究方向。在H-自由图（即不包含固定图H作为导出子图的图）中，当H包含一个环、一个度数至少为四的顶点、或一个连通分量中出现两个度数至少为三的顶点时，该问题已知仍为NP困难和APX困难的。对于其余情形（即H的每个分量均为一条路径或一根细分爪），最大独立集问题的复杂度仍广泛开放，仅对于少数小图H（如P5、P6、爪或叉图）存在多项式时间可解性结果。我们证明：对于任意图H，若在H-自由图中最大独立集问题尚未被证明是APX困难和SUBEXP困难的，则该问题在此类图中可接受拟多项式时间近似方案和次指数时间精确算法。我们的算法同样适用于更一般的加权场景，即输入图附带顶点权函数，且目标是最大化独立集的总权重。