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找出一个两两非相邻的顶点集合，使其基数尽可能大。对于一般图而言，这一经典问题已知为NP难问题，且对于任意ε>0，其近似比难以达到n^{1-ε}。因此，研究不同图类中最大独立集问题的复杂性，以期获得更优的可解性结果，已成为一个活跃的研究方向。在H-free图（即不含固定图H作为诱导子图的图）中，当H包含一个环、一个度数至少为4的顶点，或同一连通分量中两个度数至少为3的顶点时，该问题已知仍为NP难和APX难。对于其余情况（即H的每个连通分量均为路径或细分爪图），最大独立集的复杂性仍广泛未知，仅对少数小图H（如P₅、P₆、爪图或叉图）存在多项式时间可解性结果。我们证明：对于每个使得H-free图中最大独立集问题尚未被证明为APX难和SUBEXP难的图H，该类图上的问题具有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设置，此时输入图附有顶点权函数，目标为最大化独立集的总权值。