  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.
