Parameterized inapproximability of independent set in H-free graphs

We study the Independent Set problem in H-free graphs, i.e., graphs excluding some fixed graph H as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halldórsson [SODA 1995] showed that for every δ> 0 the Independent Set problem has a polynomial-time (d-12+δ)-approximation algorithm in K1,d-free graphs. We extend this result by showing that Ka,b-free graphs admit a polynomial-timeO(α(G) 1-1/a) -approximation, where α(G) is the size of a maximum independent set in G. Furthermore, we complement the result of Halldórsson by showing that for some γ= Θ (d/ log d) , there is no polynomial-time γ-approximation algorithm for these graphs, unless NP = ZPP. Bonnet et al. [Algorithmica 2020] showed that Independent Set parameterized by the size k of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least 4, (2) the star K1 , 4, and (3) any tree with two vertices of degree at least 3 at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken conditions (1) and (2) that G does not contain a cycle of constant length at least 5 or K1 , 5). First, under the ETH, there is no f(k) · no(k/logk) algorithm for any computable function f. Then, under the deterministic Gap-ETH, there is a constant δ> 0 such that no δ-approximation can be computed in f(k) · nO(1) time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime f(k)·no(k). Finally, we consider the parameterization by the excluded graph H, and show that under the ETH, Independent Set has no no(α(H)) algorithm in H-free graphs. Also, we prove that there is no d/ ko(1)-approximation algorithm for K1,d-free graphs with runtime f(d, k) · nO(1), under the deterministic Gap-ETH.