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-1/2+δ ) -approximation algorithm in K_1,d -free graphs. We extend this result by showing that K_a,b -free graphs admit a polynomial-time 𝒪(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 K_1,4 , 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 K_1,5 ). First, under the ETH, there is no f(k) · n^o(k/log k) 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) · n^O(1) time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime f(k) · n^o(√(k)) . Finally, we consider the parameterization by the excluded graph H , and show that under the ETH, Independent Set has no n^o(α (H)) algorithm in H -free graphs. Also, we prove that there is no d/k^o(1) -approximation algorithm for K_1,d -free graphs with runtime f(d,k) · n^𝒪(1) , under the deterministic Gap-ETH.