Parameterized Inapproximability of Independent Set in H-Free Graphs

Abstract

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 \(\delta >0\) the Independent Set problem has a polynomial-time \((\frac{d-1}{2}+\delta )\)-approximation algorithm in \(K_{1,d}\)-free graphs. We extend this result by showing that \(K_{a,b}\)-free graphs admit a polynomial-time\({\mathcal {O}}(\alpha (G)^{1-1/a})\)-approximation, where \(\alpha (G)\) is the size of a maximum independent set in G. Furthermore, we complement the result of Halldórsson by showing that for some \(\gamma =\Theta (d/\log d),\) there is no polynomial-time \(\gamma \)-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 \(f(k) \cdot n^{o(k/\log k)}\) algorithm for any computable function f. Then, under the deterministic Gap-ETH, there is a constant \(\delta >0\) such that no \(\delta \)-approximation can be computed in \(f(k) \cdot n^{O(1)}\) time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime \(f(k) \cdot n^{o(\sqrt{k})}\). Finally, we consider the parameterization by the excluded graph H, and show that under the ETH, Independent Set has no \(n^{o(\alpha (H))}\) 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 \(f(d,k) \cdot n^{{\mathcal {O}}(1)}\), under the deterministic Gap-ETH.