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. prove several inapproximability results both for polynomial-time and parameterized algorithms. Halldórsson [SODA 1995] showed that every $$\delta >0$$ has a $$(\frac{d-1}{2}+\delta )$$ -approximation algorithm $$K_{1,d}$$ -free graphs. extend this result by showing $$K_{a,b}$$ admit $${\mathcal {O}}(\alpha (G)^{1-1/a})$$ -approximation, where $$\alpha (G)$$ is size of maximum independent set G. Furthermore, we complement $$\gamma =\Theta (d/\log d),$$ there no $$ these unless NP = ZPP. Bonnet et al. [Algorithmica 2020] k W[1]-hard on which do not contain (1) cycle constant length at least 4, (2) star $$K_{1,4}$$ , (3) any tree with two vertices degree 3 distance. strengthen proving three under different complexity assumptions almost same class (we weaken conditions G does 5 or $$K_{1,5}$$ ). First, ETH, $$f(k) \cdot n^{o(k/\log k)}$$ computable function f. Then, deterministic Gap-ETH, such can be computed n^{O(1)}$$ time. Also, stronger randomized Gap-ETH approximation runtime n^{o(\sqrt{k})}$$ . Finally, consider parameterization excluded H, show $$n^{o(\alpha (H))}$$ $$d/k^{o(1)}$$ $$f(d,k) n^{{\mathcal {O}}(1)}$$ Gap-ETH.