Improved FPT algorithms for weighted independent set in bull-free graphs

Abstract

Very recently, Thomassé, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O(k^5)}\cdot n^9$. In this article we improve this running time to $2^{O(k^2)}\cdot n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O(k^5)$ to $O(k^2)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p-1$ for $p\ge 3$, we speed up the running time to $2^{O(k\cdot k^{\frac{1}{p-1}})}\cdot n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p\ge 3$, Weighted Independent Set cannot be solved in time $2^{o(k)}\cdot n^{O(1)}$ in the class of $\{bull,C_4,\dots ,C_{2p-1}\}$-free graphs unless the ETH fails.

15 pages