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

TL;DR

This paper improves fixed-parameter tractable algorithms for the weighted independent set problem in bull-free graphs, achieving faster running times and tighter kernels, with matching lower bounds indicating near-optimality.

Contribution

It presents improved FPT algorithms with better running times and kernel sizes for weighted independent set in bull-free graphs, and establishes tight lower bounds for certain subclasses.

Findings

Reduced the algorithm running time from 2^{O(k^5)} to 2^{O(k^2)}

Improved the Turing-kernel size from O(k^5) to O(k^2)

Proved tight lower bounds for subclasses without small holes

Abstract

Very recently, Thomass\'e, 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 $p \geq 3$, 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 $p \geq 3$, Weighted Independent Set cannot be solved in time $2^{o(k)} \cdot n^{O(1)}$ in the class of $\{bull,C_4,\ldots,C_{2p-1}\}$-free graphs unless the ETH fails.