A faster algorithm for Vertex Cover parameterized by solution size

TL;DR

This paper introduces a significantly faster fixed-parameter algorithm for Vertex Cover that leverages a potential function tracking both solution size and LP relaxation value, improving runtime over previous methods.

Contribution

The authors develop a new algorithm with improved runtime $O^*(1.25284^k)$ for Vertex Cover, using a potential function that considers both solution size and LP relaxation, and introduce novel branching techniques.

Findings

Runtime improved from $O^*(1.2738^k)$ to $O^*(1.25284^k)$

Utilizes potential function tracking $k$ and LP relaxation value

Develops new branching steps for local obstructions

Abstract

We describe a new algorithm for vertex cover with runtime $O^*(1.25284^k)$, where $k$ is the size of the desired solution and $O^*$ hides polynomial factors in the input size. This improves over previous runtime of $O^*(1.2738^k)$ due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential function which simultaneously tracks $k$ as well as the optimal value $\lambda$ of the vertex cover LP relaxation. This approach also allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both $k$ and $\mu = k - \lambda$ are decreased at each step. There can be local obstructions in the graph that prevent $\mu$ from decreasing in this process; we develop a number of novel branching steps to handle these situations.