New Branching Rules: Improvements on Independent Set and Vertex Cover in Sparse Graphs

TL;DR

This paper introduces new branching techniques that significantly improve the efficiency of algorithms for maximum independent set and vertex cover problems in sparse graphs with degree at most 3.

Contribution

The paper presents two novel branching methods and improved algorithms with better time complexities for maximum independent set and vertex cover in degree-bounded graphs.

Findings

Maximum independent set algorithm with $O^*(1.0919^n)$ time complexity

Vertex cover decision algorithm with $O^*(1.1923^k)$ time complexity

Introduction of branching on a bottle and 4-cycle techniques

Abstract

We present an $O^*(1.0919^n)$-time algorithm for finding a maximum independent set in an $n$-vertex graph with degree bounded by 3, which improves the previously known algorithm of running time $O^*(1.0977^n)$ by Bourgeois, Escoffier and Paschos [IWPEC 2008]. We also present an $O^*(1.1923^k)$-time algorithm to decide if a graph with degree bounded by 3 has a vertex cover of size $k$, which improves the previously known algorithm of running time $O^*(1.1939^k)$ by Chen, Kanj and Xia [ISAAC 2003]. Two new branching techniques, \emph{branching on a bottle} and \emph{branching on a 4-cycle}, are introduced, which help us to design simple and fast algorithms for the maximum independent set and minimum vertex cover problems and avoid tedious branching rules.