Exact algorithms for maximum weighted independent set on sparse graphs

TL;DR

This paper introduces new reduction rules and an efficient exact algorithm for the maximum weighted independent set problem, particularly optimized for sparse graphs with low average degree, improving previous bounds.

Contribution

It presents a novel algorithm with measure-and-conquer analysis that achieves faster running times on sparse graphs, especially those with average degree at most 3.

Findings

Algorithm runs in $O^*(1.1443^n)$ time for graphs with average degree ≤ 3

The algorithm is polynomial space efficient

Improves previous bounds for cubic graphs

Abstract

The maximum independent set problem is one of the most important problems in graph algorithms and has been extensively studied in the line of research on the worst-case analysis of exact algorithms for NP-hard problems. In the weighted version, each vertex in the graph is associated with a weight and we are going to find an independent set of maximum total vertex weight. In this paper, we design several reduction rules and a fast exact algorithm for the maximum weighted independent set problem, and use the measure-and-conquer technique to analyze the running time bound of the algorithm. Our algorithm works on general weighted graphs and it has a good running time bound on sparse graphs. If the graph has an average degree at most 3, our algorithm runs in $O^*(1.1443^n)$ time and polynomial space, improving previous running time bounds for the problem in cubic graphs using polynomial space.