Small Maximal Independent Sets and Faster Exact Graph Coloring

TL;DR

This paper introduces tight bounds on the number of small maximal independent sets in graphs and leverages these bounds to develop a faster exact algorithm for computing the graph's chromatic number, improving previous methods.

Contribution

The paper provides tight bounds on small maximal independent sets and uses them to significantly improve the time complexity of exact graph coloring algorithms.

Findings

Bound on the number of small maximal independent sets in graphs.

An improved algorithm for exact graph coloring with complexity ~2.4150^n.

Bounds are tight for certain parameter ranges.

Abstract

We show that, for any n-vertex graph G and integer parameter k, there are at most 3^{4k-n}4^{n-3k} maximal independent sets I \subset G with |I| <= k, and that all such sets can be listed in time O(3^{4k-n} 4^{n-3k}). These bounds are tight when n/4 <= k <= n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3 + 3^{4/3}/4)^n) ~= 2.4150^n, improving a previous O((1+3^{1/3})^n) ~= 2.4422^n algorithm of Lawler (1976).