Counting independent sets via Divide Measure and Conquer method

TL;DR

This paper introduces a novel Divide Measure and Conquer algorithm for counting independent sets in graphs, achieving improved exponential time bounds for subcubic and general graphs, and potentially impacting other combinatorial problems.

Contribution

It presents a new Divide Measure and Conquer method, combining existing techniques to efficiently count independent sets with polynomial space complexity.

Findings

Time complexity of $O^*(1.1394^n)$ for subcubic graphs

Time complexity of $O^*(1.2369^n)$ for general graphs

Enhanced graph coloring algorithm using the new method as a subroutine

Abstract

In this paper we give an algorithm for counting the number of all independent sets in a given graph which works in time $O^*(1.1394^n)$ for subcubic graphs and in time $O^*(1.2369^n)$ for general graphs, where $n$ is the number of vertices in the instance graph, and polynomial space. The result comes from combining two well known methods "Divide and Conquer" and "Measure and Conquer". We introduce this new concept of Divide, Measure and Conquer method and expect it will find applications in other problems. The algorithm of Bj\"orklund, Husfeldt and Koivisto for graph colouring with our algorithm used as a subroutine has complexity $O^*(2.2369^n)$ and is currently the fastest graph colouring algorithm in polynomial space.