ALLSAT compressed with wildcards: All, or all maximum independent sets

TL;DR

This paper presents a method to efficiently generate all maximum independent sets in a graph using odd cycle covers and wildcards for compact representation, significantly improving enumeration speed.

Contribution

It introduces a novel approach combining odd cycle covers and wildcards to efficiently enumerate and compactly represent all maximum independent sets.

Findings

Enumeration time is polynomial in the size of the graph and the number of anticliques.

Wildcards enable compact clustering of anticliques.

The method applies to arbitrary graphs for generating all anticliques.

Abstract

An odd cycle cover is a vertex set whose removal makes a graph bipartite. We show that if a $k$-element odd cycle cover of a graph with w vertices is known then all $N$ maximum anticliques (= independent sets) can be generated in time $O(2^k w^3 + N w^2))$. Generating ${\it all}\ N'$ anticliques (maximum or not) is easier and works for arbitrary graphs in time $O(N'w^2)$. In fact the use of wildcards allows to compactly generate the anticliques in clusters.