Enumerating All Maximal Clique-Partitions of an Undirected Graph

TL;DR

This paper presents an improved algorithm for enumerating all maximal clique-partitions of an undirected graph, reducing search space and avoiding duplicates, with applications in resource allocation and anti-unification.

Contribution

The paper introduces a novel algorithm that efficiently enumerates all maximal clique-partitions by applying criteria to eliminate duplicates and reduce search space.

Findings

The new algorithm guarantees enumeration of each maximal clique-partition exactly once.

It significantly reduces the search space compared to previous methods.

The algorithm is applicable to resource allocation and anti-unification problems.

Abstract

We address the problem of enumerating all maximal clique-partitions of an undirected graph and present an algorithm based on the observation that every maximal clique-partition can be produced from the maximal clique-cover of the graph by assigning the vertices shared among maximal cliques, to belong to only one clique. This simple algorithm has the following drawbacks: (1) the search space is very large; (2) it finds some clique-partitions which are not maximal; and (3) some clique-partitions are found more than once. We propose two criteria to avoid these drawbacks. The outcome is an algorithm that explores a much smaller search space and guarantees that every maximal clique-partition is computed only once. The algorithm can be used in problems such as anti-unification with proximity relations or in resource allocation tasks when one looks for several alternative ways to allocate resources.