Efficient Enumeration of the Optimal Solutions to the Correlation Clustering problem

TL;DR

This paper introduces an efficient method to enumerate all optimal solutions for the correlation clustering problem, addressing the challenge of exploring diverse solutions in NP-hard scenarios.

Contribution

The paper presents a novel enumeration approach combining exhaustive search with neighborhood strategies to fully explore the solution space of correlation clustering.

Findings

Method effectively retrieves all optimal solutions in middle-sized networks.

Approach improves understanding of solution diversity in correlation clustering.

Results confirm computational efficiency and practical usefulness.

Abstract

According to the structural balance theory, a signed graph is considered structurally balanced when it can be partitioned into a number of modules such that positive and negative edges are respectively located inside and between the modules. In practice, real-world networks are rarely structurally balanced, though. In this case, one may want to measure the magnitude of their imbalance, and to identify the set of edges causing this imbalance. The correlation clustering (CC) problem precisely consists in looking for the signed graph partition having the least imbalance. Recently, it has been shown that the space of the optimal solutions of the CC problem can be constituted of numerous and diverse optimal solutions. Yet, this space is difficult to explore, as the CC problem is NP-hard, and exact approaches do not scale well even when looking for a single optimal solution. To alleviate this issue, in this work we propose an efficient enumeration method allowing to retrieve the complete space of optimal solutions of the CC problem. It combines an exhaustive enumeration strategy with neighborhoods of varying sizes, to achieve computational effectiveness. Results obtained for middle-sized networks confirm the usefulness of our method.