Efficient enumeration of the optimal solutions to the correlation clustering problem

• Published in: Journal of global optimization Vol. 86; no. 2; pp. 355 - 391
• Main Authors: Arınık, Nejat, Figueiredo, Rosa, Labatut, Vincent
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2023; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Clustering; Computer Science; Enumeration; Linear programming; Mathematical programming; Mathematics; Mathematics and Statistics; Methods; Modules; Operations Research; Operations Research/Decision Theory; Optimization; Partitions (mathematics); Real Functions; Social networks; Signed graph; Local search; Optimal solution space; Enumeration strategy; Correlation clustering; Structural balance; Signed Graph; Local Search; Optimal Solution Space; Correlation Clustering; Enumeration Strategy; Structural Balance
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-023-01270-3

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.