Concise integer linear programming formulation for clique partitioning problems

Miyuki Koshimura, Emi Watanabe, Yuko Sakurai, Makoto Yokoo

Research output: Contribution to journalArticlepeer-review


A Clique Partitioning Problem (CPP) finds an optimal partition of a given edge-weighted undirected graph, such that the sum of the weights is maximized. This general graph problem has a wide range of real-world applications, including correlation clustering, group technology, community detection, and coalition structure generation. Although a CPP is NP-hard, due to the recent advance of Integer Linear Programming (ILP) solvers, we can solve reasonably large problem instances by formulating a CPP as an ILP instance. The first ILP formulation was introduced by Grötschel and Wakabayashi (Mathematical Programming, 45(1-3), 59–96, 1989). Recently, Miyauchi et al. (2018) proposed a more concise ILP formulation that can significantly reduce transitivity constraints as compared to previously introduced models. In this paper, we introduce a series of concise ILP formulations that can reduce even more transitivity constraints. We theoretically evaluate the amount of reduction based on a simple model in which edge signs (positive/negative) are chosen independently. We show that the reduction can be up to 50% (dependent of the ratio of negative edges) and experimentally evaluate the amount of reduction and the performance of our proposed formulation using a variety of graph data sets. Experimental evaluations show that the reduction can exceed 50% (where edge signs can be correlated), and our formulation outperforms the existing state-of-the-art formulations both in terms of memory usage and computational time for most problem instances.

Original languageEnglish
Pages (from-to)99-115
Number of pages17
Issue number1-2
Publication statusPublished - Apr 2022

All Science Journal Classification (ASJC) codes

  • Software
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Artificial Intelligence


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