TY - JOUR

T1 - Concise integer linear programming formulation for clique partitioning problems

AU - Koshimura, Miyuki

AU - Watanabe, Emi

AU - Sakurai, Yuko

AU - Yokoo, Makoto

N1 - Publisher Copyright:
© 2022, The Author(s).

PY - 2022/4

Y1 - 2022/4

N2 - 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.

AB - 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.

KW - Clique partitioning problem

KW - Integer linear programming

KW - Transitivity constraints

UR - http://www.scopus.com/inward/record.url?scp=85128748170&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85128748170&partnerID=8YFLogxK

U2 - 10.1007/s10601-022-09326-z

DO - 10.1007/s10601-022-09326-z

M3 - Article

AN - SCOPUS:85128748170

SN - 1383-7133

VL - 27

SP - 99

EP - 115

JO - Constraints

JF - Constraints

IS - 1-2

ER -