Lagrangian relaxation and pegging test for the clique partitioning problem

Authors:
Noriyoshi Sukegawa;Yoshitsugu Yamamoto;Liyuan Zhang
Affiliations:
Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Tokyo, Japan 152-8552;Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Japan 305-8573;Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Japan 305-8573
Venue:
Advances in Data Analysis and Classification
Year:
2013

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Abstract

The clique partitioning problem is an NP-hard combinatorial optimization problem with applications to data analysis such as clustering. Though a binary integer linear programming formulation has been known for years, one needs to deal with a huge number of variables and constraints when solving a large instance. In this paper, we propose a size reduction algorithm which is based on the Lagrangian relaxation and the pegging test, and verify its validity through numerical experiments. We modify the conventional subgradient method in order to manage the high dimensionality of the Lagrangian multipliers, and also make an improvement on the ordinary pegging test by taking advantage of the structural property of the clique partitioning problem.