Clique Relaxations in Social Network Analysis: The Maximum k-Plex Problem

Authors:
Balabhaskar Balasundaram;Sergiy Butenko;Illya V. Hicks
Affiliations:
School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma 74078;Department of Industrial and Systems Engineering, Texas A&M University, College Station, Texas 77843;Computational and Applied Mathematics Department, Rice University, Houston, Texas 77005
Venue:
Operations Research
Year:
2011

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Abstract

This paper introduces and studies the maximum k-plex problem, which arises in social network analysis and has wider applicability in several important areas employing graph-based data mining. After establishing NP-completeness of the decision version of the problem on arbitrary graphs, an integer programming formulation is presented, followed by a polyhedral study to identify combinatorial valid inequalities and facets. A branch-and-cut algorithm is implemented and tested on proposed benchmark instances. An algorithmic approach is developed exploiting the graph-theoretic properties of a k-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.