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An Ellipsoidal Bounding Scheme for the Quasi-Clique Number of a Graph

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  • Zhuqi Miao

    (Center for Health Systems Innovation, Oklahoma State University, Stillwater, Oklahoma 74078)

  • Balabhaskar Balasundaram

    (School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma 74078)

Abstract

A γ -quasi-clique in a simple undirected graph refers to a subset of vertices that induces a subgraph with edge density at least γ. When γ equals one, this definition corresponds to a classical clique. When γ is less than one, it relaxes the requirement of all possible edges by the clique definition. Quasi-clique detection has been used in graph-based data mining to find dense clusters, especially in large-scale error-prone data sets in which the clique model can be overly restrictive. The maximum γ -quasi-clique problem , seeking a γ-quasi-clique of maximum cardinality in the given graph, can be formulated as an optimization problem with a linear objective function and a single quadratic constraint in binary variables. This article investigates the Lagrangian dual of this formulation and develops an upper-bounding technique using the geometry of ellipsoids to bound the Lagrangian dual. The tightness of the upper bound is compared with those obtained from multiple mixed-integer programming formulations of the problem via experiments on benchmark instances.

Suggested Citation

  • Zhuqi Miao & Balabhaskar Balasundaram, 2020. "An Ellipsoidal Bounding Scheme for the Quasi-Clique Number of a Graph," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 763-778, July.
    • Handle: RePEc:inm:orijoc:v:32:y:3:i:2020:p:763-778
    DOI: 10.1287/ijoc.2019.0922
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    References listed on IDEAS

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