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Towards effective exact methods for the Maximum Balanced Biclique Problem in bipartite graphs

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  • Zhou, Yi
  • Rossi, André
  • Hao, Jin-Kao

Abstract

The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP in bipartite graphs. First, an Upper Bound Propagation (UBP) procedure to pre-compute an upper bound involving each vertex is introduced. Then we extend a simple Branch-and-Bound (B&B) algorithm by integrating the pre-computed upper bounds. Based on UBP, we also study a new integer linear programming model of MBBP which is more compact than an existing formulation (Dawande, Keskinocak, Swaminathan, & Tayur, 2001). We introduce new valid inequalities induced from the upper bounds to tighten these mathematical formulations for MBBP. Experiments with random bipartite graphs demonstrate the efficiency of the extended B&B algorithm and the valid inequalities generated on demand. Further tests with 30 real-life instances show that, for at least three very large graphs, the new approaches improve the computational time with four orders of magnitude compared to the original B&B.

Suggested Citation

  • Zhou, Yi & Rossi, André & Hao, Jin-Kao, 2018. "Towards effective exact methods for the Maximum Balanced Biclique Problem in bipartite graphs," European Journal of Operational Research, Elsevier, vol. 269(3), pages 834-843.
    • Handle: RePEc:eee:ejores:v:269:y:2018:i:3:p:834-843
    DOI: 10.1016/j.ejor.2018.03.010
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    1. Balabhaskar Balasundaram & Sergiy Butenko & Illya V. Hicks, 2011. "Clique Relaxations in Social Network Analysis: The Maximum k -Plex Problem," Operations Research, INFORMS, vol. 59(1), pages 133-142, February.
    2. Wu, Qinghua & Hao, Jin-Kao, 2015. "A review on algorithms for maximum clique problems," European Journal of Operational Research, Elsevier, vol. 242(3), pages 693-709.
    3. Zhou, Yi & Hao, Jin-Kao & Goëffon, Adrien, 2017. "PUSH: A generalized operator for the Maximum Vertex Weight Clique Problem," European Journal of Operational Research, Elsevier, vol. 257(1), pages 41-54.
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