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Why Is Maximum Clique Often Easy in Practice?

Author

Listed:
  • Jose L. Walteros

    (Department of Industrial and Systems Engineering, University at Buffalo, Buffalo, New York 14260)

  • Austin Buchanan

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

Abstract

To this day, the maximum clique problem remains a computationally challenging problem. Indeed, despite researchers’ best efforts, there exist unsolved benchmark instances with 1,000 vertices. However, relatively simple algorithms solve real-life instances with millions of vertices in a few seconds. Why is this the case? Why is the problem apparently so easy in many naturally occurring networks? In this paper, we provide an explanation. First, we observe that the graph’s clique number ω is very near to the graph’s degeneracy d in most real-life instances. This observation motivates a main contribution of this paper, which is an algorithm for the maximum clique problem that runs in time polynomial in the size of the graph, but exponential in the gap g ≔ ( d + 1 ) − ω between the clique number ω and its degeneracy-based upper bound d +1. When this gap g can be treated as a constant, as is often the case for real-life graphs, the proposed algorithm runs in time O ( d m ) = O ( m 1.5 ) . This provides a rigorous explanation for the apparent easiness of these instances despite the intractability of the problem in the worst case. Further, our implementation of the proposed algorithm is actually practical—competitive with the best approaches from the literature.

Suggested Citation

  • Jose L. Walteros & Austin Buchanan, 2020. "Why Is Maximum Clique Often Easy in Practice?," Operations Research, INFORMS, vol. 68(6), pages 1866-1895, November.
    • Handle: RePEc:inm:oropre:v:68:y:2020:i:6:p:1866-1895
    DOI: 10.1287/opre.2019.1970
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    References listed on IDEAS

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    1. Anurag Verma & Austin Buchanan & Sergiy Butenko, 2015. "Solving the Maximum Clique and Vertex Coloring Problems on Very Large Sparse Networks," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 164-177, February.
    2. Felix Lieder & Fatemeh Rad & Florian Jarre, 2015. "Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques," Computational Optimization and Applications, Springer, vol. 61(3), pages 669-688, July.
    3. M. W. P. Savelsbergh, 1994. "Preprocessing and Probing Techniques for Mixed Integer Programming Problems," INFORMS Journal on Computing, INFORMS, vol. 6(4), pages 445-454, November.
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    Cited by:

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    3. San Segundo, Pablo & Furini, Fabio & Álvarez, David & Pardalos, Panos M., 2023. "CliSAT: A new exact algorithm for hard maximum clique problems," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1008-1025.
    4. Yajun Lu & Hosseinali Salemi & Balabhaskar Balasundaram & Austin Buchanan, 2022. "On Fault-Tolerant Low-Diameter Clusters in Graphs," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 3181-3199, November.
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    6. S. Raghavan & Rui Zhang, 2022. "Rapid Influence Maximization on Social Networks: The Positive Influence Dominating Set Problem," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1345-1365, May.

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