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SMC-BRB: an algorithm for the maximum clique problem over large and sparse graphs with the upper bound via \(s^+\)-index

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Abstract

The Maximum Clique Problem (MCP) is a classic NP-hard problem, which has the goal of finding the largest possible clique. It is known to have direct applications in a wide spectrum of fields such as data association problems appearing in bioinformatics and computational biology, computer vision and robotics. Solutions like using brute force, backtracking and branch and bound are designed to deal with the maximum problem. In the branch and bound method, a branch is pruned if the currently found largest clique is better than its upper bound. However, the upper bound obtained by current methods is often not close enough to the \(\omega (G)\), leading to large inefficient search space. This paper discusses the branch and bound procedure to solve the maximum clique problem in large and sparse graphs and proposes a new efficient branch and bound maximum clique algorithm named SMC-BRB. SMC-BRB solves the maximum clique problem in heuristic search stage and exact search stage. It simultaneously utilizes the \(s^+\)-index and color-based upper bound in heuristic search stage, which effectively reduces the number of branches in the exact search stage. This method is beneficial to the solution of MCP because it provides a scale reduction on heuristic search stage. Experimental results show that SMC-BRB has better performance than the state-of-the-art algorithm MC-BRB, which demonstrates the efficiency of the proposed approach.

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Data availability

Datasets used in this paper are all available in public repositories.

Notes

  1. http://snap.stanford.edu/.

  2. http://networkrepository.com/.

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Authors and Affiliations

  1. College of Computer Science, Chongqing University, Chongqing, 400044, China

    Mingqiang Zhou, Qianqian Zeng & Ping Guo

  2. College Key Laboratory of Software Theory and Technology, Chongqing, 400044, China

    Mingqiang Zhou & Ping Guo

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  1. Mingqiang Zhou
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Correspondence to Mingqiang Zhou.

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Zhou, M., Zeng, Q. & Guo, P. SMC-BRB: an algorithm for the maximum clique problem over large and sparse graphs with the upper bound via \(s^+\)-index. J Supercomput 79, 8026–8047 (2023). https://doi.org/10.1007/s11227-022-04982-7

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  • DOI: https://doi.org/10.1007/s11227-022-04982-7

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