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Approximation algorithms for the capacitated correlation clustering problem with penalties

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Abstract

This paper considers the capacitated correlation clustering problem with penalties (CCorCwP), which is a new generalization of the correlation clustering problem. In this problem, we are given a complete graph, each edge is either positive or negative. Moreover, there is an upper bound on the number of vertices in each cluster, and each vertex has a penalty cost. The goal is to penalize some vertices and select a clustering of the remain vertices, so as to minimize the sum of the number of positive cut edges, the number of negative non-cut edges and the penalty costs. In this paper we present an integer programming, linear programming relaxation and two polynomial time algorithms for the CCorCwP. Given parameter \(\delta \in (0,4/9]\), the first algorithm is a \(\left( 8/(4-5\delta ), 8/\delta \right) \)-bi-criteria approximation algorithm for the CCorCPwP, which means that the number of vertices in each cluster does not exceed \(8/(4-5\delta )\) times the upper bound, and the output objective function value of the algorithm does not exceed \(8/\delta \) times the optimal value. The second one is based on above bi-criteria approximation, and we prove that the second algorithm can achieve a constant approximation ratio for some special instances of the CCorCwP.

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Acknowledgements

The first author is supported by National Natural Science Foundation of China (No. 12101594) and the Project funded by China Postdoctoral Science Foundation (No. 2021M693337). The third author is supported by National Natural Science Foundation of China (No. 11871081). The fourth author is supported by National Natural Science Foundation of China (No. 11801310).

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Authors and Affiliations

  1. Institute of Mathematics, Hebei University of Technology, Tianjin, 300401, People’s Republic of China

    Sai Ji

  2. Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Gaidi Li

  3. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China

    Dongmei Zhang

  4. School of Mathematics and Statistics, Linyi University, Linyi, 276005, People’s Republic of China

    Xianzhao Zhang

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  1. Sai Ji
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  2. Gaidi Li
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  3. Dongmei Zhang
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  4. Xianzhao Zhang
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Correspondence to Gaidi Li.

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A preliminary version of this paper appeared in Proceedings of the 15th International Conference on Algorithmic Applications in Management, pp. 15–26, 2021.

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Ji, S., Li, G., Zhang, D. et al. Approximation algorithms for the capacitated correlation clustering problem with penalties. J Comb Optim 45, 12 (2023). https://doi.org/10.1007/s10878-022-00930-6

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