Approximation algorithms for the capacitated correlation clustering problem with penalties

• Published in: Journal of combinatorial optimization Vol. 45; no. 1; p. 12
• Main Authors: Ji, Sai, Li, Gaidi, Zhang, Dongmei, Zhang, Xianzhao
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2023; Springer Nature B.V
• Subjects: Algorithms; Apexes; Approximation; Clustering; Combinatorics; Convex and Discrete Geometry; Correlation; Costs; Criteria; Fines & penalties; Integer programming; Linear programming; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Polynomials; Theory of Computation; Upper bounds; LP-rounding; Penalties; Approximation algorithm; Capacitated; Correlation clustering
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-022-00930-6

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 δ ∈ ( 0 , 4 / 9 ] , the first algorithm is a 8 / ( 4 - 5 δ ) , 8 / δ -bi-criteria approximation algorithm for the CCorCPwP, which means that the number of vertices in each cluster does not exceed 8 / ( 4 - 5 δ ) times the upper bound, and the output objective function value of the algorithm does not exceed 8 / δ 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.