Fast algorithms for supermodular and non-supermodular minimization via bi-criteria strategy

• Published in: Journal of combinatorial optimization Vol. 44; no. 5; pp. 3549 - 3574
• Main Authors: Zhang, Xiaojuan, Liu, Qian, Li, Min, Zhou, Yang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2022; Springer Nature B.V
• Subjects: Algorithms; Approximation; Combinatorics; Convex and Discrete Geometry; Criteria; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Theory of Computation; Supermodular; 68Q25; Non-supermodular; Bi-criteria; 68W10; Fuzzy; means problem; 90C27; 68W25; Group sparse linear regression problem
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-022-00914-6

In this paper, we concentrate on exploring fast algorithms for the minimization of a non-increasing supermodular or non-supermodular function f subject to a cardinality constraint. As for the non-supermodular minimization problem with the weak supermodularity ratio r , we can obtain a ( 1 + ϵ ) -approximation algorithm with adaptivity O ( n ϵ log r n · f ( ∅ ) ϵ · OPT ) under the bi-criteria strategy, where OPT denotes the optimal objective value of the problem. That is, instead of selecting at most k elements on behalf of the constraint, the cardinality of the output may reach to k r log f ( ∅ ) ϵ · OPT . Moreover, for the supermodular minimization problem, we propose two ( 1 + ϵ ) -approximation algorithms for which the output solution X is of size | X 0 | + O k log f ( X 0 ) ϵ · OPT . The adaptivities of this two algorithms are O log 2 n · log f ( X 0 ) ϵ · OPT and O log n · log f ( X 0 ) ϵ · OPT , where X 0 is an input set and OPT is the optimal value. Applications to group sparse linear regression problems and fuzzy C -means problems are studied at the end.