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

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 ) -app...

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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
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Summary: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.
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ISSN:1382-6905
1573-2886
DOI:10.1007/s10878-022-00914-6