Fast algorithms for supermodular and non-supermodular minimization via bi-criteria strategy - Zhang, Xiaojuan; Liu, Qian; Li, Min; Zhou, Yang (Journal of combinatorial optimization, 2022)
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.