A Bicriteria Approximation Algorithm for Minimum Submodular Cost Partial Multi-Cover Problem

Abstract

This paper studies randomized approximation algorithm for a variant of the set cover problem called minimum submodular cost partial multi-cover (SCPMC).In a partial set cover problem, the goal is to find a minimum cost sub-collection of sets covering at least a required fraction of elements. In a multi-cover problem, each element e has a covering requirement r e , and the goal is to find a minimum cost sub-collection of sets S ′ which fully covers all elements, where an element e is fully covered by S ′ if e belongs to at least r e sets of S ′ . In a minimum submodular cost set cover problem (SCSC), the cost function on sub-collection of sets is submodular and the goal is to find a set cover with the minimum cost.The SCPMC problem studied in this paper is a combination of the above three problems, in which the cost function on sub-collection of sets is submodular and the goal is to find a minimum cost sub-collection of sets which fully covers at least q-percentage of all elements. Previous work shows that such a combination enormously increases the difficulty of studies, even when the cost function is linear.In this paper, assuming that the maximum covering requirement r max = max e r e is a constant and the cost function is nonnegative, monotone nondecreasing, and submodular, we give the first randomized bicriteria algorithm for SCPMC the output of which fully covers at least (q−ε)-percentage of all elements and the performance ratio is O(b/ε) with a high probability, where b = max e f re and f is the maximum number of sets containing a common element. The algorithm is based on a novel non-linear program. Furthermore, in the case when the * Corresponding Author: Zhao Zhang, hxhzz@sina.com. covering requirement r ≡ 1, a bicriteria O(f /ε)-approximation can be achieved even when monotonicity requirement is dropped off from the cost function.