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

This paper studies randomized approximation algorithm for a variant of the set cover problem called minimum submodular cost partial multi-cover (SCPMC), in which each element $e$ has a covering requirement $r_e$ and a profit $p_e$, and the cost function on sub-collection of sets is submodular, the goal is to find a minimum cost sub-collection of sets which fully covers at least $q$-percentage of total profit, where an element $e$ is fully covered by sub-collection $S'$ if and only if it belongs to at least $r_e$ sets of $\mathcal S'$. 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 $(q-\varepsilon)$-percentage of all elements and the performance ratio is $O(b/\varepsilon)$ with a high probability, where $b=\max_e\binom{f}{r_{e}}$ 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 covering requirement $r\equiv 1$, a bicriteria $O(f/\varepsilon)$-approximation can be achieved even when monotonicity requirement is dropped off from the cost function.