Approximation Algorithm for the Partial Set Multi-Cover Problem

Partial set cover problem and set multi-cover problem are two generalizations of set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set $E$, a collection of sets $\mathcal S\subseteq 2^E$, a total covering ratio $q$ which is a constant between 0 and 1, each set $S\in\mathcal S$ is associated with a cost $cS$, each element $e\in E$ is associated with a covering requirement $re$, the goal is to find a minimum cost sub-collection $\mathcal S'\subseteq\mathcal S$ to fully cover at least $q|E|$ elements, where element $e$ is fully covered if it belongs to at least $re$ sets of $\mathcal S'$. Denote by $r{\max}=\max{re\colon e\in E}$ the maximum covering requirement. We present an $(O(\frac{r{\max}\log^{2n}{\varepsilon}),1-\varepsilon)$-bicriteria} approximation algorithm, that is, the output of our algorithm has cost at most $O(\frac{r_{\max}\log² n}{\varepsilon})$ times of the optimal value while the number of fully covered elements is at least $(1-\varepsilon)q|E|$.