Approximation Algorithm for the Partial Set Multi-Cover Problem

Abstract: 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 $\mathcal S\subseteq 2<sup>E$, a total covering ratio $q$ which is a constant between 0 and 1, each set $S\in\mathcal S$ is associated with a cost $c_S$, each element $e\in E$ is associated with a covering requirement $r_e$, the goal is to find a minimum cost sub-collection $\mathcal S&#39;\subseteq\mathcal S$ to fully cover at least $q|E|$ elements, where element $e$ is fully covered if it belongs to at least $r_e$ sets of $\mathcal S&#39;$. Denote by $r_{\max}=\max{r_e\colon e\in E}$ the maximum covering requirement. We present an $$(O(\frac{r_{\max}\log<sup>2n}{\varepsilon}),1-\varepsilon)$$-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 $(1-\varepsilon)q|E|$.