Approximation Algorithm for the Partial Set Multi-Cover Problem

Partial set cover problem and set multi-cover problem are two generalizations of the 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, 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 \({\mathcal {S}}'\subseteq {\mathcal {S}}\) to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least \(r_e\) sets of \({\mathcal {S}}'\). Denote by \(r_{\max }=\max \{r_e:e\in E\}\) the maximum covering requirement. We present an \((O (r_{\max }\log ^2n(1+\ln (\frac{1}{\varepsilon })+\frac{1-q}{\varepsilon q})),1-\varepsilon )\)-bicriteria approximation algorithm, that is, the output of our algorithm has cost \(O(r_{\max }\log ^2 n(1+\ln (\frac{1}{\varepsilon })+\frac{1-q}{\varepsilon q}))\) times of the optimal value while the number of fully covered elements is at least \((1-\varepsilon )q|E|\).