Primal Dual Algorithm for Partial Set Multi-cover

Abstract

In a minimum partial set multi-cover problem (MinPSMC), given an element set E, a collection of subsets $\mathcal S \subseteq 2^E$, a cost $w_S$ on each set $S\in \mathcal S$, a covering requirement $r_e$ for each element $e\in E$, and an integer k, the goal is to find a sub-collection $\mathcal F \subseteq \mathcal S$ to fully cover at least k elements such that the cost of $\mathcal F$ is as small as possible, where element e is fully covered by $\mathcal F$ if it belongs to at least $r_e$ sets of $\mathcal F$. On the application side, the problem has its background in the seed selection problem in a social network. On the theoretical side, it is a natural combination of the minimum partial (single) set cover problem (MinPSC) and the minimum set multi-cover problem (MinSMC). Although both MinPSC and MinSMC admit good approximations whose performance ratios match those lower bounds for the classic set cover problem, previous studies show that theoretical study on MinPSMC is quite challenging. In this paper, we prove that MinPSMC cannot be approximated within factor $O(n^\frac{1}{2(\log \log n)^c})$ under the ETH assumption. A primal dual algorithm for MinPSMC is presented with a guaranteed performance ratio $O(\sqrt{n})$ when $r_{\max }$ and f are constants, where $r_{\max } =\max _{e\in E} r_e$ is the maximum covering requirement and f is the maximum frequency of elements (that is the maximum number of sets containing a common element). We also improve the ratio for a restricted version of MinPSMC which possesses a graph-type structure.