Local ratio method on partial set multi-cover

Abstract

In this paper, we study the minimum partial set multi-cover problem (PSMC). Given an element set E, a collection of subsets ${\mathcal {S}}\subseteq 2^E$, a cost $c_S$ on each set $S\in {\mathcal {S}}$, a covering requirement $r_e$ for each element $e\in E$, and an integer k, the PSMC problem 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 ${\mathcal {F}}$. This paper presents an approximation algorithm using local ratio method achieving performance ratio $\max \left\{ \frac{\Delta }{k}\left( \frac{1}{f-r_{\min }}+\frac{r_{\max }}{r_{\min }}\right) ,\frac{1}{\rho }+\frac{f}{r_{\min }}+\frac{1}{r_{\max }}-\frac{1}{\rho r_{\max }}-1,\frac{1}{\rho }\right\}$, where $\Delta$ is the size of a maximum set, f is the maximum number of sets containing a common element, $\rho$ is the minimum percentage of elements required to be fully covered during iterations of the algorithm, and $r_{\max }$ and $r_{\min }$ are the maximum and the minimum covering requirement, respectively. In particular, when $r_{\max }$ is a constant, the first term can be omitted. Notice that our ratio coincides with the classic ratio f for both the set multi-cover problem (in which case $k=|E|$) and the partial set single-cover problem (in which case $r_{\max }=1$). However, when $k<|E|$ and $r_{\max }>1$, the ratio might be as large as $\Theta (n)$. This result shows an interesting “shock wave like” feature of approximating PSMC. The purpose of this paper is trying to arouse some interest in such a feature and attract more work on this challenging problem.