In this paper, we study the minimum partial set multi-cover problem (PSMC). Given an element set E, a collection of subsets S⊆2E\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {S}}\subseteq 2^E$$\end{document}, a cost cS\documentclass[12pt]{minimal}
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\begin{document}$$c_S$$\end{document} on each set S∈S\documentclass[12pt]{minimal}
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\begin{document}$$S\in {\mathcal {S}}$$\end{document}, a covering requirement re\documentclass[12pt]{minimal}
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\begin{document}$$r_e$$\end{document} for each element e∈E\documentclass[12pt]{minimal}
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\begin{document}$$e\in E$$\end{document}, and an integer k, the PSMC problem is to find a sub-collection F⊆S\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}\subseteq {\mathcal {S}}$$\end{document} to fully cover at least k elements such that the cost of F\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}$$\end{document} is as small as possible, where element e is fully covered by F\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}$$\end{document} if it belongs to at least re\documentclass[12pt]{minimal}
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\begin{document}$$r_e$$\end{document} sets of F\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {F}}$$\end{document}. This paper presents an approximation algorithm using local ratio method achieving performance ratio maxΔk1f-rmin+rmaxrmin,1ρ+frmin+1rmax-1ρrmax-1,1ρ\documentclass[12pt]{minimal}
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\begin{document}$$\max \left\{ \frac{\Delta }{k}\left( \frac{1}{f-r_{\min }}+\frac{r_{\max }}{r_{\min }}\right) ,\frac{1}{\rho }+\frac{f}{r_{\min }}+\frac{1}{r_{\max }}-\frac{1}{\rho r_{\max }}-1,\frac{1}{\rho }\right\} $$\end{document}, where Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} is the size of a maximum set, f is the maximum number of sets containing a common element, ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} is the minimum percentage of elements required to be fully covered during iterations of the algorithm, and rmax\documentclass[12pt]{minimal}
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\begin{document}$$r_{\max }$$\end{document} and rmin\documentclass[12pt]{minimal}
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\begin{document}$$r_{\min }$$\end{document} are the maximum and the minimum covering requirement, respectively. In particular, when rmax\documentclass[12pt]{minimal}
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\begin{document}$$r_{\max }$$\end{document} 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|\documentclass[12pt]{minimal}
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\begin{document}$$k=|E|$$\end{document}) and the partial set single-cover problem (in which case rmax=1\documentclass[12pt]{minimal}
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\begin{document}$$r_{\max }=1$$\end{document}). However, when k<|E|\documentclass[12pt]{minimal}
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\begin{document}$$k<|E|$$\end{document} and rmax>1\documentclass[12pt]{minimal}
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\begin{document}$$r_{\max }>1$$\end{document}, the ratio might be as large as Θ(n)\documentclass[12pt]{minimal}
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\begin{document}$$\Theta (n)$$\end{document}. 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.
