Abstract
In a minimum partial set multi-cover problem (MinPSMC), given an element set X, a collection of subsets \({\mathcal {S}} \subseteq 2^X\), a cost \(c_S\) on each set \(S\in {\mathcal {S}}\), a covering requirement \(r_x\) for each element \(x\in X\), 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 x is fully covered by \({\mathcal {F}}\) if it belongs to at least \(r_x\) sets of \({\mathcal {F}}\). Recently, it was proved that MinPSMC is at least as hard as the densest k-subgraph problem. The question is: how about the problem in some geometric settings? In this paper, we consider the MinPSMC problem in which X is a set of points on the plane and \({\mathcal {S}}\) is a set of unit squares (MinPSMC-US). Under the assumption that \(r_x=f_x\) for every \(x\in X\), where \(f_x=|\{S\in {\mathcal {S}}:x\in S\}|\) is the number of sets containing element x, we design an approximation algorithm achieving approximation ratio \((1+\varepsilon )\) for MinPSMC-US.
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This research work is supported in part by NSFC (11901533, U20A2068, 11771013), ZJNSFC (LD19A010001), and NSF (1907472).
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Ran, Y., Huang, X., Zhang, Z. et al. Approximation algorithm for minimum partial multi-cover under a geometric setting. Optim Lett 16, 667–680 (2022). https://doi.org/10.1007/s11590-021-01746-9
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DOI: https://doi.org/10.1007/s11590-021-01746-9