Abstract
In this paper, we introduce the k-prize-collecting minimum power cover problem (k-PCPC). In this problem, we are given a point set V, a sensor set S on a plane and a parameter k with \(k\le |V|\). Each sensor can adjust its power and the covering range of sensor s with power p(D(s, r(s))) is a disk D(s, r(s)), where r(s) is the radius of disk D(s, r(s)) and \(p(D(s,r(s)))=c\cdot r(s)^{\alpha }\). The k-PCPC determines a disk set \(\mathcal {F}\) such that at least k points are covered, where the center of any disk in \(\mathcal {F}\) is a sensor. The objective is to minimize the total power of the disk set \(\mathcal {F}\) plus the penalty of R, where R is the set of points that are not covered by \(\mathcal {F}\). This problem generalizes the well-known minimum power cover problem, minimum power partial cover problem and prize collecting minimum power cover problem. Our main result is to present a novel two-phase primal-dual algorithm for the k-PCPC with an approximation ratio of at most \(3^{\alpha }\).
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The work is supported in part by the National Natural Science Foundation of China [No. 12071417], Program for Excellent Young Talents of Yunnan University, Training Program of National Science Fund for Distinguished Young Scholars, and IRTSTYN.
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Liu, X., Li, W. & Xie, R. A primal-dual approximation algorithm for the k-prize-collecting minimum power cover problem. Optim Lett 16, 2373–2385 (2022). https://doi.org/10.1007/s11590-021-01831-z
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DOI: https://doi.org/10.1007/s11590-021-01831-z