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A note on the minimum power partial cover problem on the plane
Journal of Combinatorial Optimization  (IF1.262),  Pub Date : 2022-06-11, DOI: 10.1007/s10878-022-00869-8
Han Dai, Bin Deng, Weidong Li, Xiaofei Liu

Given a set of n points and a set of m sensors on the plane, each sensor s can adjust its power p(s) and the covering range which is a disk of radius r(s) satisfying \(p(s)=c\cdot r(s)^{\alpha }\). The minimum power partial cover problem, introduced by Freund (Proceedings of international workshop on approximation and online algorithms, pp 137–150. 2011. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.737.1320), is to determine the power assignment on every sensor such that at least k (\(k\le n\)) points are covered and the total power consumption is minimized. By generalizing the method in Li (Journal of Com. Opti.2020. https://doi.org/10.1007/s10878-020-00567-3) whose approximation ratio is \(3^{\alpha }\) and enlarging the radius of each disk in the relaxed independent set, we present an \(O(\alpha )\)-approximation algorithm.