A note on the minimum power partial cover problem on the plane

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 . 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 <= 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.