Abstract
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.
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References
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Acknowledgements
We thank the two anonymous reviewers for their valuable comments and constructive suggestions, which helped us significantly improve the quality of our work. The work is supported in part by the National Natural Science Foundation of China [No. 12071417].
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Dai, H., Deng, B., Li, W. et al. A note on the minimum power partial cover problem on the plane. J Comb Optim 44, 970–978 (2022). https://doi.org/10.1007/s10878-022-00869-8
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DOI: https://doi.org/10.1007/s10878-022-00869-8