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≤|V|\documentclass[12pt]{minimal}
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\begin{document}$$k\le |V|$$\end{document}. 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·r(s)α\documentclass[12pt]{minimal}
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\begin{document}$$p(D(s,r(s)))=c\cdot r(s)^{\alpha }$$\end{document}. The k-PCPC determines a disk set F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} such that at least k points are covered, where the center of any disk in F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} is a sensor. The objective is to minimize the total power of the disk set F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} plus the penalty of R, where R is the set of points that are not covered by F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}. 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 3α\documentclass[12pt]{minimal}
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\begin{document}$$3^{\alpha }$$\end{document}.
