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 |. 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 . The k-PCPC determines a disk set F such that at least k points are covered, where the center of any disk in F is a sensor. The objective is to minimize the total power of the disk set F plus the penalty of R, where R is the set of points that are not covered by 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 α .

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