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Covering Points with Pairs of Concentric Disks

In this paper, we study the following problem motivated by applications in wireless local area networks. We are given a set of m pairs of concentric disks in ddimensional space, d ∈ {1, 2}, where each pair consists of one disk with radius one and the other with radius two. We are also given a set of n points such that the union of the m pairs of disks covers all the n points. The goal is to select exactly one disk from each pair such that every point is covered by at least one disk and the number of points covered by at least one disk with radius one is maximized; we refer to this as the sDiskCover problem. When d = 1 (i.e., we have m pairs of intervals on the real line), we give an exact algorithm that solves the sDiskCover problem in O(mn) time. We also consider a special case of the problem for d = 1, and show that it can be solved in O(mn) time. For d = 2, we prove that the sDiskCover problem is NP-hard.