Abstract
We consider a geometric optimization problem that arises in sensor network design. Given a polygon P (possibly with holes) with n vertices, a set Y of m points representing sensors, and an integer k, 1 ≤ k ≤ m. The goal is to assign a sensing range, r i , to each of the sensors y i ∈ Y, such that each point p ∈ P is covered by at least k sensors, and the cost, \(\sum_i r_i^\alpha\), of the assignment is minimized, where α is a constant.
In this paper, we assume that α= 2, that is, find a set of disks centered at points of Y, such that (i) each point in P is covered by at least k disks, and (ii) the sum of the areas of the disks is minimized. We present, for any constant k ≥ 1, a polynomial-time c 1-approximation algorithm for this problem, where c 1 = c 1(k) is a constant. The discrete version, where one has to cover a given set of n points, X, by disks centered at points of Y, arises as a subproblem. We present a polynomial-time c 2-approximation algorithm for this problem, where c 2 = c 2(k) is a constant.
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Abu-Affash, A.K., Carmi, P., Katz, M.J., Morgenstern, G. (2011). Multi Cover of a Polygon Minimizing the Sum of Areas. In: Katoh, N., Kumar, A. (eds) WALCOM: Algorithms and Computation. WALCOM 2011. Lecture Notes in Computer Science, vol 6552. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19094-0_15
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DOI: https://doi.org/10.1007/978-3-642-19094-0_15
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