Multi Cover of a Polygon Minimizing the Sum of Areas

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, ri, to each of the sensors yi ∈ Y, such that each point p ∈ P is covered by at least k sensors, and the cost, Σi rαi, 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 c1-approximation algorithm for this problem, where c1 = c1(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 c2-approximation algorithm for this problem, where c2 = c2(k) is a constant.