Multi Cover of a Polygon Minimizing the Sum of Areas

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, i r α i , 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 polynomialtime 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.

A constant-factor approximation for multi-covering with disks

“…Abu-Affash et al [1] gave an O(k) approximation for the problem using mainly geometric ideas. Bar-Yehuda and Rawitz [3] gave another algorithm that achieves the same approximation factor of O(k), using an analysis based on the local ratio technique.…”

“…There is a considerable amount of work on clustering and covering problems related to the MCMC problem, and we refer the reader to the previous papers for a detailed survey [1,3]. Here, we offer a view of some of that work from the standpoint of techniques that may be applicable to the problem at hand.…”

“…Our technique is closely related to the geometric approach of Abu-Affash et al [1] and builds on their ideas. For each x ∈ X, consider the disk δ(y k (x), ||y k (x) − x||2) of radius ||y k (x) − x||2 centered at y k (x), the k-th nearest neighbor of x in Y .…”

“…We call this the non-uniform minimum-cost multi cover problem (non-uniform MCMC problem). Abu-Affash et al [1] consider the version of this problem where κ(x) = k, ∀x ∈ X, where k > 0 is some given integer. We will refer to this as the uniform MCMC problem, or following [1], simply as the MCMC problem.…”

“…Abu-Affash et al [1] consider the version of this problem where κ(x) = k, ∀x ∈ X, where k > 0 is some given integer. We will refer to this as the uniform MCMC problem, or following [1], simply as the MCMC problem. We are interested in designing a polynomial time algorithm that outputs a feasible solution (a κcover) whose cost is at most some factor f ≥ 1 times the cost of an optimal solution.…”