Range assignment of base-stations maximizing coverage area without interference

Abstract

We study the problem of assigning non-overlapping geometric objects centered at a given set of points such that the sum of area covered by them is maximized. If the points are placed on a straight-line and the objects are disks, then the problem is solvable in polynomial time. However, we show that the problem is NP-hard even for simplest objects like disks or squares in R 2 . Eppstein [CCCG, pages 260-265, 2016] proposed a polynomial time algorithm for maximizing the sum of radii (or perimeter) of non-overlapping balls or disks when the points are arbitrarily placed on a plane. We show that Eppstein's algorithm for maximizing sum of perimeter of the disks in R 2 gives a 2-approximation solution for the sum of area maximization problem. We propose a PTAS for our problem. These approximation results are extendible to higher dimensions. All these approximation results hold for the area maximization problem by regular convex polygons with even number of edges centered at the given points.

Faster Algorithms for some Optimization Problems on Collinear Points

“…We want to assign to every p i ∈ P a radius r i such that the disks with the given radii do not overlap and their total area, or equivalently r 2 i , is as large as possible. Acharyya et al [1] showed how to obtain such an assignment in O(n 2 ) time. We show how to obtain such an assignment in linear time.…”

“…We briefly review the O(n 2 )-time algorithm of Acharyya et al [1]. First, compute a set D of disks centered at points of P , which is the superset of every optimal solution.…”

“…, p n , and maximize the total area. Since the maximum weight independent set of m intervals that are given in sorted order of their left endpoints can be computed in O(m) time [8], the time complexity of the above algorithm is essentially dominated by the size of D. Acharyya et al [1] showed how to compute such a set D of size Θ(n 2 ) and order the corresponding intervals in O(n 2 ) time. Therefore, the total running time of their algorithm is O(n 2 ).…”

“…These three problems are solvable in polynomial time in 1-dimension. Both Problem 1 and Problem 2 are NP-hard in dimension d, for every d 2, and both have a PTAS [2,3,4].…”

“…Acharyya et al [2] showed that Problem 1 can be solved in O(n 2 ) time. Eppstein [7] proved that an alternate version of this problem, where the goal is to maximize the sum of the radii, can be solved in O(n 2−1/d ) time for any constant dimension d. Bilò et al [4] showed that Problem 2 is solvable in polynomial time by reducing it to an integer linear program with a totally unimodular constraint matrix.…”