Covering a Set of Points with a Minimum Number of Equal Disks via Simulated Annealing

Abstract

This paper considers the following problem. Given n points in the plane, what is the minimum number of disks of radius r needed to cover all n points? A point is covered if it lies inside at least one disk. The problem is equivalent to the unit disc cover problem. It is known to be NP-hard. To solve the problem, we propose a stochastic optimization algorithm of estimated time complexity O(n$^\wedge$2). First, the original problem is converted into an unconstrained optimization problem by introducing an objective function, called energy, in such a way that if a configuration of disks minimizes the energy, then it necessarily covers all points and does this with a minimum number of disks. Thus, the original problem is reduced to finding a configuration of disks that minimizes the energy. To solve this optimization problem, we propose a Monte Carlo simulation algorithm based on the simulated annealing technique. Computer experiments are performed demonstrating the ability of the algorithm to find configurations with minimum energy, hence solving the original problem of covering the points with minimum number of disks. By adjusting the parameters of the simulation, we can increase the probability that the found configuration is a solution.