Dispersion problem on a convex polygon

Abstract

Given a set P={p1,p2,…,pn} of n points in R2 and a positive integer k(≤n), we wish to find a subset S of P of size k such that the cost of a subset S, cost(S)=min⁡{d(p,q)|p,q∈S}, is maximized, where d(p,q) is the Euclidean distance between two points p and q. The problem is called the max-min k-dispersion problem. In this article, we consider the max-min k-dispersion problem, where a given set P of n points are vertices of a convex polygon. We refer to this variant of the problem as the convex k-dispersion problem.We propose an 1.733-factor approximation algorithm for the convex k-dispersion problem. In addition, we study the convex k-dispersion problem for k=4. We propose an iterative algorithm that returns an optimal solution of size 4 in O(n3) time.