Dispersion problem on a convex polygon

Given a set $P=\{p_1,p_2,\dots ,p_n\}$ of n points in $R^2$ and a positive integer k $(\le n)$, we wish to find a subset S of P of size k such that the cost of a subset S, $cost\left(S\right)=\min \left\{d\right(p,q\left)\right|p,q\in S\}$, 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 $O\left(n^3\right)$ time.