Exact Algorithms for the Max-Min Dispersion Problem

Abstract

Given a set P of n elements, and a function d that assigns a non-negative real number d(p, q) for each pair of elements $p,q \in P$, we want to find a subset $S\subseteq P$ with $|S|=k$ such that $\mathop {\mathrm {cost}}(S)= \min \{ d(p,q) \mid p,q \in S\}$ is maximized. This is the max-min k-dispersion problem. In this paper, exact algorithms for the max-min k-dispersion problem are studied. We first show the max-min k-dispersion problem can be solved in $O(n^{\omega k/3} \log n)$ time. Then, we show two special cases in which we can solve the problem quickly. Namely, we study the cases where a set of n points lie on a line and where a set of n points lie on a circle (and the distance is measured by the shortest arc length on the circle). We obtain O(n)-time algorithms after sorting.