Exact Algorithms for the Max-Min Dispersion Problem

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 is an element of P, we want to find a subset S. P with vertical bar S vertical bar = k such that cost(S) = min{d(p, q) vertical bar p, q is an element of 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.