An Approximation Algorithm for the 2-Dispersion Problem

Let P be a set of points on the plane, and d(p, q) be the distance between a pair of points p, q in P. For a point p∈P and a subset S ⊂ P with |S|≥3, the 2-dispersion cost, denoted by cost2(p, S), of p with respect to S is the sum of (1) the distance from p to the nearest point in S\setminus{p} and (2) the distance from p to the second nearest point in S\setminus{p}. The 2-dispersion cost cost2(S) of S ⊂ P with |S|≥3 is minp∈S{cost2(p, S)}. Given a set P of n points and an integer k we wish to compute k point subset S of P with maximum cost2(S). In this paper we give a simple 1/({4\sqrt{3}}) approximation algorithm for the problem.