Approximation Algorithms For The Dispersion Problems in a Metric Space

Abstract

In this article, we consider the c-dispersion problem in a metric space (X, d).Let P = {p 1 , p 2 , . . . , p n } be a set of n points in a metric space (X, d). For each point p ∈ P and S ⊆ P , we define cost c (p, S) as the sum of distances from p to the nearest c points in S \ {p}, where c ≥ 1 is a fixed integer. We define cost c (S) = min p∈S {cost c (p, S)} for S ⊆ P . In the c-dispersion problem, a set P of n points in a metric space (X, d) and a positive integer k ∈ [c + 1, n] are given. The objective is to find a subset S ⊆ P of size k such that cost c (S) is maximized.We propose a simple polynomial time greedy algorithm that produces a 2c-factor approximation result for the c-dispersion problem in a metric space. The best known result for the c-dispersion problem in the Euclidean metric space (X, d) is 2c 2 , where P ⊆ R 2 and the distance function is Euclidean distance [ Amano, K. and Nakano, S. I., Away from Rivals, CCCG, pp.68-71, 2018 ]. We also prove that the c-dispersion problem in a metric space is W [1]-hard.