Approximation Algorithms For The Dispersion Problems in a Metric Space
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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.
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