Max-Min k-Dispersion on a Convex Polygon
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In this paper, we consider the following k-dispersion problem. Given a set S of n points placed in the plane in a convex position, and an integer k (0<k<n), the objective is to compute a subset S'⊂ S such that |S'|=k and the minimum distance between a pair of points in S' is maximized. Based on the bounded search tree method we propose an exact fixed-parameter algorithm in O(2^k(n^2log n+n(log^2 n)(log k))) time, for this problem, where k is the parameter. The proposed exact algorithm is better than the current best exact exponential algorithm [n^O(√(k))-time algorithm by Akagi et al.,(2018)] whenever k<clog^2n for some constant c. We then present an O(logn)-time 1/2√(2)-approximation algorithm for the problem when k=3 if the points are given in convex position order.
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