A linear time approximation scheme for computing geometric maximum k-star

Facility dispersion seeks to locate the facilities as far away from each other as possible, which has attracted a multitude of research attention recently due to the pressing need on dispersing facilities in various scenarios. In this paper, as a facility dispersion problem, the geometric maximum k-star problem is considered. Given a set P of n points in the Euclidean plane, a k-star is defined as selecting k points from P and linking k − 1 points to the center point. The maximum k-star problem asks to compute a k-star on P with the maximum total length over its k − 1 edges. A linear time approximation scheme is proposed for this problem. It approximates the maximum k-star within a factor of (1 + ) in O(n + 1/ 4 log 1/ ) time for any > 0. To the best of the authors’ knowledge, this work presents the first linear time approximation scheme on the facility dispersion problems.