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 $${(1+\epsilon)}$$ in $${O(n+1/\epsilon^4 \log 1/\epsilon)}$$ time for any $${\epsilon >0 }$$ . To the best of the authors’ knowledge, this work presents the first linear time approximation scheme on the facility dispersion problems. Copyright Springer Science+Business Media, LLC. 2013