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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${(1+\epsilon)}$$</EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${O(n+1/\epsilon^4 \log 1/\epsilon)}$$</EquationSource> </InlineEquation> time for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\epsilon >0 }$$</EquationSource> </InlineEquation>. 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