Dispersion in Unit Disks

We present two new approximation algorithms with (improved) constant ratios for selecting $n$ points in $n$ unit disks such that the minimum pairwise distance among the points is maximized. (I) A very simple $O(n log{n})$-time algorithm with ratio $0.5110$ for disjoint unit disks. In combination with an algorithm of Cabello~cite{Ca07}, it yields a $O(n^2)$-time algorithm with ratio of $0.4487$ for dispersion in $n$ not necessarily disjoint unit disks. (II) A more sophisticated LP-based algorithm with ratio $0.6495$ for disjoint unit disks that uses a linear number of variables and constraints, and runs in polynomial time. The algorithm introduces a novel technique which combines linear programming and projections for approximating distances. The previous best approximation ratio for disjoint unit disks was $frac{1}{2}$. Our results give a partial answer to an open question raised by Cabello~cite{Ca07}, who asked whether $frac{1}{2}$ could be improved