Dispersion in disks

We present three new approximation algorithms with improved constant ratios for selecting $n$ points in $n$ disks such that the minimum pairwise distance among the points is maximized. (1) A very simple $O(n\log n)$-time algorithm with ratio $0.511$ for disjoint unit disks. (2) An LP-based algorithm with ratio $0.707$ for disjoint disks of arbitrary radii that uses a linear number of variables and constraints, and runs in polynomial time. (3) A hybrid algorithm with ratio either $0.4487$ or $0.4674$ for (not necessarily disjoint) unit disks that uses an algorithm of Cabello in combination with either the simple $O(n\log n)$-time algorithm or the LP-based algorithm. The LP algorithm can be extended for disjoint balls of arbitrary radii in $\RR^d$, for any (fixed) dimension $d$, while preserving the features of the planar algorithm. The algorithm introduces a novel technique which combines linear programming and projections for approximating Euclidean distances. The previous best approximation ratio for dispersion in disjoint disks, even when all disks have the same radius, was $1/2$. Our results give a partial answer to an open question raised by Cabello, who asked whether the ratio $1/2$ could be improved.