Dispersion in Unit Disks

Abstract

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.