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.
Dispersion problem
linear programming
approximation algorithm
299-310
Regular Paper
Adrian
Dumitrescu
Adrian Dumitrescu
Minghui
Jiang
Minghui Jiang
10.4230/LIPIcs.STACS.2010.2464
Creative Commons Attribution-NoDerivs 3.0 Unported license
https://creativecommons.org/licenses/by-nd/3.0/legalcode