Dispersion in disks

TL;DR

This paper introduces three new approximation algorithms with improved ratios for selecting points in disks to maximize minimum pairwise distance, advancing beyond the previous 1/2 ratio and addressing an open problem.

Contribution

The paper presents novel algorithms with better approximation ratios for dispersion in disks, including LP-based and hybrid methods, extending to higher dimensions.

Findings

Achieved a 0.511 ratio for disjoint unit disks in O(n log n) time.

Developed an LP-based algorithm with ratio 0.707 for disks of arbitrary radii.

Created a hybrid algorithm with ratios 0.4487 or 0.4674 for general unit disks.

Abstract

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.