A Lower Bound on the Area of a 3-Coloured Disk Packing

Details A Lower Bound on the Area of a 3-Coloured Disk Packing A Lower Bound on the Area of a 3-Coloured Disk Packing

2008-04-08

arxiv.org/abs/0804.1173v1

Computer Science

Computational Geometry

15 pages (11 pages + 4 page appendix), 12 figures

Scientific paper

Given a set of unit-disks in the plane with union area $A$, what fraction of $A$ can be covered by selecting a pairwise disjoint subset of the disks? Rado conjectured 1/4 and proved $1/4.41$. Motivated by the problem of channel-assignment for wireless access points, in which use of 3 channels is a standard practice, we consider a variant where the selected subset of disks must be 3-colourable with disks of the same colour pairwise-disjoint. For this variant of the problem, we conjecture that it is always possible to cover at least $1/1.41$ of the union area and prove $1/2.09$. We also provide an $O(n^2)$ algorithm to select a subset achieving a $1/2.77$ bound.

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