Decomposition of Multiple Coverings into More Parts

TL;DR

This paper proves that multiple coverings of the plane with translates of a centrally symmetric convex polygon can be decomposed into simpler coverings, improving previous bounds and motivated by sensor network monitoring applications.

Contribution

It establishes a linear bound for decomposing multiple coverings of the plane with convex polygons, advancing beyond the quadratic bounds of prior work.

Findings

Decomposition of multiple coverings into more parts is possible with a linear bound.

Improves previous quadratic upper bounds for covering decompositions.

Results are motivated by sensor network monitoring problems.

Abstract

We prove that for every centrally symmetric convex polygon Q, there exists a constant alpha such that any alpha*k-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Toth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery lifetime.