Packing-constrained point coverings

TL;DR

This paper investigates the packing-constrained point covering problem, establishing bounds on the minimum number of points that cannot all be covered by a packing of unit disks, highlighting its higher complexity class.

Contribution

It provides the first bounds on the minimum number of points in the PC^2 problem, suggesting its complexity exceeds standard packing and covering problems.

Findings

Bounds established: 11 <= N <= 55

Indicates PC^2 belongs to a higher complexity class

Highlights disparity in bounds as symptomatic of complexity

Abstract

In the packing-constrained point covering problem, PC^2, one seeks configurations of points in the plane that cannot all be covered by a packing arrangement of unit disks. We consider in particular the problem of finding the minimum number of points N for which such a configuration exists and obtain the bounds 11 <= N <= 55. The disparity of these bounds is symptomatic, we believe, of the fact that PC^2 belongs in a higher complexity class than the standard packing and covering problems.