Covering grids with multiplicity

Abstract

Given a finite grid in R 2 , how many lines are needed to cover all but one point at least k times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We solve this problem for various types of grids, in particular showing the tightness of the Ball-Serra bound when one side is much larger than the other. In other cases, we prove new lower bounds that improve upon Ball-Serra and provide an asymptotic answer for almost all grids. For the standard grid {0, . . . , n − 1} × {0, . . . , n − 1}, we prove nontrivial upper and lower bounds on the number of lines needed. To prove our results, we combine linear programming duality with some combinatorial arguments.