Packing, Hitting, and Colouring Squares

TL;DR

This paper establishes new bounds on the ratio of hitting points to disjoint squares and relates chromatic and clique numbers for non-axis-aligned squares, filling a gap in geometric combinatorics research.

Contribution

It provides the first non-trivial bounds for the / u ratio for arbitrary squares and relates this to chromatic and clique numbers, extending known results beyond axis-aligned cases.

Findings

Upper bound of 6 for / u ratio for unit squares

Upper bound of 10 for / u ratio for squares of varying sizes

Established bounds for / u and / ratios for non-axis-aligned squares

Abstract

Given a finite family of squares in the plane, the packing problem asks for the maximum number $\nu$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $\tau$ of points hitting all of them. Clearly, $\tau \ge \nu$. Both problems are known to be NP-hard, even for families of axis-parallel unit squares. The main results of this work provide the first non-trivial bounds for the $\tau / \nu$ ratio for not necessarily axis-parallel squares. We establish an upper bound of $6$ for unit squares and $10$ for squares of varying sizes. The worst ratios we can provide with examples are $3$ and $4$, respectively. For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval $[\frac{3}{2},2]$ for unit squares and $[\frac{3}{2},4]$ for squares of varying sizes. The methods we introduced for the $\tau/\nu$ ratio can also be used to relate the chromatic number $\chi$ and clique number $\omega$ of squares by bounding the $\chi/\omega$ ratio by $6$ for unit squares and $9$ for squares of varying sizes. The $\tau / \nu$ and $\chi/\omega$ ratios have already been bounded before by a constant for "fat" objects, the fattest and simplest of which are disks and squares. However, while disks have received significant attention, specific bounds for squares have remained essentially unexplored. This work intends to fill this gap.