Packing, Hitting, and Colouring Squares

TL;DR

The paper studies the packing number $\nu$ and hitting number $\tau$ for finite families of squares, with emphasis on non-axis-aligned cases and the ratio $\tau/\nu$. It develops a hitting-degeneracy framework built on local homothety and hole-patching to bound $\tau$ in terms of $\nu$ and to relate the chromatic and clique numbers. Main results give $\tau \le 6\nu$ for unit squares and $\tau \le 10\nu$ for squares of varying sizes, with lower bounds $\tau/\nu \ge 3$ and $4$; and bounds on the $\chi/\omega$ ratio $\chi/\omega \le 6$ for unit squares and $\le 9$ for varying sizes. These findings fill a gap in the theory for squares, extend fat-object bounds, and suggest avenues for improved axis-parallel bounds and further questions about Helly-type properties.

Abstract

Given a finite family of squares in the plane, the packing problem asks for the maximum number $ν$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $τ$ of points hitting all of them. Clearly, $τ\ge ν$. 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 $τ/ ν$ 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 $τ/ν$ ratio can also be used to relate the chromatic number $χ$ and clique number $ω$ of squares by bounding the $χ/ω$ ratio by $6$ for unit squares and $9$ for squares of varying sizes. The $τ/ ν$ and $χ/ω$ 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.

Figures (18)

• Figure 1: Three pairwise intersecting unit squares.
• Figure 2: An axis-parallel square and the domain of the centres of its neighbours.
• Figure 3: A "hole" not covered by any of three disks, but "patched": the three vertices of the triangle hit all unit squares having their centre in the dark (blue) hole as well.
• Figure 4: The outer and inner squares of two intersecting convex sets with slimness $\mathop{\mathrm{\rho}}\nolimits_{\infty}$ and inner unit squares.
• Figure 5: Illustration of Lemma \ref{['lem:outside']}.

Theorems & Definitions (32)

• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4: Pach, 1980 1980_Pach
• proof