Packing, Hitting, and Colouring Squares
Marco Caoduro, Andr\'as Seb\H{o}

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 of pairwise disjoint squares among them, while the hitting problem for the minimum number of points hitting all of them. Clearly, . 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 ratio for not necessarily axis-parallel squares. We establish an upper bound of for unit squares and for squares of varying sizes. The worst ratios we can provide with examples are and , respectively. For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval for unit squares and for squares of varying sizes. The methods we introduced for the ratio can also be…
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Taxonomy
TopicsOptimization and Packing Problems · graph theory and CDMA systems · Graph Labeling and Dimension Problems
