Piercing axis-parallel boxes
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Date
2018
Authors
Chudnovsky, Maria
Spirkl, Sophie
Zerbib, Shira
Journal Title
Journal ISSN
Volume Title
Publisher
The Electronic Journal of Combinatorics
Abstract
Let F be a finite family of axis-parallel boxes in Rd such that F contains no k + 1 pairwise disjoint boxes. We prove that if F contains a subfamily M of k pairwise disjoint boxes with the property that for every F E F and M E M with F ∩ M ≠ 6= Ø, either F contains a corner of M or M contains 2d-1 corners of F, then F can be pierced by O(k) points. One consequence of this result is that if d = 2 and the ratio between any of the side lengths of any box is bounded by a constant, then F can be pierced by O(k) points. We further show that if for each two intersecting boxes in F a corner of one is contained in the other, then F can be pierced by at
most O(k log log(k)) points, and in the special case where F contains only cubes this bound improves to O(k).
Description
Keywords
axis-parallel boxes, hitting set, piercing, packing and covering, matching and covering