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Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets

Authors Tim Planken , Torsten Ueckerdt

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Tim Planken
  • University of Birmingham, UK
Torsten Ueckerdt
  • Karlsruhe Institute of Technology, Germany

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Tim Planken and Torsten Ueckerdt. Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.74

Abstract

A range family ℛ is a family of subsets of ℝ^d, like all halfplanes, or all unit disks. Given a range family ℛ, we consider the m-uniform range capturing hypergraphs ℋ(V,ℛ,m) whose vertex-sets V are finite sets of points in ℝ^d with any m vertices forming a hyperedge e whenever e = V ∩ R for some R ∈ ℛ. Given additionally an integer k ≥ 2, we seek to find the minimum m = m_ℛ(k) such that every ℋ(V,ℛ,m) admits a polychromatic k-coloring of its vertices, that is, where every hyperedge contains at least one point of each color. Clearly, m_ℛ(k) ≥ k and the gold standard is an upper bound m_ℛ(k) = O(k) that is linear in k.
A t-shallow hitting set in ℋ(V,ℛ,m) is a subset S ⊆ V such that 1 ≤ |e ∩ S| ≤ t for each hyperedge e; i.e., every hyperedge is hit at least once but at most t times by S. We show for several range families ℛ the existence of t-shallow hitting sets in every ℋ(V,ℛ,m) with t being a constant only depending on ℛ. This in particular proves that m_ℛ(k) ≤ tk = O(k) in such cases, improving previous polynomial bounds in k. Particularly, we prove this for the range families of all axis-aligned strips in ℝ^d, all bottomless and topless rectangles in ℝ², and for all unit-height axis-aligned rectangles in ℝ².

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypergraphs
Keywords
  • geometric hypergraphs
  • range spaces
  • polychromatic coloring
  • shallow hitting sets

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