Colorful Strips

Greg Aloupis, Jean Cardinal, Sébastien Collette, Shinji Imahori, Matias Korman, Stefan Langerman, Oded Schwartz, Shakhar Smorodinsky, Perouz Taslakian

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k-1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k-1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.

Original languageEnglish
Pages (from-to)327-339
Number of pages13
JournalGraphs and Combinatorics
Volume27
Issue number3
DOIs
Publication statusPublished - 2011 May 1
Externally publishedYes

Keywords

  • Computational geometry
  • Covering decomposition
  • Hypergraph coloring
  • Lovász local lemma

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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  • Cite this

    Aloupis, G., Cardinal, J., Collette, S., Imahori, S., Korman, M., Langerman, S., Schwartz, O., Smorodinsky, S., & Taslakian, P. (2011). Colorful Strips. Graphs and Combinatorics, 27(3), 327-339. https://doi.org/10.1007/s00373-011-1014-5