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Matching random colored points with rectangles

Author

Listed:
  • Josué Corujo

    (Université de Strasbourg)

  • David Flores-Peñaloza

    (Universidad Nacional Autónoma de México)

  • Clemens Huemer

    (Universitat Politècnica de Catalunya)

  • Pablo Pérez-Lantero

    (Universidad de Santiago de Chile (USACH))

  • Carlos Seara

    (Universitat Politècnica de Catalunya)

Abstract

Given $$n>0$$ n > 0 , let $$S\subset [0,1]^2$$ S ⊂ [ 0 , 1 ] 2 be a set of n points, chosen uniformly at random. Let $$R\cup B$$ R ∪ B be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random variable M(n) equal to the number of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed axis-aligned rectangles that cover exactly two points of S of the same color, and is strong in the sense that all of its rectangles are pairwise disjoint. We prove that almost surely $$M(n)\ge 0.83\,n$$ M ( n ) ≥ 0.83 n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm that runs over the random point set as a Markov chain.

Suggested Citation

  • Josué Corujo & David Flores-Peñaloza & Clemens Huemer & Pablo Pérez-Lantero & Carlos Seara, 2023. "Matching random colored points with rectangles," Journal of Combinatorial Optimization, Springer, vol. 45(2), pages 1-18, March.
    • Handle: RePEc:spr:jcomop:v:45:y:2023:i:2:d:10.1007_s10878-023-01010-z
    DOI: 10.1007/s10878-023-01010-z
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    References listed on IDEAS

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    1. L. E. Caraballo & C. Ochoa & P. Pérez-Lantero & J. Rojas-Ledesma, 2017. "Matching colored points with rectangles," Journal of Combinatorial Optimization, Springer, vol. 33(2), pages 403-421, February.
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