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Matching colored points with rectangles

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Abstract

Let S be a point set in the plane such that each of its elements is colored either red or blue. A matching of S with rectangles is any set of pairwise-disjoint axis-aligned closed rectangles such that each rectangle contains exactly two points of S. Such a matching is monochromatic if every rectangle contains points of the same color, and is bichromatic if every rectangle contains points of different colors. We study the following two problems: (1) Find a maximum monochromatic matching of S with rectangles. (2) Find a maximum bichromatic matching of S with rectangles. For each problem we provide a polynomial-time approximation algorithm that constructs a matching with at least 1 / 4 of the number of rectangles of an optimal matching. We show that the first problem is \(\mathsf {NP}\)-hard even if either the matching rectangles are restricted to axis-aligned segments or S is in general position, that is, no two points of S share the same x or y coordinate. We further show that the second problem is also \(\mathsf {NP}\)-hard, even if S is in general position. These \(\mathsf {NP}\)-hardness results follow by showing that deciding the existence of a matching that covers all points is \(\mathsf {NP}\)-complete in each case. Additionally, we prove that it is \(\mathsf {NP}\)-complete to decide the existence of a matching with rectangles that cover all points in the case where all the points have the same color, solving an open problem of Bereg et al. (Comput Geom 42(2):93–108, 2009).

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Notes

  1. The associated graph is the bipartite graph with vertices the variables and the clauses, and there exists an edge between a variable and a clause if and only if the variable participates in the clause.

  2. If D(a, b) is a box, then its interior is the interior of the box. Otherwise, if D(a, b) is a segment, then its interior is the set \(D(a,b){\setminus }\{a,b\}\).

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Acknowledgments

C. Ochoa is supported by CONICYT-PCHA/Doctorado Nacional/ 2013-63130161 (Chile). P. Pérez-Lantero is supported by project Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003 (Chile). J. Rojas-Ledesma is supported by CONICYT-PCHA/Doctorado Nacional/2013-63130209 (Chile). The authors would like to thank anonymous referees for helpful comments and suggestions.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    L. E. Caraballo

  2. Departamento de Ciencias de la Computación (DCC), Universidad de Chile, Santiago, Chile

    C. Ochoa & J. Rojas-Ledesma

  3. Escuela de Ingeniería Civil en Informática, Universidad de Valparaíso, Valparaiso, Chile

    P. Pérez-Lantero

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  1. L. E. Caraballo
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  2. C. Ochoa
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  3. P. Pérez-Lantero
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  4. J. Rojas-Ledesma
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Correspondence to P. Pérez-Lantero.

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Caraballo, L.E., Ochoa, C., Pérez-Lantero, P. et al. Matching colored points with rectangles. J Comb Optim 33, 403–421 (2017). https://doi.org/10.1007/s10878-015-9971-x

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  • DOI: https://doi.org/10.1007/s10878-015-9971-x

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