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Jump Number of Two-Directional Orthogonal Ray Graphs

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Integer Programming and Combinatoral Optimization (IPCO 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6655))

Abstract

We model maximum cross-free matchings and minimum biclique covers of two-directional orthogonal ray graphs (2-dorgs ) as maximum independent sets and minimum hitting sets of an associated family of rectangles in the plane, respectively. We then compute the corresponding maximum independent set using linear programming and uncrossing techniques. This procedure motivates an efficient combinatorial algorithm to find a cross-free matching and a biclique cover of the same cardinality, proving the corresponding min-max relation.

We connect this min-max relation with the work of Györi, [19] Lubiw [23], and Frank and Jordán [16] on seemingly unrelated problems. Our result can be seen as a non-trivial application of Frank and Jordán’s Theorem.

As a direct consequence, we obtain the first polynomial algorithm for the jump number problem on 2-dorgs. For the subclass of convex graphs, our approach is a vast improvement over previous algorithms. Additionally, we prove that the weighted maximum cross-free matching problem is NP-complete for 2-dorgs and give polynomial algorithms for some subclasses.

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Author information

Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, USA

    José A. Soto

  2. Operation Research Center, MIT, Cambridge, MA, USA

    Claudio Telha

Authors
  1. José A. Soto
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  2. Claudio Telha
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Editor information

Editors and Affiliations

  1. Mathematical Sciences Department, IBM T. J. Watson Research Center, 10598, Yorktown Heights, NY, USA

    Oktay Günlük

  2. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600, Eindhoven, MB, The Netherlands

    Gerhard J. Woeginger

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Soto, J.A., Telha, C. (2011). Jump Number of Two-Directional Orthogonal Ray Graphs. In: Günlük, O., Woeginger, G.J. (eds) Integer Programming and Combinatoral Optimization. IPCO 2011. Lecture Notes in Computer Science, vol 6655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20807-2_31

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  • DOI: https://doi.org/10.1007/978-3-642-20807-2_31

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20806-5

  • Online ISBN: 978-3-642-20807-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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