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Approximating the Maximum Rectilinear Crossing Number

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Book cover Computing and Combinatorics (COCOON 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9797))

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Abstract

Drawing a graph in a way that minimizes the number of edge-crossings is a well-studied problem. Recently there has been work characterizing both the minimum and maximum number of edge-crossings possible in various graph classes, assuming rectilinear (straight-line) edges. In this paper, we investigate the algorithmic problem of maximizing the number of edge-crossings over all rectilinear drawings a graph. We show that this problem is NP-hard and lies in \(\exists \mathbb {R}\). We give a nontrivial derandomization of the natural randomized 1/3-approximation algorithm, which generalizes to a weighted setting as well as to an ordering constraint satisfaction problem. We evaluate these algorithms and other heuristics in simulation.

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Notes

  1. 1.

    For ease of exposition, we refer simply to the number of crossings for the remainder of the proof, although exactly the same analysis applies in the weighted setting, whether the weights of crossings are based on underlying edge weights or are permitted to be completely arbitrary, as in an OCSP.

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Acknowledgements

This work was supported in part by PSC-CUNY Research Award 67665-00 45, CUNY Collaborative Incentive Research Grant (CIRG 21) 2153, and a Research in the Classroom Idea Grant.

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Authors and Affiliations

  1. The Graduate Center of the City University of New York, New York, USA

    Samuel Bald, Matthew P. Johnson & Ou Liu

  2. Lehman College, City University of New York, New York, USA

    Matthew P. Johnson

Authors
  1. Samuel Bald
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  2. Matthew P. Johnson
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  3. Ou Liu
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Corresponding author

Correspondence to Samuel Bald .

Editor information

Editors and Affiliations

  1. Virginia Commonwealth Univ , Richmond, Virginia, USA

    Thang N. Dinh

  2. University of Florida , Gainesville, Alabama, USA

    My T. Thai

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Bald, S., Johnson, M.P., Liu, O. (2016). Approximating the Maximum Rectilinear Crossing Number. In: Dinh, T., Thai, M. (eds) Computing and Combinatorics . COCOON 2016. Lecture Notes in Computer Science(), vol 9797. Springer, Cham. https://doi.org/10.1007/978-3-319-42634-1_37

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  • DOI: https://doi.org/10.1007/978-3-319-42634-1_37

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42633-4

  • Online ISBN: 978-3-319-42634-1

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