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    Crossing minimization in perturbed drawings

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    Author
    Fulek, Radoslav
    Tóth, Csaba D.
    Affiliation
    Univ Arizona
    Issue Date
    2020-06-12
    Keywords
    Computational Theory and Mathematics
    Control and Optimization
    Applied Mathematics
    Discrete Mathematics and Combinatorics
    Computer Science Applications
    
    Metadata
    Show full item record
    Publisher
    SPRINGER
    Citation
    Fulek, R., Tóth, C.D. Crossing minimization in perturbed drawings. J Comb Optim 40, 279–302 (2020). https://doi.org/10.1007/s10878-020-00586-0
    Journal
    JOURNAL OF COMBINATORIAL OPTIMIZATION
    Rights
    Copyright © Springer Science+Business Media, LLC, part of Springer Nature 2020.
    Collection Information
    This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at repository@u.library.arizona.edu.
    Abstract
    Due to data compression or low resolution, nearby vertices and edges of a graph drawn in the plane may be bundled to a common node or arc. We model such a "compromised" drawing by a piecewise linear map phi:G -> R-2. We wish to perturb phi by an arbitrarily small epsilon>0 into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An epsilon-perturbation, for every epsilon>0, is given by a piecewise linear map psi epsilon:G -> R-2 with ||phi-psi epsilon||<epsilon is the uniform norm (i.e.,supnorm). We present a polynomial-time solution for this optimization problem whenGis a cycle and the map phi has nospurs(i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) whenGis an arbitrary graph and phi has no spurs, and (ii) when phi may have spurs andGis a cycle or a union of disjoint paths.
    Note
    12 month embargo; published online: 12 June 2020
    ISSN
    1382-6905
    DOI
    10.1007/s10878-020-00586-0
    Version
    Final accepted manuscript
    ae974a485f413a2113503eed53cd6c53
    10.1007/s10878-020-00586-0
    Scopus Count
    Collections
    UA Faculty Publications

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