Abstract
In this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s–t paths, cycles, matchings, and κ-factors for κ ∈ {1,2}. We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k 1 − ε for any ε > 0, where k is the number of crossings in G. We then show that the problems are fixed-parameter tractable if we use the number of crossings in the given graph as the parameter. Finally we present a simple but effective heuristic for spanning trees.
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References
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Halperin, D.: Arrangements. In: Handbook of Discrete and Computational Geometry, ch. 24, pp. 529–562. CRC Press, Boca Raton (2004)
Jansen, K., Woeginger, G.J.: The complexity of detecting crossingfree configurations in the plane. BIT 33, 580–595 (1993)
Knauer, C., Schramm, É., Spillner, A., Wolff, A.: Configurations with few crossings in topological graphs. Techical Report 2005-24, Universität Karlsruhe (September 2005), http://www.ubka.uni-karlsruhe.de/cgi-bin/psview?document=/ira/2005/24
Kratochvíl, J., Lubiw, A., Nešetřil, J.: Noncrossing subgraphs in topological layouts. SIAM J. Disc. Math. 4(2), 223–244 (1991)
Micali, S., Vazirani, V.V.: An \( {O} (\sqrt{|V|} |{E}|)\) algorithm for finding maximum matching in general graphs. In: Proc. IEEE Symp. Found. Comp. Sci., pp. 17–27 (1980)
Rendl, F., Woeginger, G.: Reconstructing sets of orthogonal line segments in the plane. Discrete Mathematics 119, 167–174 (1993)
Tutte, W.T.: A short proof of the factor theorem for finite graphs. Canad. J. Math. 6, 347–352 (1954)
Vazirani, V.V.: A theory of alternating paths and blossoms for proving correctness of the O (| V |1/2 | E |) general graph matching algorithm. Combinatorica 14, 71–91 (1994)
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Knauer, C., Schramm, É., Spillner, A., Wolff, A. (2005). Configurations with Few Crossings in Topological Graphs. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_61
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DOI: https://doi.org/10.1007/11602613_61
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
Online ISBN: 978-3-540-32426-3
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