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Optimality Program in Segment and String Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11159))

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Abstract

Planar graphs are known to allow subexponential algorithms running in time \(2^{O(\sqrt{n})}\) or \(2^{O(\sqrt{n} \log n)}\) for most of the paradigmatic problems, while the brute-force time \(2^{\varTheta (n)}\) is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in \(2^{O(n^{2/3}\log n)}\) by Fox and Pach [SODA’11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time \(2^{O(n^{2/3}\log ^{O(1)}n)}\) on string graphs while an algorithm running in time \(2^{o(n)}\) for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker lower bound, excluding a \(2^{o(n^{2/3})}\) algorithm (under the ETH). The construction exploits the celebrated Erdős-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set, but not to Min Dominating Set and Min Independent Dominating Set.

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Notes

  1. 1.

    Full proofs of theorems marked with (\(\star \)) can be found in [6].

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Author information

Authors and Affiliations

  1. ENS Lyon, LIP, Lyon, France

    Édouard Bonnet

  2. Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland

    Paweł Rzążewski

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  1. Édouard Bonnet
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  2. Paweł Rzążewski
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Correspondence to Paweł Rzążewski .

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Editors and Affiliations

  1. Universität Rostock, Rostock, Germany

    Andreas Brandstädt

  2. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Ekkehard Köhler

  3. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Klaus Meer

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Bonnet, É., Rzążewski, P. (2018). Optimality Program in Segment and String Graphs. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_7

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  • DOI: https://doi.org/10.1007/978-3-030-00256-5_7

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