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Ranking and Drawing in Subexponential Time

  • Henning Fernau
  • Fedor V. Fomin
  • Daniel Lokshtanov
  • Matthias Mnich
  • Geevarghese Philip
  • Saket Saurabh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)

Abstract

In this paper we obtain parameterized subexponential-time algorithms for p -Kemeny Aggregation (p-KAGG) — a problem in social choice theory — and for p -One-Sided Crossing Minimization (p-OSCM) – a problem in graph drawing (see the introduction for definitions). These algorithms run in time \(\mathcal{O}^{*}(2^{\mathcal{O}(\sqrt{k}{\rm log} k)})\), where k is the parameter, and significantly improve the previous best algorithms with running times \(\cal{O}^{*}\)(1.403 k ) and \(\cal{O}^{*}\)(1.4656 k ), respectively. We also study natural “above-guarantee” versions of these problems and show them to be fixed parameter tractable. In fact, we show that the above-guarantee versions of these problems are equivalent to a weighted variant of p -Directed Feedback Arc Set. Our results for the above-guarantee version of p-KAGG reveal an interesting contrast. We show that when the number of “votes” in the input to p-KAGG is odd the above guarantee version can still be solved in time \(\mathcal{O}^{*}(2^{\mathcal{O}(\sqrt{k}{\rm log} k)})\), while if it is even then the problem cannot have a subexponential time algorithm unless the exponential time hypothesis fails (equivalently, unless FPT=M[1]).

Keywords

Kemeny Aggregation One-Sided Crossing Minimization Parameterized Complexity Subexponential-time Algorithms Social Choice Theory Graph Drawing Directed Feedback Arc Set 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Henning Fernau
    • 1
  • Fedor V. Fomin
    • 2
  • Daniel Lokshtanov
    • 2
  • Matthias Mnich
    • 3
  • Geevarghese Philip
    • 4
  • Saket Saurabh
    • 4
  1. 1.Universität Trier FB 4—Abteilung InformatikTrierGermany
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.Eindhoven University of TechnologyEindhovenThe Netherlands
  4. 4.The Institute of Mathematical SciencesChennaiIndia

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