License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ICALP.2017.54
URN: urn:nbn:de:0030-drops-74790
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7479/
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Bezáková, Ivona ; Curticapean, Radu ; Dell, Holger ; Fomin, Fedor V.

Finding Detours is Fixed-Parameter Tractable

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LIPIcs-ICALP-2017-54.pdf (0.6 MB)

Abstract

We consider the following natural "above guarantee" parameterization of the classical longest path problem: For given vertices s and t of a graph G, and an integer k, the longest detour problem asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that the longest detour problem is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k.

Furthermore, we study a related problem, exact detour, that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k * poly(n), and a deterministic algorithm with running time about 6.745^k * poly(n), showing that this problem is FPT as well. Our algorithms for the exact detour problem apply to both undirected and directed graphs.

BibTeX - Entry

@InProceedings{bezkov_et_al:LIPIcs:2017:7479,
  author =	{Ivona Bez{\'a}kov{\'a} and Radu Curticapean and Holger Dell and Fedor V. Fomin},
  title =	{{Finding Detours is Fixed-Parameter Tractable}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7479},
  URN =		{urn:nbn:de:0030-drops-74790},
  doi =		{10.4230/LIPIcs.ICALP.2017.54},
  annote =	{Keywords: longest path, fixed-parameter tractable algorithms, above-guarantee parameterization, graph minors}
}

Keywords: longest path, fixed-parameter tractable algorithms, above-guarantee parameterization, graph minors
Collection: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 07.07.2017

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