License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2020.38
URN: urn:nbn:de:0030-drops-118992
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11899/
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Kratsch, Stefan ; Nelles, Florian

Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths

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LIPIcs-STACS-2020-38.pdf (0.6 MB)


Abstract

Computing all-pairs shortest paths is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the 𝒪(n³) time algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist for the vertex-weighted version if fast matrix multiplication may be used. Yuster (SODA 2009) gave an algorithm running in time 𝒪(n^2.842), but no combinatorial, truly subcubic algorithm is known.
Motivated by the recent framework of efficient parameterized algorithms (or "FPT in P"), we investigate the influence of the graph parameters clique-width (cw) and modular-width (mw) on the running times of algorithms for solving ALL-PAIRS SHORTEST PATHS. We obtain efficient (and combinatorial) parameterized algorithms on non-negative vertex-weighted graphs of times 𝒪(cw²n²), resp. 𝒪(mw²n + n²). If fast matrix multiplication is allowed then the latter can be improved to 𝒪(mw^{1.842} n + n²) using the algorithm of Yuster as a black box. The algorithm relative to modular-width is adaptive, meaning that the running time matches the best unparameterized algorithm for parameter value mw equal to n, and they outperform them already for mw ∈ 𝒪(n^{1 - ε}) for any ε > 0.

BibTeX - Entry

@InProceedings{kratsch_et_al:LIPIcs:2020:11899,
  author =	{Stefan Kratsch and Florian Nelles},
  title =	{{Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{38:1--38:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11899},
  URN =		{urn:nbn:de:0030-drops-118992},
  doi =		{10.4230/LIPIcs.STACS.2020.38},
  annote =	{Keywords: All-pairs shortest Paths, efficient parameterized Algorithms, parameterized Complexity, Clique-width, Modular-width}
}

Keywords: All-pairs shortest Paths, efficient parameterized Algorithms, parameterized Complexity, Clique-width, Modular-width
Collection: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue Date: 2020
Date of publication: 04.03.2020


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