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Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths

Authors Stefan Kratsch , Florian Nelles

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LIPIcs.STACS.2020.38.pdf
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Stefan Kratsch
  • Humboldt-Universität zu Berlin, Germany
Florian Nelles
  • Humboldt-Universität zu Berlin, Germany

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Stefan Kratsch and Florian Nelles. Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.38

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 ?(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 ?(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 ?(cw²n²), resp. ?(mw²n + n²). If fast matrix multiplication is allowed then the latter can be improved to ?(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 ∈ ?(n^{1 - ε}) for any ε > 0.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Shortest paths
Keywords
  • All-pairs shortest Paths
  • efficient parameterized Algorithms
  • parameterized Complexity
  • Clique-width
  • Modular-width

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