eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-03-04
38:1
38:15
10.4230/LIPIcs.STACS.2020.38
article
Efficient Parameterized Algorithms for Computing All-Pairs Shortest Paths
Kratsch, Stefan
1
https://orcid.org/0000-0002-0193-7239
Nelles, Florian
1
Humboldt-Universität zu Berlin, Germany
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol154-stacs2020/LIPIcs.STACS.2020.38/LIPIcs.STACS.2020.38.pdf
All-pairs shortest Paths
efficient parameterized Algorithms
parameterized Complexity
Clique-width
Modular-width