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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

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.

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)

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@InProceedings{kratsch_et_al:LIPIcs.STACS.2020.38, author = {Kratsch, Stefan and Nelles, Florian}, 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 = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.38}, 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} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We study the influence of a graph parameter called modular-width on the time complexity for optimally solving well-known polynomial problems such as Maximum Matching, Triangle Counting, and Maximum s-t Vertex-Capacitated Flow. The modular-width of a graph depends on its (unique) modular decomposition tree, and can be computed in linear time O(n+m) for graphs with n vertices and m edges. Modular decompositions are an important tool for graph algorithms, e.g., for linear-time recognition of certain graph classes.
Throughout, we obtain efficient parameterized algorithms of running times O(f(mw)n+m), O(n+f(mw)m) , or O(f(mw)+n+m) for low polynomial functions f and graphs of modular-width mw. Our algorithm for Maximum Matching, running in time O(mw^2 log mw n+m), is both faster and simpler than the recent O(mw^4n+m) time algorithm of Coudert et al. (SODA 2018). For several other problems, e.g., Triangle Counting and Maximum b-Matching, we give adaptive algorithms, meaning that their running times match the best unparameterized algorithms for worst-case modular-width of mw=Theta(n) and they outperform them already for mw=o(n), until reaching linear time for mw=O(1).

Stefan Kratsch and Florian Nelles. Efficient and Adaptive Parameterized Algorithms on Modular Decompositions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kratsch_et_al:LIPIcs.ESA.2018.55, author = {Kratsch, Stefan and Nelles, Florian}, title = {{Efficient and Adaptive Parameterized Algorithms on Modular Decompositions}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {55:1--55:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.55}, URN = {urn:nbn:de:0030-drops-95187}, doi = {10.4230/LIPIcs.ESA.2018.55}, annote = {Keywords: efficient parameterized algorithms, modular-width, adaptive algorithms} }

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