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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

A rotor walk in a directed graph can be thought of as a deterministic version of a Markov Chain, where a pebble moves from vertex to vertex following a simple rule until a terminal vertex, or sink, has been reached. The ARRIVAL problem, as defined by Dohrau et al. [Dohrau et al., 2017], consists in determining which sink will be reached. While the walk itself can take an exponential number of steps, this problem belongs to the complexity class NP ∩ co-NP without being known to be in P. In this work, we define a class of directed graphs, namely tree-like multigraphs, which are multigraphs having the global shape of an undirected tree. We prove that in this class, ARRIVAL can be solved in almost linear time, while the number of steps of a rotor walk can still be exponential. Then, we give an application of this result to solve some deterministic analogs of stochastic models (e.g., Markovian decision processes, Stochastic Games).

David Auger, Pierre Coucheney, and Loric Duhazé. Polynomial Time Algorithm for ARRIVAL on Tree-Like Multigraphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{auger_et_al:LIPIcs.MFCS.2022.12, author = {Auger, David and Coucheney, Pierre and Duhaz\'{e}, Loric}, title = {{Polynomial Time Algorithm for ARRIVAL on Tree-Like Multigraphs}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.12}, URN = {urn:nbn:de:0030-drops-168103}, doi = {10.4230/LIPIcs.MFCS.2022.12}, annote = {Keywords: Rotor-routing, Rotor Walk, Reachability Problem, Game Theory, Tree-like Multigraph} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We present a generic strategy improvement algorithm (GSIA) to find an optimal strategy of simple stochastic games (SSG). We prove the correctness of GSIA, and derive a general complexity bound, which implies and improves on the results of several articles. First, we remove the assumption that the SSG is stopping, which is usually obtained by a polynomial blowup of the game. Second, we prove a tight bound on the denominator of the values associated to a strategy, and use it to prove that all strategy improvement algorithms are in fact fixed parameter tractable in the number r of random vertices. All known strategy improvement algorithms can be seen as instances of GSIA, which allows to analyze the complexity of converge from below by Condon [Condon, 1993] and to propose a class of algorithms generalising Gimbert and Horn’s algorithm [Gimbert and Horn, 2008; Gimbert and Horn, 2009]. These algorithms terminate in at most r! iterations, and for binary SSGs, they do less iterations than the current best deterministic algorithm given by Ibsen-Jensen and Miltersen [Ibsen-Jensen and Miltersen, 2012].

David Auger, Xavier Badin de Montjoye, and Yann Strozecki. A Generic Strategy Improvement Method for Simple Stochastic Games. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{auger_et_al:LIPIcs.MFCS.2021.12, author = {Auger, David and Badin de Montjoye, Xavier and Strozecki, Yann}, title = {{A Generic Strategy Improvement Method for Simple Stochastic Games}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.12}, URN = {urn:nbn:de:0030-drops-144524}, doi = {10.4230/LIPIcs.MFCS.2021.12}, annote = {Keywords: Simple Stochastic Games, Strategy Improvement, Parametrized Complexity, Stopping, Meta Algorithm, f-strategy} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

The best algorithm so far for solving Simple Stochastic Games is Ludwig’s randomized algorithm [Ludwig, 1995] which works in expected 2^{O(sqrt{n})} time. We first give a simpler iterative variant of this algorithm, using Bland’s rule from the simplex algorithm, which uses exponentially less random bits than Ludwig’s version. Then, we show how to adapt this method to the algorithm of Gimbert and Horn [Gimbert and Horn, 2008] whose worst case complexity is O(k!), where k is the number of random nodes. Our algorithm has an expected running time of 2^{O(k)}, and works for general random nodes with arbitrary outdegree and probability distribution on outgoing arcs.

David Auger, Pierre Coucheney, and Yann Strozecki. Solving Simple Stochastic Games with Few Random Nodes Faster Using Bland’s Rule. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{auger_et_al:LIPIcs.STACS.2019.9, author = {Auger, David and Coucheney, Pierre and Strozecki, Yann}, title = {{Solving Simple Stochastic Games with Few Random Nodes Faster Using Bland’s Rule}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.9}, URN = {urn:nbn:de:0030-drops-102488}, doi = {10.4230/LIPIcs.STACS.2019.9}, annote = {Keywords: simple stochastic games, randomized algorithm, parametrized complexity, strategy improvement, Bland’s rule} }