Document Open Access Logo

Polynomial Time Algorithm for ARRIVAL on Tree-Like Multigraphs

Authors David Auger , Pierre Coucheney, Loric Duhazé

Thumbnail PDF


  • Filesize: 1.16 MB
  • 14 pages

Document Identifiers

Author Details

David Auger
  • DAVID Lab.,UVSQ, Université Paris Saclay, 45 avenue des Etats-Unis, 78000, Versailles, France
Pierre Coucheney
  • DAVID Lab.,UVSQ, Université Paris Saclay, 45 avenue des Etats-Unis, 78000, Versailles, France
Loric Duhazé
  • DAVID Lab., UVSQ, Université Paris Saclay, 45 avenue des Etats-Unis, 78000, Versailles, France
  • LISN, Université Paris Saclay, France


We want to thank Johanne Cohen for her advises and attentive review.

Cite AsGet BibTex

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)


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).

Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Rotor-routing
  • Rotor Walk
  • Reachability Problem
  • Game Theory
  • Tree-like Multigraph


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. David Auger, Pierre Coucheney, and Loric Duhaze. Polynomial time algorithm for arrival on tree-like multigraphs. arXiv preprint arXiv:2204.13151, 2022. Google Scholar
  2. David Auger, Pierre Coucheney, and Yann Strozecki. Finding optimal strategies of almost acyclic simple stochastic games. In International Conference on Theory and Applications of Models of Computation, pages 67-85. Springer, 2014. Google Scholar
  3. 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, 2019. Google Scholar
  4. Anders Björner, László Lovász, and Peter W Shor. Chip-firing games on graphs. European Journal of Combinatorics, 12(4):283-291, 1991. Google Scholar
  5. Richard P Brent and Paul Zimmermann. Modern computer arithmetic, volume 18. Cambridge University Press, 2010. Google Scholar
  6. Swee Hong Chan, Lila Greco, Lionel Levine, and Peter Li. Random walks with local memory. Journal of Statistical Physics, 184(1):1-28, 2021. Google Scholar
  7. Joshua N Cooper and Joel Spencer. Simulating a random walk with constant error. Combinatorics, Probability and Computing, 15(6):815-822, 2006. Google Scholar
  8. Jérôme Dohrau, Bernd Gärtner, Manuel Kohler, Jiří Matoušek, and Emo Welzl. Arrival: A zero-player graph game in NP ∩ coNP. In A journey through discrete mathematics, pages 367-374. Springer, 2017. Google Scholar
  9. Tobias Friedrich and Thomas Sauerwald. The cover time of deterministic random walks. In International Computing and Combinatorics Conference, pages 130-139. Springer, 2010. Google Scholar
  10. Bernd Gärtner, Thomas Dueholm Hansen, Pavel Hubácek, Karel Král, Hagar Mosaad, and Veronika Slívová. Arrival: Next stop in cls. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  11. Bernd Gärtner, Sebastian Haslebacher, and Hung P. Hoang. A Subexponential Algorithm for ARRIVAL. In ICALP 2021, volume 198, pages 69:1-69:14, 2021. Google Scholar
  12. Giuliano Pezzolo Giacaglia, Lionel Levine, James Propp, and Linda Zayas-Palmer. Local-to-global principles for rotor walk. arXiv preprint arXiv:1107.4442, 2011. Google Scholar
  13. Nir Halman. Simple stochastic games, parity games, mean payoff games and discounted payoff games are all LP-type problems. Algorithmica, 49(1):37-50, 2007. Google Scholar
  14. David Harvey and Joris Van Der Hoeven. Integer multiplication in time O(n log n). Annals of Mathematics, 193(2):563-617, 2021. Google Scholar
  15. Alexander E. Holroyd, Lionel Levine, Karola Mészáros, Yuyal Peres, James Propp, and David B. Wilson. Chip-Firing and Rotor-Routing on Directed Graphs, pages 331-364. Springer, 2008. Google Scholar
  16. AM Povolotsky, VB Priezzhev, and RR Shcherbakov. Dynamics of eulerian walkers. Physical review E, 58(5):5449, 1998. Google Scholar
  17. Vyatcheslav B Priezzhev, Deepak Dhar, Abhishek Dhar, and Supriya Krishnamurthy. Eulerian walkers as a model of self-organized criticality. Physical Review Letters, 77(25):5079, 1996. Google Scholar
  18. Rahul Savani, Matthias Mnich, Martin Gairing, and John Fearnley. Reachability switching games. Logical Methods in Computer Science, 17, 2021. Google Scholar
  19. Vladimir Yanovski, Israel A Wagner, and Alfred M Bruckstein. A distributed ant algorithm for protect efficiently patrolling a network. Algorithmica, 37(3):165-186, 2003. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail