A Subexponential Algorithm for ARRIVAL

Authors Bernd Gärtner , Sebastian Haslebacher , Hung P. Hoang

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Bernd Gärtner
  • Institute of Theoretical Computer Science, Department of Computer Science, ETH Zürich, Switzerland
Sebastian Haslebacher
  • Department of Computer Science, ETH Zürich, Switzerland
Hung P. Hoang
  • Institute of Theoretical Computer Science, Department of Computer Science, ETH Zürich, Switzerland


We thank Günter Rote for pointing out an error in an earlier version of the manuscript.

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Bernd Gärtner, Sebastian Haslebacher, and Hung P. Hoang. A Subexponential Algorithm for ARRIVAL. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The ARRIVAL problem is to decide the fate of a train moving along the edges of a directed graph, according to a simple (deterministic) pseudorandom walk. The problem is in NP∩coNP but not known to be in 𝖯. The currently best algorithms have runtime 2^Θ(n) where n is the number of vertices. This is not much better than just performing the pseudorandom walk. We develop a subexponential algorithm with runtime 2^O(√nlog n). We also give a polynomial-time algorithm if the graph is almost acyclic. Both results are derived from a new general approach to solve ARRIVAL instances.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Pseudorandom walks
  • reachability
  • graph games
  • switching systems


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