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ARRIVAL: A Zero-Player Graph Game in NP ∩ coNP

In: A Journey Through Discrete Mathematics

Author

Listed:
  • Jérôme Dohrau

    (Institute of Theoretical Computer Science, ETH Zürich, Department of Computer Science)

  • Bernd Gärtner

    (Institute of Theoretical Computer Science, ETH Zürich, Department of Computer Science)

  • Manuel Kohler

    (Institute of Theoretical Computer Science, ETH Zürich, Department of Computer Science)

  • Jiří Matoušek

    (Institute of Theoretical Computer Science, ETH Zürich, Department of Computer Science
    Charles University, Malostranské nám. 25, Department of Applied Mathematics)

  • Emo Welzl

    (Institute of Theoretical Computer Science, ETH Zürich, Department of Computer Science)

Abstract

Suppose that a train is running along a railway network, starting from a designated origin, with the goal of reaching a designated destination. The network, however, is of a special nature: every time the train traverses a switch, the switch will change its position immediately afterwards. Hence, the next time the train traverses the same switch, the other direction will be taken, so that directions alternate with each traversal of the switch. Given a network with origin and destination, what is the complexity of deciding whether the train, starting at the origin, will eventually reach the destination? It is easy to see that this problem can be solved in exponential time, but we are not aware of any polynomial-time method. In this short paper, we prove that the problem is in NP ∩ coNP. This raises the question whether we have just failed to find a (simple) polynomial-time solution, or whether the complexity status is more subtle, as for some other well-known (two-player) graph games (Halman, Algorithmica 49(1):37–50, 2007).

Suggested Citation

  • Jérôme Dohrau & Bernd Gärtner & Manuel Kohler & Jiří Matoušek & Emo Welzl, 2017. "ARRIVAL: A Zero-Player Graph Game in NP ∩ coNP," Springer Books, in: Martin Loebl & Jaroslav Nešetřil & Robin Thomas (ed.), A Journey Through Discrete Mathematics, pages 367-374, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-44479-6_14
    DOI: 10.1007/978-3-319-44479-6_14
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