A Journey Through Discrete Mathematics pp 367-374 | Cite as

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

## 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).

## Notes

### Acknowledgements

We thank the referees for valuable comments and Rico Zenklusen for constructive discussions.

## References

- 1.A. Condon, The complexity of stochastic games. Inf. Comput.
**96**(2), 203–224 (1992)MathSciNetCrossRefMATHGoogle Scholar - 2.N. Halman, Simple stochastic games, parity games, mean payoff games and discounted payoff games are all LP-type problems. Algorithmica
**49**(1), 37–50 (2007)MathSciNetCrossRefMATHGoogle Scholar - 3.M. Jurdziński, Deciding the winner in parity games is in UP ∩ co-UP. Inf. Process. Lett.
**68**(3), 119–124 (1998)MathSciNetCrossRefMATHGoogle Scholar - 4.C.S. Karthik, Did the train reach its destination: the complexity of finding a witness (2016). https//arxiv.org/abs/1609.03840Google Scholar
- 5.B. Katz, I. Rutter, G. Woeginger, An algorithmic study of switch graphs. Acta Inf.
**49**(5), 295–312 (2012)MathSciNetCrossRefMATHGoogle Scholar - 6.B. Korte, J. Vygen,
*Combinatorial Optimization: Theory and Algorithms*, 5th edn. (Springer, Berlin, 2012)CrossRefMATHGoogle Scholar