Mittagsseminar (by J. Lengler, K. Bringmann, B. Gärtner, M. Hoffmann, R. Kyng, D.Steurer, V. Traub)

Date and Time: Tuesday, March 03, 2026, 12:15 pm

Duration: 30 minutes

Location: OAT S15

Speaker: Matthias Mnich (Technische Universität Hamburg)

Improved algorithms for the ARRIVAL problem

In 2014, the ARRIVAL problem was conceived at ETH Zürich. Dohrau, Gärtner, Kohler, Matouˇsek and Welzl describe it as follows: "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?” They devised an algorithm for ARRIVAL which is exponential in the number $n$ of switches. As they also showed the problem to lie in $NP ∩ co-NP$, a natural question to ask is whether ARRIVAL is in $P$. The currently fastest algorithm, by Gärtner, Haslebacher and Hoang, solves Arrival in subexponential time $n^{O(\sqrt{n})$. We report progress on the ARRIVAL problem by presenting asymptotically faster algorithms for sparse instances, dense instances, and general instances of ARRIVAL.

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