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arXiv · 2102.06427

A Subexponential Algorithm for ARRIVAL

Bernd Gärtner·Sebastian Haslebacher·Hung P. Hoang
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Abstract

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 $NP \cap coNP$ but not known to be in $P$. The currently best algorithms have runtime $2^{\Theta(n)}$ 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 $2^{O(\sqrt{n}\log n)}$. 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.

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Bernd Gärtner, Sebastian Haslebacher, Hung P. Hoang. 2021-02-12. A Subexponential Algorithm for ARRIVAL. https://arxiv.org/abs/2102.06427

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