# A Subexponential Algorithm for ARRIVAL

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(√(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.

READ FULL TEXT