A simple lower bound for ARRIVAL

Abstract

The ARRIVIAL problem introduced by Dohrau, G\"artner, Kohler, Matou\v{s}ek and Welzl concerns a train moving on a directed graph proceeding along outward edges according to the position of 'switches' at each vertex, which in turn are toggled whenever the train passes through them. The problem asks whether the train every reaches a designated destination vertex. It is known that ARRIVAL is contained in UP $\cap$ coUP, while the previously best published lower bound is that it is NL-hard. In this note we provide a simple reduction to the $\mathsf{DIGICOMP}_\mathsf{EXP}$ problem considered by Aaronson. It follows in particular that ARRIVAL is both CC-hard and PL-hard.