NP-hardness of shortest path problems in networks with non-FIFO time-dependent travel times

We study the complexity of earliest arrival problems on time-dependent networks with non-FIFO travel time functions, i.e. when departing later might lead to an earlier arrival. In this paper, we present a simple proof of the weak NP-hardness of the problem for travel time functions defined on integers. This simplifies and reproduces an earlier result from Orda and Rom. Our proof generalizes to travel time functions defined on rational numbers and also implies that, in this case, the problem becomes harder, i.e. is strongly NP-hard. As arbitrary functions are impractical for applications, we also study a more realistic problem model where travel time functions are piecewise linear and represented by a sequence of breakpoints with integer coordinates. We show that this problem formulation is strongly NP-hard, too. As an intermediate step for this proof, we also show the strong NP-completeness of SubsetProduct on rational numbers.