Unfortunately, it is found in this paper that this is not the case. It is shown that continuous-time versions of the problems satisfying the smoothness conditions of control theory can have a finite but very large number of feasible solutions. The same happens for the discrete-time case. These difficulties arise even with the simplest versions of the problem (with unique origin-destination paths, perfect information, and deterministic travel times.)
The paper also shows that the continuous-time feasibility problem is NP-hard, and that if we restrict our attention to (practical) problems whose data can be described with a finite number of bits (e.g., in discrete time), then the problem is NP-complete. These results are established by showing that the problem instances of interest can be related to the Directed Hamiltonian Path problem by a polynomial tranformation.