A seminal result of [Fleischer et al., 2004], [Karakostas and Kolliopoulos, 2004] and [Yang and Huang, 2004] states that system optimal multi-commodity static network flows are always implementable as tolled Wardrop equilibrium flows even if users have heterogeneous value-of-time sensitivities. Their proof uses LP-duality to characterize the general implementability of network flows by tolls. For the much more complex setting of dynamic flows, [Graf et al., 2025] identified necessary and sufficient conditions for a dynamic s-d flow to be implementable as a tolled dynamic equilibrium. They used the machinery of (infinite-dimensional) strong duality to obtain their characterizations. Their work, however, does not answer the question of whether system optimal dynamic network flows are implementable by tolls.

We consider dynamic network flows and study the following question: Which dynamic edge flows can be implemented as tolled dynamic equilibrium flows? We study this question for the “heterogeneous-user” model, where the flow particles are partitioned into populations having different valuations of travel time and money spent. As our main result, we give the first characterization of this type of implementability showing that for single-source single-destination networks and heterogeneous users, a dynamic edge flow is implementable by tolls if and only if the induced subgraph of the edge flow contains no cycle of positive length containing the destination. For the proof of this result we make several technical contributions: We formulate a novel infinite dimensional optimization problem, where the goal is to minimize the weighted travel times with respect to the fixed network loading induced by the given edge flow. Using the recently introduced concept of parameterized network loadings (cf. [23]), we prove existence of optimal solutions, strong duality, and a characterization of special optimal solutions for which an inequality is tight. These results are then all used for the proof of the above mentioned main characterization.