Consider the problem of maximizing the revenue generated by tolls set on a subset of arcs of a transportation network, where origin-destination flows (commodities) are assigned to shortest paths with respect to the sum of tolls and initial costs. This work is concerned with a particular case of the above problem, in which all toll arcs are connected and constitute a path, as occurs on highways. As toll levels are usually computed using the highway entry-exit points, a complete toll subgraph is considered, where each toll arc corresponds to a toll subpath. The problem is modelled as a linear mixed integer program, and proved to be NP-hard. Several classes of valid inequalities are proposed, which strengthen important constraints of the initial model. Their efficiency is first shown theoretically, as these are facet defining for the restricted one and two commodity problems. Numerical tests also highlight the practical efficiency of the valid inequalities for the multi-commodity case. Finally, we point out the links between this problem and a more classical design and pricing problem in economics.

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