Local User Cost Equilibrium: a bush-based algorithm for traffic assignment

This article presents a new algorithm for traffic assignment, called Local User Cost Equilibrium (LUCE), which iteratively solves a sequence of user-equilibrium problems associated with flows exiting from a node. The method is based on the idea of assigning users directed towards each destination separately; these flows form a bush, i.e. an acyclic sub-graph that connects every node to that destination. For each node, the algorithm considers the arcs of its forward star as the set of travel alternatives available to users and seeks a deterministic equilibrium of flows towards the same destination. The cost function associated with each of these local route choices expresses the average impedance to reaching the destination if a user continues the trip on a particular arc. The method is local' in an analytical sense, because the cost function is linearised at the current flow pattern, as if it was independent from the other splitting rates of the same node. The method is also local' in a topological sense, as nodes are processed through a polynomial visit of the current bush, inspired by dynamic programming. The node problem is formulated as a quadratic program in terms of destination-specific flows. We prove that its solution recursively applied in topological order provides a descent direction with respect to the sum-integral objective function of traffic assignment. The local equilibrium problem at nodes is solved through a greedy algorithm resembling the ad-hoc method used to compute shortest hyperpaths in transit assignment. The latter is the main contribution of this article. The main advantage of LUCE is to achieve a fast convergence rate that compares favourably with the existing methods, and to implicitly assign the demand flow of each origin-destination pair on several paths at once.