The Minimum Road Trips Problem

Road networks can be represented as partially directed graphs; directed edges are one-way road segments and undirected edges can be traversed in either direction. Vehicle trips are simply paths from some starting vertex to some ending vertex that must agree with the direction of any directed edge taken. A loop detector is a device that counts the number of vehicles that cross an edge during some time period. Loop detectors are typically present on only a subset of the edges in the network. The basic problem we are interested in is to determine the minimum number of trips (simple paths) needed to explain all of the loop detector count measurements. We also consider a dynamic version of the problem in which time is discretized and vehicles move one edge per time step. In this case, the loop detectors provide traffic counts for each time step and the goal is again to determine the fewest number of trips needed to explain the data. Trips are now specified by a path and a starting time.