A feedback neural network approach to communication routing problems is developed, with emphasis on multiple shortest path problems, with several requests for transmissions between distinct start and end nodes. The basic ingredients are a set of Potts neurons for each request, with interactions designed to minimize path lengths and prevent overloading of network arcs. The topological nature of the problem is conveniently handled using a propagator matrix approach. Although the constraints are global, the algorithmic steps are based entirely on local information, facilitating distributed implementations. In the polynomially solvable single-request case, the approach reduces to a fuzzy version of the Bellman-Ford algorithm. The method is evaluated for synthetic problems of varying sizes and load levels, by comparing to exact solutions from a branch-and-bound method, or to approximate solutions from a simple heuristic. With very few exceptions, the Potts approach gives high-quality legal solutions. The computational demand scales merely as the product of the numbers of requests, nodes, and arcs.