Algebraic theory on shortest paths for all flows

Abstract

As a mathematical model for the passenger routing problem for ticketing in a railway network, we consider a shortest path problem for a directed graph with edges labeled with a cost and a capacity. The problem is to push flow f from a specified source to all other vertices with the minimum cost for all f values. If there are t different capacity values, we can solve the single source shortest path problem for all f t times in O(tm+tnlog n)=O(m2) time when t=m. We improve this time to O(cmn) if edge costs are non-negative integers bounded by c.