On the Quickest Flow Problem in Dynamic Networks - A Parametric Min-Cost Flow Approach

We consider the quickest flow problem in dynamic networks with a single source s and a single sink t: given a flow F , find the minimum time needed to send F from s to t, and the corresponding optimal flow over time. We introduce new mathematical formulations and derive optimality conditions for the quickest flow problem. Based on the optimality conditions, we develop a new cost-scaling algorithm that leverages the parametric nature of the problem. The algorithm solves the quickest flow problem with integer arc costs in O(nm log(n/m) log(nC)) time, where n, m, and C are the number of nodes, arcs, and the maximum arc cost, respectively. Our algorithm runs in the same time bound as the cost-scaling algorithm by Goldberg and Tarjan [10, 11] for solving the min-cost flow problem. This result shows for the first time that the quickest flow problem can be solved within the same time bound as the min-cost flow problem. As a consequence, our algorithm will remain one of the fastest unless the quickest flow problem can be shown to be simpler than the min-cost flow problem. †Operations Research Center, MIT, Cambridge, MA, 02139; [email protected] ‡Department of Electrical Engineering and Computer Science, and Operations Research Center, MIT, Cambridge, MA, 02139; [email protected]