Relaxation Methods for Minimum Cost Network Flow

We view the optimal single commodity network flow problem with linear arc costs and its dual as a pair of monotropic programming problems, i.e. problems of minimizing the separable sum of scalar extended real-valued convex functions over a subspace. For such problems directions of cost improvement can be selected from among a finite set of directions--the elementary vectors of the constraint subspace. The classical primal simplex, dual simplex, and primal-dual methods turn out to be particular implementations of this idea. This paper considers alternate implementations leading to new dual descent algorithms which are conceptually related to coordinate descent and Gauss-Seidel relaxation methods for unconstrained optimization or solution of equations. Contrary to primal simplex and primal-dual methods, these algorithms admit a natural extension to network problems with strictly convex arc costs. Our first coded implementation of relaxation methods is compared with mature state-of-the-art primal simplex and primal-dual codes and is found to be substantially faster on most types of network flow problems of practical interest. This work has been supported by the National Science Foundation under Grant NSF-ECS-8217668. Many thanks are due to Tom Magnanti who supplied us with the primal-dual code KILTER and to Michael Grigoriadis who supplied us with the primal simplex code RNET for the purposes of comparative testing. They also, together with John Mulvey and Bob Gallager, clarified several questions for us. Support provided by Alphatech, Inc. is also gratefully acknowledged. **The authors are with the Laboratory for Information and Decision Systems and the Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.