A Computational Study of Cost Reoptimization for Min-Cost Flow Problems

In the last two decades, a number of algorithms for the linear single-commodity min-cost flow (MCF) problem have been proposed, and several efficient codes are available that implement different variants of the algorithms. The practical significance of the algorithms has been tested by comparing the time required by their implementations for solving from-scratch instances of MCF, of different classes, as the size of the problem (number of nodes and arcs) increases. However, in many applications several closely related instances of MCF have to be sequentially solved, so that reoptimization techniques can be used to speed up computations, and the most attractive algorithm is the one that minimizes the total time required to solve all the instances in the sequence. In this paper we compare the performances of four different efficient implementations of algorithms for MCF under cost reoptimization in the context of decomposition algorithms for the multicommodity min-cost flow (MMCF) problem, showing that for some classes of instances the relative performances of the codes doing from-scratch optimization do not accurately predict the relative performances when reoptimization is used. Because the best solver depends both on the class and on the size of the instance, this work also shows the usefulness of a standard interface for MCF problem solvers that we have proposed and implemented.

[1]  David K. Smith.  Network Flows: Theory, Algorithms, and Applications , 1994 .

[2]  V G Andrew,et al.  AN EFFICIENT IMPLEMENTATION OF A SCALING MINIMUM-COST FLOW ALGORITHM , 1997 .

[3]  R. Helgason,et al.  Chapter 2 Primal simplex algorithms for minimum cost network flows , 1995 .

[4]  Dimitri P. Bertsekas,et al.  Dual coordinate step methods for linear network flow problems , 1988, Math. Program..

[5]  Dimitri P. Bertsekas,et al.  RELAX-IV : a faster version of the RELAX code for solving minimum cost flow problems , 1994 .

[6]  Andrew V. Goldberg,et al.  Finding Minimum-Cost Circulations by Successive Approximation , 1990, Math. Oper. Res..

[7]  D. Bertsekas,et al.  The relax codes for linear minimum cost network flow problems , 1988 .

[8]  Dimitri P. Bertsekas,et al.  Network optimization : continuous and discrete models , 1998 .

[9]  Richard S. Barr,et al.  Network Reoptimization Algorithms: A Statistically Designed Comparison , 1993, INFORMS J. Comput..

[10]  T. B. Boffey.  Linear Network Optimization: Algorithms and Codes , 1994 .

[11]  R. Kraut,et al.  Vehicle scheduling in public transit and Lagrangean pricing , 1998 .

[12]  Teodor Gabriel Crainic,et al.  Bundle-based relaxation methods for multicommodity capacitated fixed charge network design , 2001, Discret. Appl. Math..

[13]  Giorgio Gallo,et al.  A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems , 1999, INFORMS J. Comput..

[14]  J. Castro,et al.  An implementation of linear and nonlinear multicommodity network flows , 1996 .

[15]  Marinus Veldhorst.  A bibliography on network flow problems , 1991 .

[16]  Panos M. Pardalos,et al.  Interior point algorithms for network flow problems , 1996 .

[17]  Paul Tseng,et al.  Relaxation Methods for Minimum Cost Ordinary and Generalized Network Flow Problems , 1988, Oper. Res..

[18]  Dimitri P. Bertsekas,et al.  RELAX: a computer code for minimum cost network flow problems , 1985 .

[19]  A. Löbel.  Solving Large-Scale Real-World Minimum-Cost Flow Problems by a Network Simplex Method , 1996 .