An Implementation of a Combinatorial Approximation Algorithm for Minimum-Cost Multicommodity Flow
The minimum-cost multicommodity flow problem involves si- multaneously shipping multiple commodities through a single network so that the total flow obeys arc capacity constraints and has minimum cost. Multicommodity flow problems can be expressed as linear programs, and most theoretical and practical algorithms use linear-programming algorithms specialized for the problems’ structures. Combinatorial ap- proximation algorithms in [GK96,KP95b,PST95] yield flows with costs slightly larger than the minimum cost and use capacities slightly larger than the given capacities. Theoretically, the running times of these algo- rithms are much less than that of linear-programming-based algorithms. We combine and modify the theoretical ideas in these approximation al- gorithms to yield a fast, practical implementation solving the minimum- cost multicommodity flow problem. Experimentally, the algorithm solved our problem instances (to 1% accuracy) two to three orders of magni- tude faster than the linear-programming package CPLEX [CPL95] and the linear-programming based multicommodity flow program PPRN [CN96].
Unable to display preview. Download preview PDF.
- AMO93.R. K. Ahuja, T. L. Magnanti, and J. B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, NJ, 1993.Google Scholar
- Bad91.T. Badics. genrmf. 1991. ftp://dimacs.rutgers.edu/pub/netflow/generators/network/genrmf/
- CPL95.CPLEX Optimization, Inc., Incline Village, NV. Using the CPLEX Callable Library, 4.0 edition, 1995.Google Scholar
- GOPS97.A. Goldberg, J. D. Oldham, S. Plotkin, and C. Stein. An implementation of a combinatorial approximation algorithm for minimum-cost multicommodity flow. Technical Report CS-TR-97-1600, Stanford University, December 1997.Google Scholar
- HL96.R. W. Hall and D. Lotspeich. Optimized lane assignment on an automated highway. Transportation Research—C, 4C(4):211–229, 1996.Google Scholar
- HO96.A. Haghani and S.-C. Oh. Formulation and solution of a multi-commodity, multi-modal network flow model for disaster relief operations. Transportation Research—A, 30A(3):231–250, 1996.Google Scholar
- KP95a.A. Kamath and O. Palmon. Improved interior point algorithms for exact and approximate solution of multicommodity flow problems. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 6, pages 502–511. Association for Computing Machinery, January 1995.MathSciNetGoogle Scholar
- KP95b.D. Karger and S. Plotkin. Adding multiple cost constraints to combinatorial optimization problems, with applications to multicommodity flows. In Symposium on the Theory of Computing, Vol. 27, pages 18–25. Association for Computing Machinery, ACM Press, May 1995.Google Scholar
- KV86.S. Kapoor and P. M. Vaidya. Fast algorithms for convex quadratic programming and multicommodity flows. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing, Vol. 18, pages 147–159. Association for Computing Machinery, 1986.Google Scholar
- LO91.Y. Lee and J. Orlin. GRIDGEN. 1991. ftp://dimacs.rutgers.edu/pub/netflow/generators/network/gridgen/
- LSS93.T. Leong, P. Shor, and C. Stein. Implementation of a combinatorial multicommodity flow algorithm. In David S. Johnson and Catherine C. Mc-Geoch, editors, Network Flows and Matching, Series in Discrete Mathematics and Theoretical Computer Science, Vol. 12, pages 387–405. American Mathematical Society, 1993.Google Scholar
- Rap90.J. Raphson. Analysis Æquationum Universalis, seu, Ad Æquationes Algebraicas Resolvendas Methodus Generalis, et Expedita. Prostant venales apud Abelem Swalle, London, 1690.Google Scholar
- Sch91.R. Schneur. Scaling Algorithms for Multicommodity Flow Problems and Network Flow Problems with Side Constraints. PhD thesis, MIT, Cambridge, MA, February 1991.Google Scholar