Abstract
Traditionally, flows over time are solved in time-expanded networks which contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time-expanded network. In particular, this approach usually does not lead to efficient algorithms with running time polynomial in the input size since the size of the time-expanded network is only pseudo-polynomial.
We present two different approaches for coping with this difficulty. Firstly, inspired by the work of Ford and Fulkerson on maximal s-t-flows over time (or ‘maximal dynamic s-t-flows’), we show that static, length-bounded flows lead to provably good multicommodity flows over time. These solutions not only feature a simple structure but can also be computed very efficiently in polynomial time.
Secondly, we investigate ‘condensed’ time-expanded networks which rely on a rougher discretization of time. Unfortunately, there is a natural tradeoff between the roughness of the discretization and the quality of the achievable solutions. However, we prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed time expanded network of polynomial size. In particular, this approach yields a fully polynomial time approximation scheme for the quickest multicommodity flow problem and also for more general problems.
Extended abstract; information on the full version of the paper can be obtained via the authors’ WWW-pages.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. E. Aronson. A survey of dynamic network flows. Annals of Operations Research, 20:1–66, 1989.
R. E. Burkard, K. Dlaska, and B. Klinz. The quickest flow problem. ZOR-Methods and Models of Operations Research, 37:31–58, 1993.
L. K. Fleischer. Approximating fractional multicommodity flows independent of the number of commodities. SIAM Journal on Discrete Mathematics, 13:505–520, 2000.
L. K. Fleischer and É Tardos. Efficient continuous-time dynamic network flow algorithms. Operations Research Letters, 23:71–80, 1998.
L. R. Ford and D. R. Fulkerson. Constructing maximal dynamic flows from static flows. Operations Research, 6:419–433, 1958.
L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1962.
D. Gale. Transient flows in networks. Michigan Mathematical Journal, 6:59–63, 1959.
N. Garg and J. Könemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, pages 300–309, Palo Alto, CA, 1998.
M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin, 1988.
G. Handler and I. Zang. A dual algorithm for the constrained shortest path problem. Networks, 10:293–310, 1980.
R. Hassin. Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research, 17:36–42, 1992.
B. Hoppe and É Tardos. Polynomial time algorithms for some evacuation problems. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 433–441, Arlington, VA, 1994.
B. Hoppe and É Tardos. The quickest transshipment problem. Mathematics of Operations Research, 25:36–62, 2000.
B. Klinz and G. J. Woeginger. Minimum cost dynamic flows: The series-parallel case. In E. Balas and J. Clausen, editors, Integer Programming and Combinatorial Optimization, volume 920 of Lecture Notes in Computer Science, pages 329–343. Springer, Berlin, 1995.
D. H. Lorenz and D. Raz. A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters, 28:213–219, 2001.
E. Minieka. Maximal, lexicographic, and dynamic network flows. Operations Research, 21:517–527, 1973.
C. A. Phillips. The network inhibition problem. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, pages 776–785, San Diego, CA, 1993.
W. B. Powell, P. Jaillet, and A. Odoni. Stochastic and dynamic networks and routing. In M. O. Ball, T. L. Magnanti, C. L. Monma, and G. L. Nemhauser, editors, Network Routing, volume 8 of Handbooks in Operations Research and Management Science, chapter 3, pages 141–295. North-Holland, Amsterdam, The Netherlands, 1995.
W. L. Wilkinson. An algorithm for universal maximal dynamic flows in a network. Operations Research, 19:1602–1612, 1971.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fleischer, L., Skutella, M. (2002). The Quickest Multicommodity Flow Problem. In: Cook, W.J., Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization. IPCO 2002. Lecture Notes in Computer Science, vol 2337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47867-1_4
Download citation
DOI: https://doi.org/10.1007/3-540-47867-1_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43676-8
Online ISBN: 978-3-540-47867-6
eBook Packages: Springer Book Archive