Abstract
In the single source unsplittable flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given digraph. The demand of each commodity must be routed along a single path. In a groundbreaking paper Dinitz, Garg, and Goemans [4] prove that any given (splittable) flow satisfying certain demands can be turned into an unsplittable flow with the following nice property: In the unsplittable flow, the flow value on any arc exceeds the flow value on that arc in the given flow by no more than the maximum demand.
Goemans conjectures that this result even holds in the more general context with arbitrary costs on the arcs when it is required that the cost of the unsplittable flow must not exceed the cost of the given (splittable) flow. The following is an equivalent formulation of Goemans’ conjecture: Any (splittable) flow can be written as a convex combination of unsplittable flows such that the unsplittable flows have the nice property mentioned above. We prove a slightly weaker version of this conjecture where each individual unsplittable flow occurring in the convex combination does not necessarily fulfill the original demands but rounded demands. Preliminary computational results based on our underlying algorithm support the strong version of the conjecture.
This work was supported in part by the Graduate School of Production Engineering and Logistics, North Rhine-Westphalia, by the DFG Focus Program 1126, “Algorithmic Aspects of Large and Complex Networks,” grants SK 58/4-1 and SK 58/5-3, and by the German Academic Exchange Service (DAAD).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Andrews, M., Zhang, L.: Hardness of the undirected congestion minimization problem. In: Proceedings of 37th Annual ACM Symposium on Theory of Computing, pp. 284–293. ACM Press, New York (2005)
Azar, Y., Regev, O.: Strongly polynomial algorithms for the unsplittable flow problem. In: Proceedings of the 8th Conference on Integer Programming and Combinatorial Optimization, pp. 15–29 (2001)
Chuzhoy, J., Naor, J.: New hardness results for congestion minimization and machine scheduling. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 28–34. ACM Press, New York (2004)
Dinitz, Y., Garg, N., Goemans, M.X.: On the single source unsplittable flow problem. Combinatorica 19, 17–41 (1999)
Du, J., Kolliopoulos, S.: Implementing approximation algorithms for the single-source unsplittable flow problem. Journal of Experimental Algorithmics 10, 2–3 (2005)
Goemans, M.X.: Cited as personal communication from (January 2000) [14, Section 7]
Guruswami, V., Khanna, S., Rajaraman, R., Shepherd, B., Yannakakis, M.: Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pp. 19–28. ACM Press, New York (1999)
Kleinberg, J.M.: Approximation Algorithms for Disjoint Path Problems. PhD thesis, Massachusetts Institute of Technology (May 1996)
Kolliopoulos, S.G., Stein, C.: Approximation algorithms for single-source unsplittable flow. SIAM Journal on Computing 31, 919–946 (2002)
Kolman, P., Scheideler, C.: Improved bounds for the unsplittable flow problem. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 184–193. ACM Press, New York (2002)
Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 46, 259–271 (1990)
Raghavan, P.: Probabilistic construction of deterministic algorithms: approximating packing integer programs. Journal of Computer and System Sciences 37, 130–143 (1988)
Raghavan, P., Thompson, C.D.: Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica 7, 365–374 (1987)
Skutella, M.: Approximating the single source unsplittable min-cost flow problem. Mathematical Programming 91, 493–514 (2002)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Martens, M., Salazar, F., Skutella, M. (2007). Convex Combinations of Single Source Unsplittable Flows. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_36
Download citation
DOI: https://doi.org/10.1007/978-3-540-75520-3_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75519-7
Online ISBN: 978-3-540-75520-3
eBook Packages: Computer ScienceComputer Science (R0)