# Approximation Algorithms for the Unsplittable Flow Problem

## Abstract

*unsplittable*

*flow problem*(UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the

*non-uniform capacity*case in which the edge capacities can vary arbitrarily over the graph. Our results are:

- For undirected graphs we obtain a

*O*(Δα^{-1}log^{2}*n*) approximation ratio, where*n*is the number of vertices, Δ the maximum degree, and α the expansion of the graph. Our ratio is capacity independent and improves upon the earlier*O*(Δα^{-1}(*c*_{max}/*c*_{min}) log*n*) bound [15] for large values of*c*_{max}/*c*_{min}. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm gives an*O*(Δα^{-1}log*n*) approximation, which matches the performance of the best-known algorithm [15] for this special case.- For certain strong constant-degree expanders considered by Frieze [10] we obtain an

*O*(√log*n*) approximation for the uniform capacity case, improving upon the current*O*(log*n*) approximation.- For UFP on the line and the ring, we give the first constant-factor approximation algorithms. Previous results addressed only the uniform capacity case.

- All of the above results improve if the maximum demand is bounded away from the minimum capacity.

Our results are based on randomized rounding followed by greedy alteration and are inspired by the use of this idea in recent work [21,9].

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### References

- 1.N. Alon and J. Spencer.
*The Probabilistic Method*. Wiley Interscience, New York, 1992.MATHGoogle Scholar - 2.B. Awerbuch, Y. Azar, and S. Plotkin. Throughput-competitive online routing. In
*Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science*, pp. 32–40. 1993.Google Scholar - 3.Y. Azar and O. Regev. Strongly polynomial algorithms for the unsplittable flow problem. In
*Proceedings of the 8th Integer Programming and Combinatorial Optimization Conference*. 2001.Google Scholar - 4.A. Bar-Noy, R. Bar-Yehuda, A. Freund, J. S. Naor, and B. Scheiber. A unified approach to approximating resource allocation and scheduling. In
*Proceedings of the 32nd Annual ACM Symposium on Theory of Computing*, pp. 735–744, 2000.Google Scholar - 6.P. Berman and B. DasGupta. Improvements in throughput maximization for realtime scheduling. In
*Proceedings of the 32nd Annual ACM Symposium on Theory of Computing*, pp. 680–687, 2000.Google Scholar - 7.T. Bohman and A. M. Frieze. Arc-disjoint paths in expander digraphs. In
*Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science*. 2001.Google Scholar - 8.A. Z. Broder, A. M. Frieze, and E. Upfal. Existence and construction of edgedisjoint paths on expander graphs.
*SIAM Journal on Computing*, 23(5):976–989, 1994.MATHCrossRefMathSciNetGoogle Scholar - 9.G. Calinescu, A. Chakrabarti, H. Karloff, and Y. Rabani. Improved approximation algorithms for resource allocation. In
*Proceedings of the 9th Integer Programming and Combinatorial Optimization Conference*, 2002.Google Scholar - 10.A. M. Frieze. Edge-disjoint paths on expander graphs.
*SIAM Journal on Computing*, 30(6):1790–1801, 2001.MATHCrossRefMathSciNetGoogle Scholar - 11.V. Guruswami, S. Khanna, R. Rajaraman, F. B. Shepherd, and M. Yannakakis. 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. 1999.Google Scholar - 12.J. M. Kleinberg.
*Approximation Algorithms for Disjoint Paths Problems*. Ph.D. thesis, MIT, 1996.Google Scholar - 13.J. M. Kleinberg and R. Rubinfeld. Short paths in expander graphs. In
*Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science*, pp. 86–95. 1996.Google Scholar - 14.P. Kolman and S. Scheideler. Simple on-line algorithms for the maximum disjoint paths problem. In
*Proceedings of 13th ACM Symposium on Parallel Algorithms and Architectures*. 2001.Google Scholar - 15.P. Kolman and S. Scheideler. Improved bounds for the unsplittable flow problem. In
*Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms*. 2002.Google Scholar - 16.F. T. Leighton and S. B. Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms.
*Journal of the ACM*, 46(6):787–832, 1999. (Preliminary version in*29th Annual Symposium on Foundations of Computer Science*, pages 422-431, 1988).MATHCrossRefMathSciNetGoogle Scholar - 17.C. A. Phillips, R. N. Uma, and J. Wein. Off-line admission control for general scheduling problems. In
*Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms*, pp. 879–888. 2000.Google Scholar - 18.P. Raghavan and C. D. Thompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs.
*Combinatorica*, 7(4):365–374, 1987.MATHCrossRefMathSciNetGoogle Scholar - 19.A. Srinivasan. Improved approximations for edge-disjoint paths, unsplittable flow, and related routing problems. In
*Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science*, pp. 416–425. 1997.Google Scholar - 20.A. Srinivasan. Improved approximation guarantees for packing and covering integer programs.
*SIAM J. Comput.*, 29(2):648–670, 1999.MATHCrossRefMathSciNetGoogle Scholar - 21.A. Srinivasan. New approaches to covering and packing problems. In
*Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms*, pp. 567–576. 2001.Google Scholar