Approximation Algorithms for the Unsplittable Flow Problem
 Amit Chakrabarti,
 Chandra Chekuri,
 Anuptam Gupta,
 Amit Kumar
 … show all 4 hide
Abstract
We present approximation algorithms for the 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 nonuniform 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 bestknown algorithm [15] for this special case.
 For certain strong constantdegree 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 constantfactor 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].
 N. Alon and J. Spencer. The Probabilistic Method. Wiley Interscience, New York, 1992.
 B. Awerbuch, Y. Azar, and S. Plotkin. Throughputcompetitive online routing. In Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–40. 1993.
 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.
 A. BarNoy, R. BarYehuda, 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.
 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.
 T. Bohman and A. M. Frieze. Arcdisjoint paths in expander digraphs. In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science. 2001.
 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. CrossRef
 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.
 A. M. Frieze. Edgedisjoint paths on expander graphs. SIAM Journal on Computing, 30(6):1790–1801, 2001. CrossRef
 V. Guruswami, S. Khanna, R. Rajaraman, F. B. Shepherd, and M. Yannakakis. Nearoptimal hardness results and approximation algorithms for edgedisjoint paths and related problems. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pp. 19–28. 1999.
 J. M. Kleinberg. Approximation Algorithms for Disjoint Paths Problems. Ph.D. thesis, MIT, 1996.
 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.
 P. Kolman and S. Scheideler. Simple online algorithms for the maximum disjoint paths problem. In Proceedings of 13th ACM Symposium on Parallel Algorithms and Architectures. 2001.
 P. Kolman and S. Scheideler. Improved bounds for the unsplittable flow problem. In Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms. 2002.
 F. T. Leighton and S. B. Rao. Multicommodity maxflow mincut 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 422431, 1988). CrossRef
 C. A. Phillips, R. N. Uma, and J. Wein. Offline admission control for general scheduling problems. In Proceedings of the 11th Annual ACMSIAM Symposium on Discrete Algorithms, pp. 879–888. 2000.
 P. Raghavan and C. D. Thompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica, 7(4):365–374, 1987. CrossRef
 A. Srinivasan. Improved approximations for edgedisjoint paths, unsplittable flow, and related routing problems. In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, pp. 416–425. 1997.
 A. Srinivasan. Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput., 29(2):648–670, 1999. CrossRef
 A. Srinivasan. New approaches to covering and packing problems. In Proceedings of the 12th Annual ACMSIAM Symposium on Discrete Algorithms, pp. 567–576. 2001.
 Title
 Approximation Algorithms for the Unsplittable Flow Problem
 Book Title
 Approximation Algorithms for Combinatorial Optimization
 Book Subtitle
 5th International Workshop, APPROX 2002 Rome, Italy, September 17–21, 2002 Proceedings
 Pages
 pp 5166
 Copyright
 2002
 DOI
 10.1007/3540457534_7
 Print ISBN
 9783540441861
 Online ISBN
 9783540457534
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 2462
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 Additional Links
 Topics
 Industry Sectors
 eBook Packages
 Editors

 Klaus Jansen ^{(4)}
 Stefano Leonardi ^{(5)}
 Vijay Vazirani ^{(6)}
 Editor Affiliations

 4. Institut für Informatik und praktische Mathematik, Universität Kiel
 5. Dipartimento di Informatika e Sistemistica, Universita di Roma La Sapienza
 6. College of Computing, Georgia Institute of Technology
 Authors

 Amit Chakrabarti ^{(7)}
 Chandra Chekuri ^{(8)}
 Anuptam Gupta ^{(8)}
 Amit Kumar ^{(9)}
 Author Affiliations

 7. Computer Science Dept., Princeton University, Princeton
 8. Bell Labs, Lucent Tech., USA
 9. Computer Science Dept., Cornell University, USA
Continue reading...
To view the rest of this content please follow the download PDF link above.