Mohammad Taghi Hajiaghayi
Robert D. Kleinberg
Abstract
Oblivious routing algorithms for general undirected networks were introduced by Räcke [2002], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or even in undirected networks with node capacities.
We present the first nontrivial upper bounds for both these cases, providing algorithms for k-commodity oblivious routing problems with competitive ratio O(√k log(n)) for undirected node-capacitated graphs and O(√k n1/4 log(n)) for directed graphs. In the special case that all commodities have a common source or sink, our upper bound becomes O(√n log(n)) in both cases, matching the lower bound up to a factor of log(n). The lower bound (which first appeared in Azar et al. [2003]) is obtained on a graph with very high degree. We show that, in fact, the degree of a graph is a crucial parameter for node-capacitated oblivious routing in undirected graphs, by providing an O(Δ polylog(n))-competitive oblivious routing scheme for graphs of degree Δ. For the directed case, however, we show that the lower bound of Ω(√n) still holds in low-degree graphs.
Finally, we settle an open question about routing problems in which all commodities share a common source or sink. We show that even in this simplified scenario there are networks in which no oblivious routing algorithm can achieve a competitive ratio better than Ω(log n).
- Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., and Naor, J. S. 2004. A general approach to online network optimization problems. In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), 577--586. Google Scholar
Digital Library
- Aspnes, J., Azar, Y., Fiat, A., Plotkin, S. A., and Waarts, O. 1997. On-Line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM 44, 3, 486--504. Also in Proceedings of the 25th STOC, 1993, 623--631. Google ScholarDigital Library
- Awerbuch, B., and Azar, Y. 1994. Local optimization of global objectives: Competitive distributed deadlock resolution and resource allocation. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science (FOCS), 240--249.Google Scholar
- Awerbuch, B., and Leighton, F. T. 1993. A simple local-control approximation algorithm for multicommodity flow. In Proceedings of the 34th IEEE Symposium on Foundations of Computer Science (FOCS), 459--468.Google Scholar
- Awerbuch, B., and Leighton, F. T. 1994. Improved approximation algorithms for the multi-commodity flow problem and local competitive routing in dynamic networks. In Proceedings of the 26th ACM Symposium on Theory of Computing (STOC), 487--496. Google ScholarDigital Library
- Azar, Y., Cohen, E., Fiat, A., Kaplan, H., and Räcke, H. 2003. Optimal oblivious routing in polynomial time. In Proceedings of the 35th ACM Symposium on Theory of Computing (STOC), 383--388. Google ScholarDigital Library
- Bartal, Y., and Leonardi, S. 1999. On-Line routing in all-optical networks. Theor. Comput. Sci. 221, 1-2, 19--39. Also in Proceedings of the 24th ICALP, 1997, 516--526. Google ScholarDigital Library
- Bienkowski, M., Korzeniowski, M., and Räcke, H. 2003. A practical algorithm for constructing oblivious routing schemes. In Proceedings of the 15th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 24--33. Google ScholarDigital Library
- Chekuri, C., Khanna, S., and Shepherd, B. 2004. The all-or-nothing multicommodity flow problem. In Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), 156--165. Google ScholarDigital Library
- Even, G., Naor, J. S., Schieber, B., and Sudan, M. 1998. Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20, 2, 151--174.Google ScholarCross Ref
- Garg, N., Vazirani, V. V., and Yannakakis, M. 1996. Approximate max-flow min-(multi)cut theorems and their applications. SIAM J. Comput. 25, 2, 235--251. Google ScholarDigital Library
- Garg, N., Vazirani, V. V., and Yannakakis, M. 2004. Multiway cuts in node weighted graphs. J. Algorithms 50, 1, 49--61. Google ScholarDigital Library
- Hajiaghayi, M. T., and Räcke, H. An O(&sqrt;n)-approximation algorithm for directed sparsest cut. Inf. Proc. Lett. to appear. Google ScholarDigital Library
- Harrelson, C., Hildrum, K., and Rao, S. B. 2003. A polynomial-time tree decomposition to minimize congestion. In Proceedings of the 15th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 34--43. Google ScholarDigital Library
- Klein, P. N., Plotkin, S. A., Rao, S. B., and Tardos, É. 1997. Approximation algorithms for Steiner and directed multicuts. J. Alg. 22, 2, 241--269. Google ScholarDigital Library
- Maggs, B. M., Meyer auf der Heide, F., Vöcking, B., and Westermann, M. 1997. Exploiting locality for networks of limited bandwidth. In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science (FOCS), 284--293. Google ScholarDigital Library
- Maggs, B. M., Miller, G. L., Parekh, O., Ravi, R., and Woo, S. L. M. 2005. Finding effective support-tree preconditioners. In Proceedings of the 17th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 176--185. Google ScholarDigital Library
- Räcke, H. 2002. Minimizing congestion in general networks. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS), 43--52. Google ScholarDigital Library
- Valiant, L., G., and Brebner, G. J. 1981. Universal schemes for parallel communication. In Proceedings of the 13th ACM Symposium on Theory of Computing (STOC), 263--277. Google ScholarDigital Library
Index Terms
- Oblivious routing on node-capacitated and directed graphs
Recommendations
Hop-constrained oblivious routing
STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of ComputingWe prove the existence of an oblivious routing scheme that is poly(logn)-competitive in terms of (congestion + dilation), thus resolving a well-known question in oblivious routing.
Concretely, consider an undirected network and a set of packets each ...
Oblivious routing in directed graphs with random demands
STOC '05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computingOblivious routing algorithms for general undirected networks were introduced by Räcke, and this work has led to many subsequent improvements and applications. More precisely, Räcke showed that there is an oblivious routing algorithm with polylogarithmic ...
Oblivious routing on node-capacitated and directed graphs
SODA '05: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithmsOblivious routing algorithms for general undirected networks were introduced by Räcke [17], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or ...
Comments