Skip to main content
SpringerLink
Log in

Single-Sink Multicommodity Flow with Side Constraints

  • Chapter
Research Trends in Combinatorial Optimization

Summary

In recent years, several new models for network flows have been analyzed, inspired by emerging telecommunication technologies. These include models of resilient flow, motivated by the introduction of high capacity optical links, coloured flow, motivated by Wavelength-Division-Multiplexed optical networks, unsplittable flow motivated by SONET networks, and confluent flow motivated by next-hop routing in internet protocol (IP) networks. In each model, the introduction of new side-constraints means that a max-flow min-cut theorem does not necessarily hold, even in the setting where all demands are destined to a common node (sink) in the network. In such cases, one may seek bounds on the “flow-cut gap” for the model. Such approximate max-flow min-cut theorems are a useful measure for bounding the impact of new technology on congestion in networks whose traffic flows obey these side constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aggarwal, C., Orlin, J.: On multi-route maximum flows in networks. Networks 39, 43–52 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Ahlswede, R., Cai, N., Li, S.-Y.R., Yeung, R.W.: Network information flow. IEEE Trans. Inf. Theory 46, 1204–1216 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Ahuja, R., Magnanti, T., Orlin, J.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)

    Google Scholar 

  • Anshelevich, E., Zhang, L.: Path decomposition under a new cost measure with applications to optical network design. In: European Symposium on Algorithms (ESA), pp. 28–39 (2004)

    Google Scholar 

  • Bartholdi, J.J., Orlin, J.B., Ratliff, H.D.: Cyclic scheduling via integer programs with circular ones. Oper. Res. 28, 1074–1085 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  • Baum, S., Trotter, L.E. Jr.: Integer rounding and polyhedral decomposition for totally unimodular systems. In: Henn, R., Korte, B., Oettli, W. (eds.) Optimization and Operations Research (Proceedings of Workshop Bad Honnef, 1977). Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 15–23. Springer, Berlin (1978)

    Google Scholar 

  • Beauquier, B., Hell, P., Perennes, S.: Optimal wavelength-routed multicasting. Discrete Appl. Math. 84, 15–20 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Beauquier, B., Bermond, J.C., Gargano, L., Hell, P., Perennes, S., Vaccaro, U.: Graph problems arising from wavelength-routing in all-optical networks. Theor. Comput. Sci. 233(1–2), 165–189 (2000)

    Google Scholar 

  • Bienstock, D., Muratore, G.: Strong inequalities for capacitated survivable network design problems. Math. Program., Ser. A 89, 127–147 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Brightwell, G., Shepherd, F.B.: Consultancy report: Resilience strategy for a single source-destination pair. LSE CDAM Report 96-22 (August 1996)

    Google Scholar 

  • Brightwell, G., Oriolo, G., Shepherd, B.: Reserving resilient capacity in a network. SIAM J. Discrete Math. 14, 524–539 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Brightwell, G., Oriolo, G., Shepherd, B.: Reserving resilient capacity for a single commodity with upper bound constraints. Networks 41(2), 87–96 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Chekuri, C., Khanna, S.: Edge disjoint paths revisited. In: Proc. of the ACM–SIAM Symposium on Discrete Algorithms (SODA) (2003)

    Google Scholar 

  • Chekuri, C., Khanna, S., Shepherd, F.B.: The all-or-nothing multicommodity flow problem. In: Proc. of the ACM Symposium on Theory of Computing (STOC) (2004a)

    Google Scholar 

  • Chekuri, C., Khanna, S., Shepherd, F.B.: Edge-disjoint paths in planar graphs. In: Proc. of IEEE Foundations of Computer Science (FOCS) (2004b)

    Google Scholar 

  • Chekuri, C., Claisse, P., Essiambre, R., Fortune, S., Kilper, D., Nithi, K., Lee, W., Saniee, I., Shepherd, B., Wilfong, G., White, C., Zhang, L.: Design tools for transparent optical networks. Bell Labs Techn. J. 11(2), 129–143 (2006a)

    Article  Google Scholar 

  • Chekuri, C., Khanna, S., Shepherd, F.B.: An \(O(\sqrt{n})\) approximation and integrality gap for disjoint paths and unsplittable flow. Theory Comput. 2–7, 137–146 (2006b)

    MathSciNet  Google Scholar 

  • Chen, J., Rajaraman, R., Sundaram, R.: Meet and merge: approximation algorithms for confluent flow. In: Proceedings of the 35th ACM Symposium on Theory of Computing (STOC), pp. 373–382 (2003)

    Google Scholar 

  • Chen, J., Kleinberg, R., Lovasz, L., Rajaraman, R., Sundaram, R., Vetta, A.: (Almost) tight bounds and existence theorems for confluent flows. In: Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), pp. 529–538 (2004)

    Google Scholar 

  • Cosares, S., Saniee, I.: An optimization problem related to balancing loads on SONET rings. Telecommun. Syst. 3, 165–181 (1994)

    Article  Google Scholar 

  • Costain, G., Kennedy, S., Meagher, C. (eds.): Bellairs Combinatorial Optimization Open Problems. http://www.math.mcgill.ca/~bshepherd/Bellairs/bellairs2007.pdf

  • Dinitz, Y., Garg, N., Goemans, M.: On the single-source unsplittable flow problem. Combinatorica 19, 17–41 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Donovan, P., Shepherd, F.B., Vetta, A., Wilfong, G.T.: Degree-constrained network flows. In: Proc. of the ACM Symposium on Theory of Computing (STOC) (2007)

    Google Scholar 

  • Edmonds, J.: Edge-disjoint branchings. In: Rustin, R. (ed.) Combinatorial Algorithms, Courant Computer Science Symposium 9, Monterey, CA, 1972, pp. 91–96. Academic Press, New York (1973)

    Google Scholar 

  • Fong, J., Gilbert, A.C., Kannan, S., Strauss, M.: Better alternatives to OSPF routing. Algorithmica 43(1–2), 113–131 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Ford, L.R. Jr., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)

    MATH  MathSciNet  Google Scholar 

  • Ford, L.R. Jr., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  • Fortune, S., Sweldens, W., Zhang, L.: Line system design for DWDM networks. In: Proceedings of the 11th International Telecommunications Network Strategy and Planning Symposium (Networks), pp. 315–320 (2004)

    Google Scholar 

  • Fortz, B., Thorup, M.: Optimizing OSPF/IS-IS weights in a changing world. IEEE J. Sel. Areas Commun. 20(4), 756–767 (2002)

    Article  Google Scholar 

  • Fragouli, C., Soljanin, E.: Network coding fundamentals. Found. Trends Netw. 2(1), 1–133 (2007)

    Article  Google Scholar 

  • Frank, A.: Personal communication (2005)

    Google Scholar 

  • Hoffman, A.J.: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Bellman, R., Hall, M. (eds.) Combinatorial Analysis, pp. 113–128. Am. Math. Soc., Providence (1960)

    Google Scholar 

  • Hwang, F.K.: The Mathematical Theory of Nonblocking Switching Networks. Series on Applied Mathematics, vol. 11. World Scientific, River Edge (1998)

    MATH  Google Scholar 

  • Iri, M.: A new method of solving transportation-network problems. J. Oper. Res. Soc. Jpn. 3, 27–87 (1960)

    Google Scholar 

  • Jung, H.A.: Einer Verallgemeinerung des n-fachen Zusammenhangs für Graphen. Math. Ann. 187, 95–103 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  • Kleinberg, J.: Single-source unsplittable flow. In: Proceedings of the 37th Symposium on Foundations of Computer Science (FOCS), pp. 68–77 (1996)

    Google Scholar 

  • Kleinberg, J.M.: Decision algorithms for unsplittable flow and the half-disjoint paths problem. In: Proc. of the ACM Symposium on Theory of Computing (STOC), pp. 530–539 (1998)

    Google Scholar 

  • Kleinberg, J.M.: An approximation algorithm for the disjoint paths problem in even-degree planar graphs. In: Proc. of the Foundations of Computer Science (FOCS) (2005)

    Google Scholar 

  • Kolliopoulos, S., Stein, C.: Improved approximation algorithms for unsplittable flow problems. In: Proceedings of the 38th Symposium on Foundations of Computer Science (FOCS), pp. 426–435 (1997)

    Google Scholar 

  • Leighton, F.T.: Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Kauffman, Los Altos (1992)

    MATH  Google Scholar 

  • Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. Assoc. Comput. Mach. 46(6), 787–832 (1999); Preliminary version in Proc. of the Foundations of Computer Science (FOCS) (1988)

    MATH  MathSciNet  Google Scholar 

  • Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–245 (1995); Preliminary version in Proc. of IEEE Foundations of Computer Science (FOCS) (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • McGregor, A., Shepherd, F.B.: Island hopping and path colouring, with applications to WDM network design. In: Proceedings of the 18th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA) (2007)

    Google Scholar 

  • Menger, K.: Zur allgemeinen Kurventheorie. Fund. Math. 10, 96–115 (1927)

    MATH  Google Scholar 

  • Pippenger, N.: Telephone switching networks. AMS Proc. Symp. Appl. Math. 26, 101–133 (1978)

    Google Scholar 

  • Pippenger, N.: Communication Networks, Handbook of Theoretical Computer Science (vol. A): Algorithms and Complexity. MIT Press, Cambridge (1991)

    Google Scholar 

  • Räcke, H.: Minimizing congestion in general networks. In: Proc. of the Foundations of Computer Science (FOCS) (2002)

    Google Scholar 

  • Robertson, N., Seymour, P.D.: Outline of a disjoint paths algorithm. In: Korte, B., Lovász, L., Prömel, H.J., Schrijver, A. (eds.) Paths, Flows and VLSI-Layout. Springer, New York (1990)

    Google Scholar 

  • Robertson, N., Seymour, P.D.: Graph minors XIII: the disjoint paths problem. J. Comb. Theory, Ser. B 63(1), 65–110 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)

    MATH  Google Scholar 

  • Schrijver, A., Seymour, P., Winkler, P.: The ring loading problem. SIAM J. Discrete Math. 11(1), 1–14 (1998); Reprinted in SIAM Rev. 41(1) 777–791 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Shepherd, F.B., Vetta, A.: Visualizing, finding and packing dijoins. In: Avis, D., Hertz, A., Marcotte, O. (eds.) Graph Theory and Combinatorial Optimization, pp. 219–254. Kluwer, New York (2005)

    Chapter  Google Scholar 

  • Shepherd, F.B., Wilfong, G.T.: Multilateral transport games. In: INOC (2005)

    Google Scholar 

  • Skutella, M.: Approximating the single source unsplittable min-cost flow problem. Math. Program., Ser. B 91, 493–514 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Winkler, P., Zhang, L.: Wavelength assignment and generalized interval graph coloring. In: Proc. of the ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 830–831 (2003)

    Google Scholar 

  • Yeung, R.R., Li, S.-Y.R., Cai, N., Zhang, Z.: Network Coding Theory. Now Publishers, Boston (2006)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke West, Montreal, Québec, H3A 2K6, Canada

    F. Bruce Shepherd

Authors
  1. F. Bruce Shepherd
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Industrial and Systems Engineering, Georgia Tech, 765 Ferst Drive, Atlanta, Georgia, 30332-0205, USA

    William Cook

  2. Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

    László Lovász

  3. Research Institute for Discrete Mathematics, University of Bonn, Lennéstr. 2, 53113, Bonn, Germany

    Jens Vygen

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Shepherd, F.B. (2009). Single-Sink Multicommodity Flow with Side Constraints. In: Cook, W., Lovász, L., Vygen, J. (eds) Research Trends in Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76796-1_20

Download citation

Publish with us

Policies and ethics

Search

Navigation