Skip to main content
SpringerLink
Log in
Book cover

Combinatorial Optimization pp 509–541Cite as

Multicommodity Flows and Edge-Disjoint Paths

  • Chapter
  • First Online:

  • 9623 Accesses

Part of the book series: Algorithms and Combinatorics ((AC,volume 21))

Abstract

The MULTICOMMODITY FLOW PROBLEM is a generalization of the MAXIMUM FLOW PROBLEM. Given a digraph with edge capacities, we now ask for an s-t-flow for several pairs (s, t) (we speak of several commodities ), such that the total flow through any edge does not exceed the capacity. We specify the pairs (s, t) by a second digraph; for technical reasons we have an edge from t to s when we ask for an s-t-flow.

This is a preview of subscription content, 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   54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   99.99
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

Learn about institutional subscriptions

References

General Literature

  • Frank , A. [1990]: Packing paths, circuits and cuts – a survey. In: Paths, Flows, and VLSI-Layout (B. Korte , L. Lovász , H.J. Prömel , A. Schrijver , eds.), Springer, Berlin 1990, pp. 47–100

    Google Scholar 

  • Naves , G., and Sebő , A. [2009]: Multiflow feasibility: an annotated tableau. In: Research Trends in Combinatorial Optimization (W.J. Cook , L. Lovász , J. Vygen , eds.), Springer, Berlin 2009, pp. 261–283

    Google Scholar 

  • Ripphausen-Lipa , H., Wagner , D., and Weihe , K. [1995]: Efficient algorithms for disjoint paths in planar graphs. In: Combinatorial Optimization; DIMACS Series in Discrete Mathematics and Theoretical Computer Science 20 (W. Cook , L. Lovász , P. Seymour , eds.), AMS, Providence 1995

    Google Scholar 

  • Schrijver , A. [2003]: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003, Chapters 70–76

    Google Scholar 

  • Shmoys , D.B. [1996]: Cut problems and their application to divide-and-conquer. In: Approximation Algorithms for NP-Hard Problems (D.S. Hochbaum , ed.), PWS, Boston, 1996

    Google Scholar 

Cited References

  • Aumann , Y. and Rabani , Y. [1998]: An O(logk) approximate min-cut max-flow theorem and approximation algorithm. SIAM Journal on Computing 27 (1998), 291–301

    Google Scholar 

  • Arora , S., Rao , S., and Vazirani , U. [2009]: Expander flows, geometric embeddings and graph partitioning. Journal of the ACM 56 (2009), Article 5

    Google Scholar 

  • Arora , S., Hazan , E., and Kale , S. [2004]: \(O(\sqrt{\log n})\) approximation to SPARSEST CUT in \(\tilde{O}(n^{2})\) time. Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (2004), 238–247

    Google Scholar 

  • Becker , M., and Mehlhorn , K. [1986]: Algorithms for routing in planar graphs. Acta Informatica 23 (1986), 163–176

    Google Scholar 

  • Bienstock , D., and Iyengar , G. [2006]: Solving fractional packing problems in \(O^{{\ast}}(\frac{1} {\epsilon } )\) iterations. SIAM Journal on Computing 35 (2006), 825–854

    Google Scholar 

  • Boesch , F., and Tindell , R. [1980]: Robbins ’s theorem for mixed multigraphs. American Mathematical Monthly 87 (1980), 716–719

    Google Scholar 

  • Charikar , M., Hajiaghayi , M.T., Karloff , H. and Rao , S. [2010]: ℓ 2 2 spreading metrics for vertex ordering problems. Algorithmica 56 (2010), 577–604

    Google Scholar 

  • Chekuri , C., and Khanna , S. [2007]: Edge-disjoint paths revisited. ACM Transactions on Algorithms 3 (2007), Article 46

    Google Scholar 

  • Chudak , F.A., and Eleutério , V. [2005]: Improved approximation schemes for linear programming relaxations of combinatorial optimization problems. In: Integer Programming and Combinatorial Optimization; Proceedings of the 11th International IPCO Conference; LNCS 3509 (M. Jünger , V. Kaibel , eds.), Springer, Berlin 2005, pp. 81–96

    Google Scholar 

  • Even , S., Itai , A., and Shamir , A. [1976]: On the complexity of timetable and multicommodity flow problems. SIAM Journal on Computing 5 (1976), 691–703

    Google Scholar 

  • Feige , U., and Lee , J.R. [2007]: An improved approximation ratio for the minimum linear arrangement problem. Information Processing Letters 101 (2007), 26–29

    Google Scholar 

  • Fleischer , L.K. [2000]: Approximating fractional multicommodity flow independent of the number of commodities. SIAM Journal on Discrete Mathematics 13 (2000), 505–520

    Google Scholar 

  • Ford , L.R., and Fulkerson , D.R. [1958]: A suggested computation for maximal multicommodity network flows. Management Science 5 (1958), 97–101

    Google Scholar 

  • Ford , L.R., and Fulkerson , D.R. [1962]: Flows in Networks. Princeton University Press, Princeton 1962

    Google Scholar 

  • Fortune , S., Hopcroft , J., and Wyllie , J. [1980]: The directed subgraph homeomorphism problem. Theoretical Computer Science 10 (1980), 111–121

    Google Scholar 

  • Frank , A. [1980]: On the orientation of graphs. Journal of Combinatorial Theory B 28 (1980), 251–261

    Google Scholar 

  • Frank , A. [1981]: How to make a digraph strongly connected. Combinatorica 1 (1981), 145–153

    Google Scholar 

  • Frank , A. [1985]: Edge-disjoint paths in planar graphs. Journal of Combinatorial Theory B 39 (1985), 164–178

    Google Scholar 

  • Frank , A., and Tardos , É. [1984]: Matroids from crossing families. In: Finite and Infinite Sets; Vol. I (A. Hajnal , L. Lovász , and V.T. Sós , eds.), North-Holland, Amsterdam, 1984, pp. 295–304

    Google Scholar 

  • Garg , N., and Könemann , J. [2007]: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM Journal on Computing 37 (2007), 630–652

    Google Scholar 

  • Grigoriadis , M.D., and Khachiyan , L.G. [1996]: Coordination complexity of parallel price-directive decomposition. Mathematics of Operations Research 21 (1996), 321–340

    Google Scholar 

  • Guruswami , V., Håstad , J., Manokaran , R., Raghavendra , P., and Charikar , M. [2011]: Beating the random ordering is hard: every ordering CSP is approximation resistant. SIAM Journal on Computing 40 (2011), 878–914

    Google Scholar 

  • Hansen , M.D. [1989]: Approximation algorithms for geometric embeddings in the plane with applications to parallel processing problems. Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (1989), 604–609

    Google Scholar 

  • Hirai , H. [2010]: Metric packing for K 3 + K 3. Combinatorica 30 (2010), 295–326

    Google Scholar 

  • Hu , T.C. [1963]: Multi-commodity network flows. Operations Research 11 (1963), 344–360

    Google Scholar 

  • Ibaraki , T., and Poljak , S. [1991]: Weak three-linking in Eulerian digraphs. SIAM Journal on Discrete Mathematics 4 (1991), 84–98

    Google Scholar 

  • Karakostas , G. [2008]: Faster approximation schemes for fractional multicommodity flow problems. ACM Transactions on Algorithms 4 (2008), Article 13

    Google Scholar 

  • Karp , R.M. [1972]: Reducibility among combinatorial problems. In: Complexity of Computer Computations (R.E. Miller , J.W. Thatcher , eds.), Plenum Press, New York 1972, pp. 85–103

    Google Scholar 

  • Karzanov , A.V. [1987]: Half-integral five-terminus flows. Discrete Applied Mathematics 18 (1987) 263–278

    Google Scholar 

  • Kawarabayashi , K., Kobayashi , Y., and Reed , B. [2012] The disjoint paths problem in quadratic time. Journal of Combinatorial Theory B 102 (2012), 424–435

    Google Scholar 

  • Kawarabayashi , K., and Wollan , P. [2010]: A shorter proof of the graph minor algorithm: the unique linkage theorem. Proceedings of the 42th Annual ACM Symposium on Theory of Computing (2010), 687–694

    Google Scholar 

  • Kleinberg , J. [1996]: Approximation algorithms for disjoint paths problems. PhD thesis, MIT, Cambridge 1996

    Google Scholar 

  • Leighton , T., and Rao , S. [1999]: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46 (1999), 787–832

    Google Scholar 

  • Linial , N., London , E., and Rabinovich , Y. [1995]: The geometry of graphs and some of its algorithmic applications. Combinatorica 15 (1995), 215–245

    Google Scholar 

  • Lomonosov , M. [1979]: Multiflow feasibility depending on cuts. Graph Theory Newsletter 9 (1979), 4

    Google Scholar 

  • Lovász , L. [1976]: On two minimax theorems in graph. Journal of Combinatorial Theory B 21 (1976), 96–103

    Google Scholar 

  • Lucchesi , C.L., and Younger , D.H. [1978]: A minimax relation for directed graphs. Journal of the London Mathematical Society II 17 (1978), 369–374

    Google Scholar 

  • Matsumoto , K., Nishizeki , T., and Saito , N. [1986]: Planar multicommodity flows, maximum matchings and negative cycles. SIAM Journal on Computing 15 (1986), 495–510

    Google Scholar 

  • Middendorf , M., and Pfeiffer , F. [1993]: On the complexity of the disjoint path problem. Combinatorica 13 (1993), 97–107

    Google Scholar 

  • Müller , D., Radke , K., and Vygen , J. [2011]: Faster min-max resource sharing in theory and practice. Mathematical Programming Computation 3 (2011), 1–35

    Google Scholar 

  • Nagamochi , H., and Ibaraki , T. [1989]: On max-flow min-cut and integral flow properties for multicommodity flows in directed networks. Information Processing Letters 31 (1989), 279–285

    Google Scholar 

  • Nash-Williams , C.S.J.A. [1969]: Well-balanced orientations of finite graphs and unobtrusive odd-vertex-pairings. In: Recent Progress in Combinatorics (W. Tutte , ed.), Academic Press, New York 1969, pp. 133–149

    Google Scholar 

  • Naves , G. [2009]: The hardness of routing two pairs on one face. Les cahiers Leibniz, Technical Report No. 177, Grenoble 2009

    Google Scholar 

  • Nishizeki , T., Vygen , J., and Zhou , X. [2001]: The edge-disjoint paths problem is NP-complete for series-parallel graphs. Discrete Applied Mathematics 115 (2001), 177–186

    Google Scholar 

  • Okamura , H., and Seymour , P.D. [1981]: Multicommodity flows in planar graphs. Journal of Combinatorial Theory B 31 (1981), 75–81

    Google Scholar 

  • Räcke , H. [2008]: Optimal hierarchical decompositions for congestion minimization in networks. Proceedings of the 40th Annual ACM Symposium on Theory of Computing (2008), 255–264

    Google Scholar 

  • Raghavan , P., and Thompson , C.D. [1987]: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (1987), 365–374

    Google Scholar 

  • Robertson , N., and Seymour , P.D. [1986]: Graph minors VI; Disjoint paths across a disc. Journal of Combinatorial Theory B 41 (1986), 115–138

    Google Scholar 

  • Robertson , N., and Seymour , P.D. [1995]: Graph minors XIII; The disjoint paths problem. Journal of Combinatorial Theory B 63 (1995), 65–110

    Google Scholar 

  • Rothschild , B., and Whinston , A. [1966]: Feasibility of two-commodity network flows. Operations Research 14 (1966), 1121–1129

    Google Scholar 

  • Scheffler , P. [1994]: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. Technical Report No. 396/1994, FU Berlin, Fachbereich 3 Mathematik

    Google Scholar 

  • Schwärzler , W. [2009]: On the complexity of the planar edge-disjoint paths problem with terminals on the outer boundary. Combinatorica 29 (2009), 121–126

    Google Scholar 

  • Sebő , A. [1993]: Integer plane multiflows with a fixed number of demands. Journal of Combinatorial Theory B 59 (1993), 163–171

    Google Scholar 

  • Seymour, P.D. [1980]: Four-terminus flows. Networks 10 (1980), 79–86

    Google Scholar 

  • Seymour , P.D. [1981]: On odd cuts and multicommodity flows. Proceedings of the London Mathematical Society (3) 42 (1981), 178–192

    Google Scholar 

  • Shahrokhi , F., and Matula , D.W. [1990]: The maximum concurrent flow problem. Journal of the ACM 37 (1990), 318–334

    Google Scholar 

  • Sherman , J. [2009]: Breaking the multicommodity flow barrier for \(O(\sqrt{\log n})\)-approximations to sparsest cut. Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (2009), 363–372

    Google Scholar 

  • Vygen , J. [1995]: NP-completeness of some edge-disjoint paths problems. Discrete Applied Mathematics 61 (1995), 83–90

    Google Scholar 

  • Wagner , D., and Weihe , K. [1995]: A linear-time algorithm for edge-disjoint paths in planar graphs. Combinatorica 15 (1995), 135–150

    Google Scholar 

  • Young , N. [1995]: Randomized rounding without solving the linear program. Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (1995), 170–178

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, Bonn, Germany

    Bernhard Korte & Jens Vygen

Authors
  1. Bernhard Korte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Vygen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Germany

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Korte, B., Vygen, J. (2018). Multicommodity Flows and Edge-Disjoint Paths. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-56039-6_19

Download citation

Publish with us

Policies and ethics

Search

Navigation