Advertisement

Multicommodity Flows and Edge-Disjoint Paths

  • Bernhard Korte
  • Jens Vygen
Chapter
Part of the Algorithms and Combinatorics book series (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.

References

General Literature

  1. 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. 47–100Google Scholar
  2. 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. 261–283Google Scholar
  3. 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 1995Google Scholar
  4. Schrijver, A. [2003]: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003, Chapters 70–76Google Scholar
  5. 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, 1996Google Scholar

Cited References

  1. Aumann, Y. and Rabani, Y. [1998]: An O(logk) approximate min-cut max-flow theorem and approximation algorithm. SIAM Journal on Computing 27 (1998), 291–301Google Scholar
  2. Arora, S., Rao, S., and Vazirani, U. [2009]: Expander flows, geometric embeddings and graph partitioning. Journal of the ACM 56 (2009), Article 5Google Scholar
  3. 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), 238–247Google Scholar
  4. Becker, M., and Mehlhorn, K. [1986]: Algorithms for routing in planar graphs. Acta Informatica 23 (1986), 163–176Google Scholar
  5. Bienstock, D., and Iyengar, G. [2006]: Solving fractional packing problems in \(O^{{\ast}}(\frac{1} {\epsilon } )\) iterations. SIAM Journal on Computing 35 (2006), 825–854Google Scholar
  6. Boesch, F., and Tindell, R. [1980]: Robbins’s theorem for mixed multigraphs. American Mathematical Monthly 87 (1980), 716–719Google Scholar
  7. Charikar, M., Hajiaghayi, M.T., Karloff, H. and Rao, S. [2010]: 2 2 spreading metrics for vertex ordering problems. Algorithmica 56 (2010), 577–604Google Scholar
  8. Chekuri, C., and Khanna, S. [2007]: Edge-disjoint paths revisited. ACM Transactions on Algorithms 3 (2007), Article 46Google Scholar
  9. 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. 81–96Google Scholar
  10. Even, S., Itai, A., and Shamir, A. [1976]: On the complexity of timetable and multicommodity flow problems. SIAM Journal on Computing 5 (1976), 691–703Google Scholar
  11. Feige, U., and Lee, J.R. [2007]: An improved approximation ratio for the minimum linear arrangement problem. Information Processing Letters 101 (2007), 26–29Google Scholar
  12. Fleischer, L.K. [2000]: Approximating fractional multicommodity flow independent of the number of commodities. SIAM Journal on Discrete Mathematics 13 (2000), 505–520Google Scholar
  13. Ford, L.R., and Fulkerson, D.R. [1958]: A suggested computation for maximal multicommodity network flows. Management Science 5 (1958), 97–101Google Scholar
  14. Ford, L.R., and Fulkerson, D.R. [1962]: Flows in Networks. Princeton University Press, Princeton 1962Google Scholar
  15. Fortune, S., Hopcroft, J., and Wyllie, J. [1980]: The directed subgraph homeomorphism problem. Theoretical Computer Science 10 (1980), 111–121Google Scholar
  16. Frank, A. [1980]: On the orientation of graphs. Journal of Combinatorial Theory B 28 (1980), 251–261Google Scholar
  17. Frank, A. [1981]: How to make a digraph strongly connected. Combinatorica 1 (1981), 145–153Google Scholar
  18. Frank, A. [1985]: Edge-disjoint paths in planar graphs. Journal of Combinatorial Theory B 39 (1985), 164–178Google Scholar
  19. 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. 295–304Google Scholar
  20. Garg, N., and Könemann, J. [2007]: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM Journal on Computing 37 (2007), 630–652Google Scholar
  21. Grigoriadis, M.D., and Khachiyan, L.G. [1996]: Coordination complexity of parallel price-directive decomposition. Mathematics of Operations Research 21 (1996), 321–340Google Scholar
  22. 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), 878–914Google Scholar
  23. 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), 604–609Google Scholar
  24. Hirai, H. [2010]: Metric packing for K 3 + K 3. Combinatorica 30 (2010), 295–326Google Scholar
  25. Hu, T.C. [1963]: Multi-commodity network flows. Operations Research 11 (1963), 344–360Google Scholar
  26. Ibaraki, T., and Poljak, S. [1991]: Weak three-linking in Eulerian digraphs. SIAM Journal on Discrete Mathematics 4 (1991), 84–98Google Scholar
  27. Karakostas, G. [2008]: Faster approximation schemes for fractional multicommodity flow problems. ACM Transactions on Algorithms 4 (2008), Article 13Google Scholar
  28. 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. 85–103Google Scholar
  29. Karzanov, A.V. [1987]: Half-integral five-terminus flows. Discrete Applied Mathematics 18 (1987) 263–278Google Scholar
  30. Kawarabayashi, K., Kobayashi, Y., and Reed, B. [2012] The disjoint paths problem in quadratic time. Journal of Combinatorial Theory B 102 (2012), 424–435Google Scholar
  31. 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), 687–694Google Scholar
  32. Kleinberg, J. [1996]: Approximation algorithms for disjoint paths problems. PhD thesis, MIT, Cambridge 1996Google Scholar
  33. 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), 787–832Google Scholar
  34. Linial, N., London, E., and Rabinovich, Y. [1995]: The geometry of graphs and some of its algorithmic applications. Combinatorica 15 (1995), 215–245Google Scholar
  35. Lomonosov, M. [1979]: Multiflow feasibility depending on cuts. Graph Theory Newsletter 9 (1979), 4Google Scholar
  36. Lovász, L. [1976]: On two minimax theorems in graph. Journal of Combinatorial Theory B 21 (1976), 96–103Google Scholar
  37. Lucchesi, C.L., and Younger, D.H. [1978]: A minimax relation for directed graphs. Journal of the London Mathematical Society II 17 (1978), 369–374Google Scholar
  38. Matsumoto, K., Nishizeki, T., and Saito, N. [1986]: Planar multicommodity flows, maximum matchings and negative cycles. SIAM Journal on Computing 15 (1986), 495–510Google Scholar
  39. Middendorf, M., and Pfeiffer, F. [1993]: On the complexity of the disjoint path problem. Combinatorica 13 (1993), 97–107Google Scholar
  40. Müller, D., Radke, K., and Vygen, J. [2011]: Faster min-max resource sharing in theory and practice. Mathematical Programming Computation 3 (2011), 1–35Google Scholar
  41. 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), 279–285Google Scholar
  42. 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. 133–149Google Scholar
  43. Naves, G. [2009]: The hardness of routing two pairs on one face. Les cahiers Leibniz, Technical Report No. 177, Grenoble 2009Google Scholar
  44. 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), 177–186Google Scholar
  45. Okamura, H., and Seymour, P.D. [1981]: Multicommodity flows in planar graphs. Journal of Combinatorial Theory B 31 (1981), 75–81Google Scholar
  46. Räcke, H. [2008]: Optimal hierarchical decompositions for congestion minimization in networks. Proceedings of the 40th Annual ACM Symposium on Theory of Computing (2008), 255–264Google Scholar
  47. Raghavan, P., and Thompson, C.D. [1987]: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (1987), 365–374Google Scholar
  48. Robertson, N., and Seymour, P.D. [1986]: Graph minors VI; Disjoint paths across a disc. Journal of Combinatorial Theory B 41 (1986), 115–138Google Scholar
  49. Robertson, N., and Seymour, P.D. [1995]: Graph minors XIII; The disjoint paths problem. Journal of Combinatorial Theory B 63 (1995), 65–110Google Scholar
  50. Rothschild, B., and Whinston, A. [1966]: Feasibility of two-commodity network flows. Operations Research 14 (1966), 1121–1129Google Scholar
  51. 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 MathematikGoogle Scholar
  52. Schwärzler, W. [2009]: On the complexity of the planar edge-disjoint paths problem with terminals on the outer boundary. Combinatorica 29 (2009), 121–126Google Scholar
  53. Sebő, A. [1993]: Integer plane multiflows with a fixed number of demands. Journal of Combinatorial Theory B 59 (1993), 163–171Google Scholar
  54. Seymour, P.D. [1980]: Four-terminus flows. Networks 10 (1980), 79–86Google Scholar
  55. Seymour, P.D. [1981]: On odd cuts and multicommodity flows. Proceedings of the London Mathematical Society (3) 42 (1981), 178–192Google Scholar
  56. Shahrokhi, F., and Matula, D.W. [1990]: The maximum concurrent flow problem. Journal of the ACM 37 (1990), 318–334Google Scholar
  57. 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), 363–372Google Scholar
  58. Vygen, J. [1995]: NP-completeness of some edge-disjoint paths problems. Discrete Applied Mathematics 61 (1995), 83–90Google Scholar
  59. Wagner, D., and Weihe, K. [1995]: A linear-time algorithm for edge-disjoint paths in planar graphs. Combinatorica 15 (1995), 135–150Google Scholar
  60. Young, N. [1995]: Randomized rounding without solving the linear program. Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (1995), 170–178Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  • Bernhard Korte
    • 1
  • Jens Vygen
    • 1
  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations