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Mehrgüterflüsse und kantendisjunkte Wege

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Kombinatorische Optimierung

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Zusammenfassung

Das MULTICOMMODITY-FLOW-PROBLEM ist eine Verallgemeinerung des MAXIMUM- FLOW-PROBLEMS. In einem gegebenen Digraphen mit Kantenkapazitäten möchten wir nun einen s-t-Fluss für mehrere Paare (s; t) finden (wir sprechen hier von mehreren Gütern), so dass der Gesamtfluss durch jede Kante deren Kapazität nicht übersteigt

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References

  1. Aumann, Y. und Rabani, Y.: An \(O(\log k)\) approximate min-cut max-flow theorem and approximation algorithm. SIAM Journal on Computing 27, 291–301 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Boesch, F., Tindell, R.: Robbins's theorem for mixed multigraphs. American Mathematical Monthly 87, 716–719 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Charikar, M., Hajiaghayi, M.T., Karloff, H. und Rao, S.: \(\ell _{2}^{2}\) spreading metrics for vertex ordering problems. Algorithmica 56, 577–604 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  19. Frank, A., Tardos, É.: Matroids from crossing families. In: Hajnal, A., Lovász, L., und Sós, V.T. (Hrsg.) Finite and Infinite Sets, Vol. I, S. 295–304. North-Holland, Amsterdam (1984)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Hirai, H.: Metric packing for \(K_{3}+K_{3}\). Combinatorica 30, 295–326 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  27. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (Hrsg.) Complexity of Computer Computations, S. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  31. Kleinberg, J.: Approximation algorithms for disjoint paths problems. Ph.D. thesis. MIT, Cambridge (1996)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  41. Naves, G.: The hardness of routing two pairs on one face. Les cahiers Leibniz, Technical Report No. 177, Grenoble (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  54. Sherman, J.: 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, S. 363–372 (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

Weiterführende Literatur

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

    Google Scholar 

  2. Naves, G., Sebő, A.: Multiflow feasibility: an annotated tableau. In: Cook, W.J., Lovász, L., Vygen, J. (Hrsg.) Research Trends in Combinatorial Optimization, S. 261–283. Springer, Berlin (2009)

    Chapter  Google Scholar 

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

    Google Scholar 

  4. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Kap. 70–76. Springer, Berlin (2003)

    MATH  Google Scholar 

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

    Google Scholar 

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Correspondence to Bernhard Korte or Jens Vygen .

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Korte, B., Vygen, J. (2012). Mehrgüterflüsse und kantendisjunkte Wege. In: Kombinatorische Optimierung. Springer-Lehrbuch Masterclass. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25401-7_19

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