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Annals of Combinatorics

, Volume 16, Issue 3, pp 543–569 | Cite as

On Tight Spans for Directed Distances

  • Hiroshi HiraiEmail author
  • Shungo Koichi
Article

Abstract

An extension (V, d) of a metric space (S, μ) is a metric space with \({{S \subseteq V}}\) and \({{d\mid{_S} = \mu}}\) , and is said to be tight if there is no other extension (V, d′) of (S, μ) with d′ ≤ d . Isbell and Dress independently found that every tight extension embeds isometrically into a certain metrized polyhedral complex associated with (S, μ), called the tight span. This paper develops an analogous theory for directed metrics, which are “not necessarily symmetric” distance functions satisfying the triangle inequality. We introduce a directed version of the tight span and show that it has a universal embedding property for tight extensions. Also we introduce a new natural class of extensions, called cyclically tight extensions, and we show that there also exists a certain polyhedral complex having a universal property relative to cyclically tightness. This polyhedral complex coincides with (a fiber of) the tropical polytope spanned by the column vectors of –μ, which was earlier introduced by Develin and Sturmfels. Thus this gives a tight-span interpretation to the tropical polytope generated by a nonnegative square matrix satisfying the triangle inequality. As an application, we prove the following directed version of the tree metric theorem: A directed metric μ is a directed tree metric if and only if the tropical rank of –μ is at most two. Also we describe how tight spans and tropical polytopes are applied to the study of multicommodity flows in directed networks.

Mathematics Subject Classification

52B99 05C12 

Keywords

metrics tight spans tropical polytopes multicommodity flows 

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References

  1. 1.
    Ahuja R.K., Magnanti T.L., Orlin J.B.: Network Flows—Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)zbMATHGoogle Scholar
  2. 2.
    Bandelt H.-J., Chepoi V., Karzanov A.: A characterization of minimizable metrics in the multifacility location problem. European J. Combin. 21(6), 715–725 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bandelt H.-J., Dress A.W.M.: A canonical decomposition theory for metrics on a finite set. Adv. Math. 92(1), 47–105 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Buneman P.: A note on the metric properties of trees. J. Combin. Theory Ser. B 17, 48–50 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Charikar M., Makarychev K., Makarychev Y.: Directed metrics and directed graph partitioning problems. In: (eds) In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms., pp. 51–60. ACM, New York (2006)Google Scholar
  6. 6.
    Chenchiah I.V., Rieger M.O., Zimmer J.: Gradient flows in asymmetric metric spaces. Nonlinear Anal. 71(11), 5820–5834 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chrobak M., Larmore L.L.: Generosity helps or an 11-competitive algorithm for three servers. J. Algorithms 16(2), 234–263 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Develin M., Santos F., Sturmfels B.: On the rank of a tropical matrix. In: Goodman, J.E., Pach, J., Welzl, E. (eds) Combinatorial and Computational Geometry., pp. 213–242. Cambridge University Press, Cambridge (2005)Google Scholar
  9. 9.
    Develin M., Sturmfels B.: Tropical convexity. Documenta Mathematica 9(1–27), 205–206 (2004)MathSciNetGoogle Scholar
  10. 10.
    Dress A.W.M.: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math. 53(3), 21–402 (1984)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dress A.W.M., Huber K.T., Koolen J., Moulton V., Spillner A.: Basic Phylogenetic Combinatorics. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  12. 12.
    Ford L.R. Jr., Fulkerson D.R.: Flows in Networks. Princeton University Press, Princeton (1962)zbMATHGoogle Scholar
  13. 13.
    Frank A.: On connectivity properties of Eulerian digraphs. In: Andersen, L.D., Jakobsen, I.T., Thomassen, C., Toft, B., Vestergaad, P.D. (eds) Graph Theory in Memory of G.A. Dirac (Sandbjerg 1985), pp. 179–194. North-Holland, Amsterdam (1989)Google Scholar
  14. 14.
    Herrmann S., Joswig M.: Splitting polytopes. Münster J. Math. 1, 109–141 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Herrmann, S.,Moulton, V.: Trees, tight-spans and point configuration. arXiv:1104.1538 (2011)Google Scholar
  16. 16.
    Hirai H.: A geometric study of the split decomposition. Discrete Comput. Geom. 36(2), 331–361 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hirai H.: Characterization of the distance between subtrees of a tree by the associated tight span. Ann. Combin. 10(1), 111–128 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hirai H.: Tight spans of distances and the dual fractionality of undirected multiflow problems. J. Combin. Theory Ser. B 99(6), 843–868 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hirai H.: Folder complexes and multiflow combinatorial dualities. SIAM J. Discrete Math. 25(3), 1119–1143 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hirai H., Koichi S.: On duality and fractionality of multicommodity flows in directed networks. Discrete Optim. 8(3), 428–445 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Huson D.H., Rupp R., Scornavacca C.: Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press Cambridge (2010)CrossRefGoogle Scholar
  22. 22.
    Ibaraki T., Karzanov A.V., Nagamochi H.: A fast algorithm for finding a maximum free multiflow in an inner Eulerian network and some generalizations. Combinatorica 18(1), 61–83 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Isbell J.R.: Six theorems about injective metric spaces. Comment. Math. Helv. 39, 65–76 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Karzanov A.V.: Metrics with finite sets of primitive extensions. Ann. Combin. 2(3), 211–241 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Karzanov A.V.: Minimum 0-extensions of graph metrics. European J. Combin. 19(1), 71–101 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lomonosov, M.V.: unpublished manuscript (1978)Google Scholar
  27. 27.
    Lomonosov M.V.: Combinatorial approaches to multiflow problems. Discrete Appl. Math. 11(1), 1–93 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Naor, J., Schwartz, R.: The directed circular arrangement problem. ACM Trans. Algorithms 6, Art. 47 (2010)Google Scholar
  29. 29.
    Papadopoulos, A., Théret, G.: On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. In: Papadopoulos, A. (Ed.) Handbook of Teichmüller theory, Vol. I, pp. 111–204. European Mathematical Society (EMS), Zürich (2007)Google Scholar
  30. 30.
    PatrinosA.N. , Hakimi S.L.: The distance matrix of a graph and its tree realization. Quart. Appl. Math. 30, 255–269 (1972/1973)Google Scholar
  31. 31.
    Schrijver A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer-Verlag, Berlin (2003)zbMATHGoogle Scholar
  32. 32.
    Semple C., Steel M.: Phylogenetics. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  33. 33.
    Simões-Pereira J.M.S.: A note on the tree realizability of a distance matrix. J. Combin. Theory 6, 303–310 (1969)zbMATHCrossRefGoogle Scholar
  34. 34.
    Sturmfels B., Yu J.: Classification of six-point metrics. Electron. J. Combin. 11(1),–R44 (2004)MathSciNetGoogle Scholar
  35. 35.
    Tansel, B.C., Francis, R.L., Lowe, T.J.: Location on networks: a survey. I-II. Management Sci. 29(4), 482–497; 498–511 (1983)Google Scholar
  36. 36.
    Zarecki K.A.: Constructing a tree on the basis of a set of distances between the hanging vertices. Uspekhi Mat. Nauk (in Russian) 20(6), 90–92 (1965)Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  2. 2.Department of Systems Design and EngineeringNanzan UniversitySetoJapan

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