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Subdominant Matroid Ultrametrics
Federico Ardila
Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720-5070, USA
Annals of Combinatorics 8 (4) p.379-389 December, 2004
AMS Subject Classification: 05B35, 51F99, 92B10
Given a matroid M on the ground set E, the Bergman fan , or space of M-ultrametrics, is a polyhedral complex in which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point w in , we wish to find an M-ultrametric which is closest to it in the
    The solution to this problem follows easily from the existence of the subdominant M-ultrametric: a componentwise maximum M-ultrametric which is componentwise smaller than w. A procedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry.
    When the matroid in question is the graphical matroid of the complete graph Kn, the Bergman fan parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data.
Keywords: phylogenetic trees, tropical geometry, Bergman complex, ultrametric


1. F. Ardila and C. Klivans, The Bergman complex of a matroid and phylogenetic trees, arxiv: math.CO/0311370, preprint, 2003.

2. G. Bergman, The logarithmic limit-set of an algebraic variety, Trans. Amer. Math. Soc. 157 (1971) 459--469.

3. R. Bieri and J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347 (1984) 168--195.

4. L. Billera, S. Holmes, and K. Vogtmann, Geometry of the space of phylogenetic trees, Adv. in Appl. Math. 27 (2001) 733--767.

5. A. Björner, The homology and shellability of matroids and geometric lattices, In: Matroid Applications, N. White, Ed., Cambridge University Press, Cambridge, 1992, pp. 226--283.

6. V. Chepoi and B. Fichet, ℓ-approximation via subdominants, J. Math. Psych. 44 (2000) 600--616.

7. M. Develin and B. Sturmfels, Tropical convexity, arXiv: math. MG/0308254, preprint, 2003.

8. M. Einsiedler, M. Kapranov, and D. Lind, Non-archimedean amoebas and tropical varieties, arXiv: math. AG/0408311, preprint, 2004.

9. D. Kozen, The Design and Analysis of Algorithms, Springer-Verlag, 1991.

10. J. G. Oxley, Matroid Theory, Oxford University Press, New York, 1992.

11. C. Semple and M. Steel, Phylogenetics, Oxford University Press, 2003.

12. B. Sturmfels, Solving Systems of Polynomial Equations, American Mathematical Society, Providence, 2002.

13. D. Speyer and B. Sturmfels, The tropical Grassmannian, arXiv:math.AG/0304218, preprint, 2003.

14. R.E. Tarjan, Data Structures and Network Algorithms, Society of Industrial and Applied Mathematics, 1983.