M-convex functions and tree metrics

Japan Journal of Industrial and Applied Mathematics

Volumn 21, Issue 3, 2004, Pages 391-403

  Hirai, Hiroshi ^a   Murota, Kazuo ^b,c  

^a KYOTO UNIVERSITY   (Japan)

^b UNIVERSITY OF TOKYO   (Japan)

^c JAPAN SCIENCE AND TECHNOLOGY AGENCY   (Japan)

Author keywords

Discrete Hessian matrix; Four point condition; M convex function; Tree metric

Indexed keywords

EID: 8744299918     PISSN: 09167005     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf03167590     Document Type: Article

References (21)

1. H-J. Bandelt and A. W. M. Dress, A canonical decomposition theory for metrics on a finite set. Adv. in Math., 92 (1992), 47-105.
2. J-P. Barthélemy and A. Guénoche, Trees and Proximity Representations. John Wiley, New York, 1991.
3. O.P. Buneman, The recovery of trees from measures of dissimilarity. Mathematics in the Archaeological and Historical Sciences (eds. F. R. Hodson, D. G. Kendall and P. Tautu), Edinburgh University Press, 1971, 387-395.
4. O.P. Buneman, A note on metric properties of trees. J. Combinatorial Theory, Series B, 17 (1974), 48-50.
5. M.M. Deza and M. Laurent, Geometry of Cuts and Metrics. Springer-Verlag, Berlin, 1997.
6. A.W.M. Dress, V. Moulton and W.F. Terhalle, T-theory: An overview. Eur. J. Combinatorics, 17 (1996), 161-175.
7. A.W.M. Dress and W. Terhalle, The tree of life and other affine buildings. Documenta Mathematica, Extra Vol. III (1998), 565-574 (electronic).
8. A.W.M. Dress and W. Wenzel, Valuated matroids. Adv. in Math., 93 (1992), 214-250.
9. J. Edmonds, Submodular functions, matroids and certain polyhedra. Combinatorial Structures and Their Applications (eds. R. Guy, H. Hanani, N. Sauer and J. Schönheim), Gordon and Breach, New York, 1970, 69-87.
10. J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs. Ann. Discrete Math., 1 (1977), 185-204.
11. P. Favati and F. Tardella, Convexity in nonlinear interger programming. Ricerca Operativa, 53 (1990), 3-44.
12. A. Frank, An algorithm for submodular functions on graphs. Ann. Discrete Math., 16 (1982), 97-120.
13. S. Fujishige, Theory of submodular programs: A Fenchel-type min-max theorem and subgradients of submodular functions. Math. Programming, 29 (1984), 142-155.
14. S. Fujishige, Submodular Functions and Optimization. Annals of Discrete Mathematics 47, North-Holland, Amsterdam, 1991.
15. L. Lovász, Submodular functions and convexity. Mathematical Programming-The State of the Art (eds. A. Bachem, M. Grötschel and B. Korte), Springer-Verlag, Berlin, 1983, 235-257.
16. S. Moriguchi and K. Murota, Discrete Hessian matrix for L-convex functions. METR 2004-30, Department of Mathematical Informatics, University of Tokyo, June 2004.
17. K. Murota, Convexity and Steinitz's exchange property. Adv. in Math., 124 (1996), 272-311.
18. K. Murota, Discrete Convex Analysis-An Introduction (in Japanese). Kyoritsu Publishing Co., Tokyo, 2001.
19. K. Murota, Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia, 2003.
20. K. Murota and A. Shioura, Quadratic M-convex and L-convex functions. Adv. in Appl. Math., 33 (2004), 318-341.
21. R. Nabben and R.S. Varga, A linear algebra proof that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. SIAM J. Matrix Anal. Appl., 15 (1994), 107-113