Abstract
We reveal a close relationship between quadratic M-convex functions and tree metrics: A quadratic function defined on the integer lattice points is M-convex if and only if it has a tree representation. Furthermore, a discrete analogue of the Hessian matrix is defined for functions on the integer points. A function is M-convex if and only if the negative of the ‘discrete Hessian matrix’ is a tree metric matrix at each integer point. Thus, the M-convexity of a function can be characterized by that of its local quadratic approximations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
J-P. Barthélemy and A. Guénoche, Trees and Proximity Representations. John Wiley, New York, 1991.
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.
O.P. Buneman, A note on metric properties of trees. J. Combinatorial Theory, Series B,17 (1974), 48–50.
M.M. Deza and M. Laurent, Geometry of Cuts and Metrics. Springer-Verlag, Berlin, 1997.
A.W.M. Dress, V. Moulton and W.F. Terhalle, T-theory: An overview. Eur. J. Combinatorics,17 (1996), 161–175.
A.W.M. Dress and W. Terhalle, The tree of life and other affine buildings. Documenta Mathematica, Extra Vol. III (1998), 565–574 (electronic).
A.W.M. Dress and W. Wenzel, Valuated matroids. Adv. in Math.,93 (1992), 214–250.
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.
J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs. Ann. Discrete Math.,1 (1977), 185–204.
P. Favati and F. Tardella, Convexity in nonlinear interger programming. Ricerca Operativa,53 (1990), 3–44.
A. Frank, An algorithm for submodular functions on graphs. Ann. Discrete Math.,16 (1982), 97–120.
S. Fujishige, Theory of submodular programs: A Fenchel-type min-max theorem and subgradients of submodular functions. Math. Programming,29 (1984), 142–155.
S. Fujishige, Submodular Functions and Optimization. Annals of Discrete Mathematics47, North-Holland, Amsterdam, 1991.
L. Lovász, Submodular functions and convexity. Mathematical Programming—The State of the Art (eds. A. Bachern, M. Grötschel and B. Korte), Springer-Verlag, Berlin, 1983, 235–257.
S. Moriguchi and K. Murota, Discrete Hessian matrix for L-convex functions. METR 2004–30, Department of Mathematical Informatics, University of Tokyo, June 2004.
K. Murota, Convexity and Steinitz’s exchange property. Adv. in Math.,124 (1996), 272–311.
K. Murota, Discrete Convex Analysis—An Introduction (in Japanese). Kyoritsu Publishing Co., Tokyo, 2001.
K. Murota, Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia, 2003.
K. Murota and A. Shioura, Quadratic M-convex and L-convex functions. Adv. in Appl. Math.,33 (2004), 318–341.
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
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Hira, H., Murota, K. M-convex functions and tree metrics. Japan J. Indust. Appl. Math. 21, 391–403 (2004). https://doi.org/10.1007/BF03167590
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03167590