Abstract
A theory of “convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz's exchange property. It includes linear functions on matroids, valuations on matroids, and separable concave functions on the integral base polytope. It is shown that a function ω has the exchange property if and only if it can be extended to a concave function \(\bar \omega\)such that the maximizers of (\(\bar \omega\)+any linear function) form an integral base polytope. A Fenchel-type min-max theorem and discrete separation theorems are given, which contain, e.g., Frank's discrete separation theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection.
The author thanks András Frank, Satoru Fujishige, Satoru Iwata, András Sebö and Akiyoshi Shioura for valuable discussions.
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References
Dress, A. W. M., Terhalle, W.: Well-layered maps and the maximum-degree k x k- subdeterminant of a matrix of rational functions. Appl. Math. Lett. 8 (1995) 19–23
Dress, A. W. M., Terhalle, W.: Well-layered maps — A class of greedily optimizable set functions. Appl. Math. Lett. 8 (1995) 77–80
Dress, A. W. M., Wenzel, W.: Valuated matroid: A new look at the greedy algorithm. Appl. Math. Lett. 3 (1990) 33–35
Dress, A. W. M., Wenzel, W.: Valuated matroids. Adv. Math. 93 (1992) 214–250
Frank, A.: A weighted matroid intersection algorithm. J. Algorithms 2 (1981) 328–336
Frank, A.: An algorithm for submodular functions on graphs. Ann. Disc. Math. 16 (1982) 97–120
Fujishige, S.: Lexicographically optimal base of a polymatroid with respect to a weight vector. Math. Oper. Res. 5 (1980) 186–196
Fujishige, S.: Theory of submodular programs: A Fenchel-type min-max theorem and subgradients of submodular functions. Math. Progr. 29 (1984) 142–155
Fujishige, S.: Submodular Functions and Optimization. Ann. Disc. Math. 47, North-Holland, 1991
Groenevelt, H.: Two algorithms for maximizing a separable concave function over a polymatroid feasible region. Working Paper, Grad. School Management, Univ. Rochester, 1995.
Iri, M., Tomizawa, N.: An algorithm for finding an optimal “independent assignment”. J. Oper. Res. Soc. Japan 19 (1976) 32–57
Lovász, L.: Submodular functions and convexity. In “Mathematical Programming — The State of the Art” (A. Bachem, M. Grötschel and B. Korte, eds.), Springer, 235–257, 1983
Murota, K.: Finding optimal minors of valuated bimatroids. Appl. Math. Lett. 8 (1995) 37–42
Murota, K.: Valuated matroid intersection, I: optimality criteria, II: algorithms. SIAM J. Disc. Math. 9 (1996) No.3 (to appear)
Murota, K.: Matroid valuation on independent sets. Report 95842-OR, Inst. Disc. Math., Univ. Bonn, 1995
Murota, K.: Submodular flow problem with a nonseparable cost function. Report 95843-OR, Inst. Disc. Math., Univ. Bonn, 1995
Murota, K.: Convexity and Steinitz's exchange property. Report 95848-OR, Inst. Disc. Math., Univ. Bonn, 1995
Rockafellar, R. T.: Convex Analysis. Princeton Univ. Press, 1970
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Murota, K. (1996). Convexity and Steinitz's exchange property. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_20
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DOI: https://doi.org/10.1007/3-540-61310-2_20
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