Algorithms in Discrete Convex Analysis

IEICE Transactions on Information and Systems

Volumn E83-D, Issue 3, 2000, Pages 344-352

  Murota, Kazuo ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

Discrete convex analysis; L convex function; M convex function; Matroid

Indexed keywords

FUNCTIONS; INTEGER PROGRAMMING; MATRIX ALGEBRA; SET THEORY; THEOREM PROVING;

DISCRETE CONVEX ANALYSIS; GREEDY ALGORITHMS; LOVASZ CONVEX FUNCTION; MUROTA CONVEX FUNCTION;

ALGORITHMS;

EID: 0034156590     PISSN: 09168532     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (43)

1. R.E. Bixby and W.H. Cunningham "Matroid optimization and algorithms" in Handbook of Combinatorics vol.1 eds. R.L. Graham M. Grötschel and L. Loväsz Chapter 11 pp.551-609 Eisevier Science B. V. Amsterdam 1995.
2. W.U. Cunningham and A. Frank "A primal-dual algorithm for submodular flows" Math. Operations Research vol.10 pp.251-262 1985.
3. A.W.M. Dress and W. Wenzel "Valuated matroid: A new look at the greedy algorithm" Appl. Math. Lett. vol.3 pp.33-35 1990.
4. A.W.M. Dress and W. Wenzel "Valuated matroids" Advances in Math. vol.93 pp.214-250 1992.
5. J. Edmonds "Submodular functions matroids and certain polyhedra" in Combinatorial Structures and Their Applications eds. R. Guy H. Hanani N. Sauer and J. Schönheim pp.69-87 Gordon and Breach New York 1970.
6. J. Edmonds "Matroid intersection" Annals of Discrete Math. vol.14 pp.39-49 1979.
7. P. Favati and F. Tardella "Convexity in nonlinear integer programming" Ricerca Operativa vol.53 pp.3-44 1990.
8. A. Frank "A weighted matroid intersection algorithm" J. Algorithms vol.2 pp.328-336 1981.
9. A. Frank "An algorithm for submodular functions on graphs" Annals of Discrete Math. vol.16 pp.97-120 1982.
10. A. Frank and E. Tardos "Generalized polymatroids and submodular flows" Math. Programming vol.42 pp.4895G3 1988.
11. S. Fujishige "Theory of submodular programs: A Fencheltype min-max theorem and subgradients of submodular functions" Math. Programming vol.29 pp.142-155 1984.
12. S. Fujishige "Submodular functions and optimization" Annals of Discrete Math. vol.47 North-Holland Amsterdam 1991.
13. S. Fujishige and K. Murota "Notes on L-/M-convex functions and the separation theorems" Math. Programming to appear.
14. H. Groenevelt "Two algorithms for maximizing a separable concave function over a polymatroidal feasible region" European J. Operations Research vol.54 pp.227-236 1991.
15. M. Grötschel L. Lovâsz and A. Schrijver Geometric Algorithms and Combinatorial Optimization 2nd. ed. SpringerVerlag Berlin 1993.
16. D.S. Hochbaum "Lower and upper bounds for the allocation problem and other nonlinear optimization problems" Math. Operations Research vol.19 pp.390-409 1994.
17. M. Iri and N. Tomizawa "An algorithm for finding an optimal 'independent assignment'" J. Operations Research Soc. Japan vol.19 pp.32-57 1976.
18. S. Iwata "The submodular flow problem" in Discrete Structures and Algorithms VI ed. S. Fujishige KindaiKagaku-sha pp. 127-170 Tokyo 1999.
19. S. Iwata L. Fleischer and S. Fujishige "A strongly polynomial-time algorithm for minimizing submodular functions" Discrete Mathematics and Systems Science Research Report no.99-07 Osaka University July 1999.
20. S. Iwata and M. Shigeno "Conjugate scaling technique for Fenchel-type duality in discrete convex optimization" Information Processing Society of Japan SIG Notes 98-AL65 pp.33-40 1998.
21. N. Katoh and T. Ibaraki "Resource allocation problems" in Handbook of Combinatorial Optimization vol.2 eds. D. -Z. Du and P.M. Pardalos pp.159-260 Kluwer Academic Publishers Boston/Dordrecht/London 1998.
22. E.L. Lawler "Matroid intersection algorithms" Math. Programming vol.9 pp.31-56 1975.
23. L. Lovâsz "Submodular functions and convexity" in Mathematical Programming -The State of the Art eds. A. Bachern M. Grötschel and B. Körte pp.235-257 Springer-Verlag Berlin 1983.
24. B.L. Miller "On minimizing nonseparable functions defined on the integers with an inventory application" SIAM J. Appl. Math. vol.21 pp.166-185 1971.
25. M. Minoux "Solving integer minimum cost flows with separable convex objective polynomially" Math. Programming vol.26 pp.237-239 1986.
26. K. Murota "Finding optimal minors of valuated bimatroids" Appl. Math. Lett. vol.8 pp.37-42 1995.
27. K. Murota "Valuated matroid intersection I: Optimality criteria" SIAM J. Discrete Math. vol.9 pp.545-561 1996.
28. K. Murota "Valuated matroid intersection II: Algorithms" SIAM J. Discrete Math. vol.9 pp.562-576 1996.
29. K. Murota "Submodular flow problem with a nonseparable cost function" Combinatorica vol.19 pp.87-109 1999.
30. K. Murota "Convexity and Steinitz's exchange property" Advances in Math. vol.124 pp.272-311 1996.
31. K. Murota "Discrete convex analysis" Math. Programming vol.83 pp.313-371 1998.
32. K. Murota "Discrete convex analysis" in Discrete Structures and Algorithms V ed. S. Fujishige pp.51-100
33. Kindai-Kagaku-sha Tokyo 1998.
34. K. Murota Matrices and Matroids for Systems Analysis Springer-Verlag Berlin 2000.
35. K. Murota and A. Shioura "M-convex function on generalized polymatroid" Math. Operations Research vol.24 pp.95-105 1999.
36. K. Murota and A. Shioura "Extension of M-convexity and L-convexity to polyhedral convex functions" RIMS Preprint 1227 Kyoto University March 1999.
37. K. Murota and A. Shioura "Relationship of M-/L-convex functions with discrete convex functions by Miller and by Favati-Tardella" Discrete Appl. Math. to appear.
38. R.T. Rockafellar Convex Analysis Princeton University Press Princeton NJ 1970.
39. R.T. Rockafellar Network Flows and Monotropic Optimization John Wiley & Sons New York 1984.
40. A. Shioura "Minimization of an M-convex function" Discrete Appl. Math. vol.84 pp.215-220 1998.
41. H. Whitney "On the abstract properties of linear dependence" American J. Math. vol.57 pp.509-533 1935.
42. U. Zimmermann "Minimization of some nonlinear functions over polymatroidal network flows" Annals of Discrete Math. vol.16 pp.287-309 1982.
43. Kazuo Murota received Bachelor Master and Doctor Degrees of Engineering from the University of Tokyo in 1978 1980 and 1983 respectively. He is presently a professor at Research Institute for Mathematical Sciences Kyoto University. His research interest centers around discrete mathematical methods in engineering.