This is a survey of algorithmic results in the theory of "discrete convex analysis " for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, the Fenchel min-max duality, and separation theorems. The technical development is based on matroidtheoretic concepts, in particular, submodular functions and exchange axioms. Keywords: discrete convex analysis, matroid, L-convex function, M-convex function 1
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1461
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Convex analysis
– Rockafellar
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810
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Geometric Algorithms and Combinatorial Optimization
– Grotschel, Lovasz, et al.
- 1993
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160
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Submodular functions, matroids and certain polyhedra
– Edmonds
- 1970
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100
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Submodular functions and optimization
– Fujishige
- 1991
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60
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Network Flows and Monotropic Optimization
– Rockafellar
- 1984
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59
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On the abstract properties of linear dependence
– Whitney
- 1935
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35
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A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions
– Iwata, Fleischer, et al.
- 2001
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33
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Matrices and Matroids for Systems Analysis
– Murota
- 2000
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32
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Generalized polymatroids and submodular flows
– Frank, Tardos
- 1988
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31
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Convexity and Steinitz's exchange property
– Murota
- 1996
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29
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An algorithm for submodular functions on graphs
– Frank
- 1982
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27
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Matroid intersection
– Edmonds
- 1979
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25
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Resource allocation problem
– Ibaraki, Katoh
- 1988
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22
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A weighted matroid intersection algorithm
– Frank
- 1981
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19
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Valuated matroid: A new look at the greedy algorithm
– Dress, Wenzel
- 1990
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17
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M-convex function on generalized polymatroid
– Murota, Shioura
- 1999
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15
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Notes on L-/M-convex functions and the separation theorems
– Fujishige, Murota
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14
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A primal-dual algorithm for submodular flows
– Cunningham, Frank
- 1985
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14
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Convexity in nonlinear integer programming
– Favati, Tardella
- 1990
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14
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On minimizing nonseparable functions defined on the integers with an inventory application
– Miller
- 1971
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14
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Submodular flow problem with a nonseparable cost function
– Murota
- 1999
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13
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Theory of submodular programs: A Fenchel-type min-max theorem and subgradients of submodular functions
– Fujishige
- 1984
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13
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Matroid intersection algorithms
– Lawler
- 1975
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12
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Valuated matroids
– Dress, Wenzel
- 1992
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12
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Valuated matroid intersection, I: optimality criteria
– Murota
- 1996
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11
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and upper bounds for the allocation problem and other nonlinear optimization problems
– Lower
- 1994
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10
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Minimization of an M-convex function
– Shioura
- 1998
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9
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Valuated matroid intersection, II: Algorithms
– Murota
- 1996
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8
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Extension of M-convexity and L-convexity to polyhedral convex functions
– Murota, Shioura
- 1999
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6
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Two algorithms for maximizing a separable concave function over a polymatroid feasible region
– Groenevelt
- 1991
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6
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Finding optimal minors of valuated bimatroids
– Murota
- 1995
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5
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Matroid optimization and algorithms
– Bixby, Cunningham
- 1995
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5
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An algorithm for finding an optimal `independent assignment
– Iri, Tomizawa
- 1976
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5
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Submodular functions and convexity," in Mathematical Programming --- The State of the
– Lovasz
- 1983
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5
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Solving integer minimum cost flows with separable convex objective polynomially
– Minoux
- 1986
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4
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Conjugate scaling technique for Fenchel-type duality in discrete convex optimization
– Iwata, Shigeno
- 1998
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3
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Relationship of M-/L-convex functions with discrete convex functions by Miller and by Favati--Tardella
– Murota, Shioura
- 1999
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2
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Geometric Algorithms and Combinatorial
– Grotschel, Lov'asz, et al.
- 1993
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2
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Discrete Convex Analysis (in Japanese
– Murota
- 1998
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1
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The submodular flow problem," in Discrete Structures and Algorithms
– Iwata
- 1999
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1
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Discrete convex analysis," in Discrete Structures and Algorithms
– Murota
- 1998
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1
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Minimization of some nonlinear functions over polymatroidal network flows," Annals of Discrete
– Zimmermann
- 1982
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1
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The submodular flow problem (in Japanese
– Iwata
- 1999
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