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64
Algorithms in Discrete Convex Analysis
 Math. Programming
, 2000
"... this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects. ..."
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Cited by 158 (36 self)
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this paper is to describe the f#eA damental results on M and Lconvex f#24L2A+ with special emphasis on algorithmic aspects.
Bisubmodular Function Minimization
 Mathematical Programming
, 2000
"... This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scali ..."
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Cited by 47 (4 self)
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This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scaling algorithm naturally leads to the rst combinatorial polynomialtime algorithm for testing membership in deltamatroid polyhedra. Unlike the case of matroid polyhedra, it remains open to develop a combinatorial strongly polynomial algorithm for this problem. Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 5608531, Japan (fujishig@sys.es.osakau.ac.jp). Research partly carried out while at Forschungsinstut fur Diskrete Mathematik, Universitat Bonn. y Department of Mathematical Engineering and Information Physics, University of Tokyo, Tokyo 1138656, Japan (iwata@sr3.t.utokyo.ac.jp). 1 1 Introduction Let V be a nite none...
Mconvex function on generalized polymatroid
, 1997
"... The concept of Mconvex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of DressWenzel (1990). In this paper, we extend this concept to functions on generalized polymat ..."
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Cited by 44 (23 self)
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The concept of Mconvex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of DressWenzel (1990). In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems. The restriction of a function to fx 2 ZV j x(V) = kg for k 2 Z is called a layer. We prove the Mconvexity of each layer, and reveal that the minimizers in consecutive layers are closely related. Exploiting these properties, we can solve the optimization on layers efficiently. A number of equivalent exchange axioms are given for Mconvex function on generalized polymatroid.
Notes on L/Mconvex Functions and the Separation Theorems
 Math. Programming
, 1999
"... . The concepts of Lconvex function and Mconvex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for Lconvex/concave functions and for Mconvex/concave functions as general ..."
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Cited by 25 (15 self)
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. The concepts of Lconvex function and Mconvex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for Lconvex/concave functions and for Mconvex/concave functions as generalizations of Frank's discrete separation theorem for submodular/supermodular set functions and Edmonds' matroid intersection theorem. This paper shows the equivalence between Murota's Lconvex functions and Favati and Tardella's submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis. Key words. Lconvex functions  Mconvex functions  integrally convex functions  submodularity  separation theorems 1. Introduction Convexity for discrete functions has been a continual research topic. Among others, Miller [15] was a forerunner in the early...
Extension of Mconvexity and Lconvexity to Polyhedral Convex Functions (Extended Abstract)
, 1999
"... The concepts of $\mathrm{M}$convex and $\mathrm{L}$convex functions were proposed by Murota in 1996 as two mutually conjugate classes of discrete functions over integer lattice points. M/Lconvex functions are deeply connected with the wellsolvability in nonlinear combinatorial optimization with ..."
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Cited by 18 (17 self)
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The concepts of $\mathrm{M}$convex and $\mathrm{L}$convex functions were proposed by Murota in 1996 as two mutually conjugate classes of discrete functions over integer lattice points. M/Lconvex functions are deeply connected with the wellsolvability in nonlinear combinatorial optimization with integer variables. In this paper, we extend the concept of Mconvexity and L–convexity to polyhedral convex functions, aiming at clarifying the wellbehaved structure in wellsolved nonlinear combinatorial optimization problems in real variables. The extended $\mathrm{M}/\mathrm{L}$convexity often appear in nonlinear combinatorial optimization problems with piecewiselinear convex cost. We investigate the structure of polyhedral $\mathrm{M}$convex and Lconvex functions from the dual viewpoint of analysis and combinatorics, and provide some properties and characterizations. It is also shown that polyhedral $\mathrm{M}/\mathrm{L}$convex functions have nice conjugacy relationship.
On Steepest Descent Algorithms for Discrete Convex Functions
 SIAM Journal on Optimization
, 2002
"... This paper investigates the complexity of steepest descent algorithms for two classes of discrete convex functions, Mconvex functions and Lconvex functions. ..."
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Cited by 18 (7 self)
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This paper investigates the complexity of steepest descent algorithms for two classes of discrete convex functions, Mconvex functions and Lconvex functions.
MConvex Functions on Jump Systems: A General Framework for Minsquare Graph Factor Problem
, 2004
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Minimization of an Mconvex Function
 Discrete Appl. Math
, 1998
"... We study the minimization of an Mconvex function introduced by Murota. It is shown that any vector in the domain can be easily separated from a minimizer of the function. Based on this property, we develop a polynomial time algorithm. Keywords: matroid, base polyhedron, convex function, minimizatio ..."
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Cited by 12 (7 self)
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We study the minimization of an Mconvex function introduced by Murota. It is shown that any vector in the domain can be easily separated from a minimizer of the function. Based on this property, we develop a polynomial time algorithm. Keywords: matroid, base polyhedron, convex function, minimization. 1 Introduction Mconvex function, recently introduced by Murota [8, 9, 10], is an extension of valuated matroid due to Dress and Wenzel [1, 2] as well as a quantitative generalization of (the integral points of) the base polyhedron of an integral submodular system [4]. Mconvexity is quite a natural concept appearing in many situations; linear and separableconvex functions are Mconvex, and more general Mconvex functions arise from the minimum cost flow problems with separableconvex cost functions. Mconvex function enjoys several nice properties which persuade us to regard it as "convexity" in combinatorial optimization. Let V be a finite set with cardinality n: A function f : Z V ...
A twosided discreteconcave market with bounded side payments: An approach by discrete convex analysis
, 2004
"... The marriage model due to Gale and Shapley and the assignment model due to Shapley and Shubik are standard in the theory of twosided matching markets. We give a common generalization of these models by utilizing discrete concave functions and considering bounded side payments. We show the existence ..."
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Cited by 12 (3 self)
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The marriage model due to Gale and Shapley and the assignment model due to Shapley and Shubik are standard in the theory of twosided matching markets. We give a common generalization of these models by utilizing discrete concave functions and considering bounded side payments. We show the existence of a pairwise stable outcome in our model. Our present model is a further natural extension of the model examined in our previous paper (Fujishige and Tamura [10]), and the proof of the existence of a pairwise stable outcome is even simpler than the previous one.
Discrete Convexity and Equilibria in Economies with Indivisible Goods and Money
 MATH. SOCIAL SCI
, 2000
"... We consider a variant of the standard ArrowDebreu model which contains a number of indivisible goods and one perfectly divisible good (numeraire or money). The objective of this paper is to clarify a general mathematical mechanism which guarantees the existence of an equilibrium in such a model. I ..."
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Cited by 11 (2 self)
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We consider a variant of the standard ArrowDebreu model which contains a number of indivisible goods and one perfectly divisible good (numeraire or money). The objective of this paper is to clarify a general mathematical mechanism which guarantees the existence of an equilibrium in such a model. It is shown that the crucial condition for the existence of an equilibrium is that the sets of supply and demand of indivisible goods should belong to a class of "discrete convexity." The class of generalized polymatroids provides one of the most interesting classes of discrete convexity.