<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\delta$" id="E1"><mml:mi>Δ</mml:mi></mml:math>-matroid and jump system

Kabadi, Santosh N.; Sridhar, R.

doi:10.1155/jamds.2005.95

Cited by 15 publications

(10 citation statements)

References 9 publications

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“…Aggregation is another fundamental operation. For instance, it is known that any polymatroid can be obtained as an aggregation of a matroid [14] and that any jump system can be obtained as an aggregation of a delta-matroid [16]. The second result of the present paper (Theorem 11) is that M J -convex functions are closed under aggregation.…”

Section: Introductionmentioning

confidence: 66%

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Operations on M‐Convex Functions on Jump Systems

Kobayashi¹,

Murota²,

Tanaka³

2007

SIAM J. Discrete Math.

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A jump system is a set of integer points with an exchange property, which is a generalization of a matroid, a delta-matroid, and a base polyhedron of an integral polymatroid (or a submodular system). Recently, the concept of M-convex functions on constant-parity jump systems is introduced by Murota as a class of discrete convex functions that admit a local criterion for global minimality. M-convex functions on constant-parity jump systems generalize valuated matroids, valuated delta-matroids, and M-convex functions on base polyhedra. This paper reveals that the class of M-convex functions on constant-parity jump systems is closed under a number of natural operations such as splitting, aggregation, convolution, composition, and transformation by networks. The present results generalize hitherto-known similar constructions for matroids, delta-matroids, valuated matroids, valuated delta-matroids, and M-convex functions on base polyhedra.

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Section: Introductionmentioning

confidence: 66%

“…A jump system [6] is a set of integer points with an exchange property (to be described later); see also [16], [17]. It is a generalization of a matroid [8], a delta-matroid [4], [7], [9], and a base polyhedron of an integral polymatroid (or a submodular system) [14].…”

Section: Introductionmentioning

confidence: 99%

Operations on M‐Convex Functions on Jump Systems

Kobayashi¹,

Murota²,

Tanaka³

2007

SIAM J. Discrete Math.

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“…Next we consider a generalized concept of M B -convex sets called jump systems [3] (see also [9,12]). A nonempty set J ⊆ Z V is said to be a jump system if it satisfies an exchange axiom, called the 2-step axiom: for any x, y ∈ J and for any (x, y)-increment…”

Section: Base Polyhedra and Jump Systemsmentioning

confidence: 99%

Induction of M-convex functions by linking systems

Kobayashi

Murota

2007

Discrete Applied Mathematics

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Induction (or transformation) by bipartite graphs is one of the most important operations on matroids, and it is well known that the induction of a matroid by a bipartite graph is again a matroid. As an abstract form of this fact, the induction of a matroid by a linking system is known to be a matroid.M-convex functions are quantitative extensions of matroidal structures, and they are known as discrete convex functions. As with matroids, it is known that the induction of an M-convex function by networks generates an M-convex function. As an abstract form of this fact, this paper shows that the induction of an M-convex function by linking systems generates an M-convex function. Furthermore, we show that this result also holds for M-convex functions on constant-parity jump systems. Previously known operations such as aggregation, splitting, and induction by networks can be understood as special cases of this construction.

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“…A jump system [4] is a set of integer points with an exchange property (to be described in Section 2); see also [14], [16]. It is a generalization of a matroid [6], [15], a delta-matroid [3], [5], [7], and a base polyhedron of an integral polymatroid (or a submodular system) [11].…”

Section: Introductionmentioning

confidence: 99%

M-Convex Functions on Jump Systems: A General Framework for Minsquare Graph Factor Problem

Murota¹

2006

SIAM J. Discrete Math.

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The concept of M-convex functions is generalized for functions defined on constant-parity jump systems. Such function arises from minimum weight perfect b-matchings and from a separable convex function (sum of univariate convex functions) on the degree sequences of an undirected graph. As a generalization of a recent result of Apollonio and Sebő for the minsquare factor problem, a local optimality criterion is given for minimization of an M-convex function subject to a component sum constraint.

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Δ-matroid and jump system

Cited by 15 publications

References 9 publications

Operations on M‐Convex Functions on Jump Systems

Operations on M‐Convex Functions on Jump Systems

Induction of M-convex functions by linking systems

M-Convex Functions on Jump Systems: A General Framework for Minsquare Graph Factor Problem

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