A characterization of bisubmodular functions

Ando, Kazutoshi; Fujishige, Satoru; Naitoh, Takeshi

doi:10.1016/0012-365x(94)00246-f

Cited by 24 publications

(21 citation statements)

References 6 publications

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“…We remark that the case of k = r = 1 corresponds to monotone submodular functions. In the case of k = r = 2, Ando, Fujishige, and Naito [2] have shown that these two properties give an exact characterization of the class of bisubmodular functions. In Section 3, we extend their result by showing that submodularity in every orthant and pairwise monotonicity in fact precisely characterize k-submodular functions for all k ≥ 2.…”

Section: Preliminariesmentioning

confidence: 99%

Maximizing k -Submodular Functions and Beyond

Ward

Živný

2016

ACM Trans. Algorithms

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We consider the maximization problem in the value oracle model of functions defined on k-tuples of sets that are submodular in every orthant and r-wise monotone, where k ≥ 2 and 1 ≤ r ≤ k. We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + r). For r = k, we give an analysis of a randomised greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + k/2).In the case of k = r = 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of 1/2. We show that, as in the case of submodular functions, this result is the best possible in both the value query model, and under the assumption that N P = RP .Extending a result of Ando et al., we show that for any k ≥ 3 submodularity in every orthant and pairwise monotonicity (i.e. r = 2) precisely characterize k-submodular functions. Consequently, we obtain an approximation guarantee of 1/3 (and thus independent of k) for the maximization problem of k-submodular functions.

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Section: Preliminariesmentioning

confidence: 99%

Maximizing k -Submodular Functions and Beyond

Ward

Živný

2016

ACM Trans. Algorithms

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“…We also have the following theorem (see [2,17] for special cases of bisubmodular and α-bisubmodular functions; also see [14,Proposition 4.11] for more general functions).…”

Section: Corollarymentioning

confidence: 99%

Generalized skew bisubmodularity: A characterization and a min–max theorem

Fujishige

Tanigawa

Yoshida

2014

Discrete Optimization

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Huber, Krokhin, and Powell (Proc. SODA2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain. In this paper we consider a natural generalization of the concept of skew bisubmodularity and show a connection between the generalized skew bisubmodularity and a convex extension over rectangles. We also analyze the dual polyhedra, called skew bisubmodular polyhedra, associated with generalized skew bisubmodular functions and derive a min-max theorem that characterizes the minimum value of a generalized skew bisubmodular function in terms of a minimum-norm point in the associated skew bisubmodular polyhedron.

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“…The "only if" direction is trivial; let us consider the "if" part. It is trivial for k = 1, and for the case k = 2 it was shown in [AFN96]. Suppose that k ≥ 3.…”

Section: Relation To Multimatroidsmentioning

confidence: 97%

Towards Minimizing k-Submodular Functions

Huber

Kolmogorov

2012

Lecture Notes in Computer Science

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In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k = 1 and k = 2 respectively.In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization problem over a (bi)submodular polyhedron. We define and investigate a k-submodular polyhedron and prove a Min-Max-Theorem for k-submodular functions.

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A characterization of bisubmodular functions

Cited by 24 publications

References 6 publications

Maximizing k -Submodular Functions and Beyond

Maximizing k -Submodular Functions and Beyond

Generalized skew bisubmodularity: A characterization and a min–max theorem

Towards Minimizing k-Submodular Functions

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