Bisubmodular polyhedra, simplicial divisions, and discrete convexity

Fujishige, Satoru

doi:10.1016/j.disopt.2014.02.002

Cited by 27 publications

(51 citation statements)

References 14 publications

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“…since (x (1) + x (2) )/2 = l k=1 λ k u (k) by (3.5) and u (k) ∈ N((x (1) + x (2) )/2). It follows from (3.4), (3.6), and (3.7) that 1 2 [g(x (1) ) + g(x (2) )] ≥g…”

Section: Projection Of Integrally Convex Functionsmentioning

confidence: 99%

“…By the definition of projection, we have (x (1) , y (1) ) ∈ S and (x (2) , y (2) ) ∈ S for some y (1) , y (2) ∈ Z m . Since (x (1) , y (1) ) − (x (2) , y (2) ) ∞ ≥ 2, discrete midpoint convexity (2.5) for S shows (x (1) , y (1)…”

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

“…∞ ≥ 2 and any ε > 0. By the definition of projection, there exist y (1) , y (2) (1) , y (1) ) + f (x (2) , y (2) ) − 2ε.…”

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

“…Consider the decomposition of (x (2) , z (2) ) − (x (1) , z (1) ) into vectors of {−1, 0, +1} n+1 as in (2.7): (x (2) , z (2) ) − (x (1) , z (1) (3.14) where m = (x (1) , z (1) ) − (x (2) , z (2) ) ∞ = max(2, z (2) − z (1) ). It should be clear that the components of the vector (x (i) , z (i) ) are numbered by 1, 2, .…”

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

“…This allows us to use discrete midpoint convexity (2.3) to obtain RHS of (3.18) ≥ f (x (1) , z (1) ) + (x (2) , z (2) ) 2 + f (x (1) , z (1) ) + (x (2) , z (2) ) 2 .…”

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

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Projection and convolution operations for integrally convex functions

Moriguchi

Murota

2019

Discrete Applied Mathematics

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This paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable under projection, and this is also the case with the subclasses of integrally convex functions satisfying local or global discrete midpoint convexity. As is known in the literature, the convolution of two integrally convex functions may possibly fail to be integrally convex. We show that the convolution of an integrally convex function with a separable convex function remains integrally convex. We also point out in terms of examples that the similar statement is false for integrally convex functions with local or global discrete midpoint convexity.

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“…since (x (1) + x (2) )/2 = l k=1 λ k u (k) by (3.5) and u (k) ∈ N((x (1) + x (2) )/2). It follows from (3.4), (3.6), and (3.7) that 1 2 [g(x (1) ) + g(x (2) )] ≥g…”

Section: Projection Of Integrally Convex Functionsmentioning

confidence: 99%

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

“…∞ ≥ 2 and any ε > 0. By the definition of projection, there exist y (1) , y (2) (1) , y (1) ) + f (x (2) , y (2) ) − 2ε.…”

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

“…This allows us to use discrete midpoint convexity (2.3) to obtain RHS of (3.18) ≥ f (x (1) , z (1) ) + (x (2) , z (2) ) 2 + f (x (1) , z (1) ) + (x (2) , z (2) ) 2 .…”

Section: Projection Of Discrete Midpoint Convex Functionsmentioning

confidence: 99%

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Projection and convolution operations for integrally convex functions

Moriguchi

Murota

2019

Discrete Applied Mathematics

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Discrete Convex Functions on Graphs and Their Algorithmic Applications

Hirai

2017

Combinatorial Optimization and Graph Algorithms

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The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems.

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