Discrete Convex Functions on Graphs and Their Algorithmic Applications

Hirai, Hiroshi

doi:10.1007/978-981-10-6147-9_4

Cited by 16 publications

(9 citation statements)

References 51 publications

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“…In analogy with isometric subgraphs of hypercubes, which are usually called partial cubes, we call partial Johnson graphs and partial halved-cubes the isometric subgraphs of Johnson graphs and halved-cubes, respectively. In the theory of discrete convexity [54,64], submodular functions on Boolean cubes, L # -convex and N -convex functions on Z d , products of trees, and median graphs are investigated. Lozenge functions, introduced in [9] and used in most of our proofs, can be viewed as a generalization of such discrete convex functions to general graphs.…”

Section: Discussionmentioning

confidence: 99%

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Graphs with $G^p$-connected medians

Bénéteau¹,

Chalopin²,

Chepoi³

et al. 2022

Preprint

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The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer p ≥ 2, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power G p of G. This extends some characterizations of graphs with connected medians (case p = 1) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have G 2 -connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.

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Section: Discussionmentioning

confidence: 99%

“…Lozenge functions on graphs were introduced and investigated in the context of median functions in [9]. They represent a generalization of L # -convex functions [49] and N -convex functions [55] on particular classes of graphs, both investigated in the theory of discrete convexity [54,64].…”

Section: 2mentioning

confidence: 99%

Graphs with $G^p$-connected medians

Bénéteau¹,

Chalopin²,

Chepoi³

et al. 2022

Preprint

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show abstract

“…It is also interesting to investigate the possibility of extending the present framework to some of more general discrete convexity (cf. [8,11]). Finally it should be noted that we have treated only bounded convex polyhedra.…”

Section: Discussionmentioning

confidence: 99%

Greedy systems of linear inequalities and lexicographically optimal solutions

Fujishige

2019

RAIRO-Oper. Res.

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The present note reveals the role of the concept of greedy system of linear inequalities played in connection with lexicographically optimal solutions on convex polyhedra and discrete convexity. The lexicographically optimal solutions on convex polyhedra represented by a greedy system of linear inequalities can be obtained by a greedy procedure, a special form of which is the greedy algorithm of J. Edmonds for polymatroids. We also examine when the lexicographically optimal solutions become integral. By means of the Fourier-Motzkin elimination Murota and Tamura have recently shown the existence of integral points in a polyhedron arising as a subdifferential of an integer-valued, integrally convex function due to Favati and Tardella [Murota and Tamura, Integrality of subgradients and biconjugates of integrally convex functions. Preprint arXiv:1806.00992v1 (2018)], which can be explained by our present result. A characterization of integrally convex functions is also given.Mathematics Subject Classification. 90C05, 90C27, 90C46.

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“…The motivation of this paper comes from submodular functions on modular semilattices [15,17]. This class of functions generalizes submodular set functions as well as other submodular-type functions, such as k-submodular functions, and appears from dualities in well-behaved multicommodity flow problems and related label assignment problems; see also [18].…”

Section: Minimizer Set Of Submodular Functionmentioning

confidence: 99%

A Compact Representation for Modular Semilattices and its Applications

Hirai

Nakashima

2017

Preprint

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A modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhoff-type representation theorem for modular semilattices, which says that every modular semilattice is isomorphic to the family of ideals in a certain poset with additional relations. This new poset structure, which we axiomatize in this paper, is called a PPIP (projective poset with inconsistent pairs). A PPIP is a common generalization of a PIP (poset with inconsistent pairs) and a projective ordered space. The former was introduced by Barthélemy and Constantin for establishing Birkhoff-type theorem for median semilattices, and the latter by Herrmann, Pickering, and Roddy for modular lattices. We show the Θ(n) representation complexity and a construction algorithm for PPIP-representations of (∧, ∨)-closed sets in the product L n of modular semilattice L. This generalizes the results of Hirai and Oki for a special median semilattice S k . We also investigate implicational bases for modular semilattices. Extending earlier results of Wild and Herrmann for modular lattices, we determine optimal implicational bases and develop a polynomial time recognition algorithm for modular semilattices. These results can be applied to retain the minimizer set of a submodular function on a modular semilattice.

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Discrete Convex Functions on Graphs and Their Algorithmic Applications

Cited by 16 publications

References 51 publications

Graphs with $G^p$-connected medians

Graphs with $G^p$-connected medians

Greedy systems of linear inequalities and lexicographically optimal solutions

A Compact Representation for Modular Semilattices and its Applications

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