On structures of bisubmodular polyhedra

Ando, Kazutoshi; Fujishige, Satoru

doi:10.1007/bf02592201

Cited by 29 publications

(30 citation statements)

References 13 publications

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“…Then y is an extreme point of P α ( f ), which is called the extreme point generated by L and σ . Conversely, every extreme point of P α ( f ) can be generated by some L and σ through (1). Note that y is determined by a signed, α-scaled version of the greedy algorithm by (1).…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

Fujishige

Tanigawa

2017

Math. Program.

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Huber et al. (SIAM J Comput 43:1064-1084) 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, and Huber and Krokhin (SIAM J Discrete Math 28:1828-1837, 2014) showed the oracle tractability of minimization of skew-bisubmodular functions. Fujishige et al.(Discrete Optim 12:1-9, 2014) also showed a min-max theorem that characterizes the skew-bisubmodular function minimization, but devising a combinatorial polynomial algorithm for skew-bisubmodular function minimization was left open. In the present paper we give first combinatorial (weakly and strongly) polynomial algorithms for skew-bisubmodular function minimization.

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Section: Definitions and Preliminariesmentioning

confidence: 99%

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

Fujishige

Tanigawa

2017

Math. Program.

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“…Other examples of conformal systems are ring families as they occur in the context of bisubmodular function optimization (see, e.g., [21], or [2]): A path system P ⊆ {−1, 0, 1} |E| is called signed ring family if P is closed w.r.t. reduced union and intersection on {−1, 0, 1} |E| , defined as…”

Section: Examples Of Conformal Systemsmentioning

confidence: 99%

“…Consider, for example, the max flow problem (2) in G with source s ∈ V , sink t ∈ V and capacities u ∈ R |E| + . Note that we could add an additional dummy edge e * = (t, s) of unbounded capacity u * = ∞ without changing the max-flow problem essentially.…”

Section: Regular Max Flowmentioning

confidence: 99%

“…A natural approach to solve (2) is to start with the all-zero vector x = 0, and to iteratively increase one of the variables (among those that can be increased at all) as far as possible, i.e., until some constraint e ∈ E becomes tight w.r.t. either the lower or the upper capacity bound.…”

Section: Introductionmentioning

confidence: 99%

“…either the lower or the upper capacity bound. Ford and Fulkerson suggest the following iterative algorithmic solution procedure for (2):…”

Section: Introductionmentioning

confidence: 99%

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We investigate the following greedy approach to attack linear programs of type max{1 T x | l ≤ Ax ≤ u} where A has entries in {−1, 0, 1}: The greedy algorithm starts with a feasible solution x and, iteratively, chooses an improving variable and raises it until some constraint becomes tight. In the special case, where A is the edgepath incidence matrix of some digraph G = (V, E), and l = 0, this greedy algorithm corresponds to the Ford-Fulkerson algorithm to solve the max (s, t)-flow problem in G w.r.t. edge-capacities u. It is well-known that the Ford-Fulkerson algorithm always terminates with an optimal flow, and that the number of augmentations strongly depends on the choice of paths in each iteration. The Edmonds-Karp rule that prefers paths with fewer arcs leads to a running time of at most |E| 2 augmentations. The paper investigates general types of matrices A and preference rules on the variables that make the greedy algorithm efficient. In this paper, we identify conditions that guarantee for the greedy algorithm not to cycle, and/or optimality of the greedy algorithm, and/or to yield a quadratic (in the number of rows) number of augmentations. We illustrate our approach with flow and circulation problems on regular oriented matroids.

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Making Bidirected Graphs Strongly Connected

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Sato

2019

Algorithmica

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We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time algorithm for finding an optimal solution. For the latter problem, we give a linear-time algorithm for finding a feasible solution whose size is equal to the obvious lower bound or more than that by one.

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On structures of bisubmodular polyhedra

Cited by 29 publications

References 13 publications

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

Greedy Oriented Flows

Making Bidirected Graphs Strongly Connected

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