A note on the Frank—Tardos bi-truncation algorithm for crossing-submodular functions

Naitoh, Takeshi; Fujishige, Satoru

doi:10.1007/bf01585712

Cited by 7 publications

(3 citation statements)

References 2 publications

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“…Note that the base polyhedron B(f ) may possibly be empty. The bi-truncation algorithm of Frank-Tardos [9] efficiently finds a base if exists, and proves emptiness otherwise (see also Naitoh-Fujishige [29]). …”

Section: Submodular Flowmentioning

confidence: 95%

Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*

2003

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This paper presents two fast cycle canceling algorithms for the submodular flow problem. The first uses an assignment problem whose optimal solution identifies most negative node-disjoint cycles in an auxiliary network. Canceling these cycles lexicographically makes it possible to obtain an optimal submodular flow in O(n 4 h log(nC)) time, which almost matches the current fastest weakly polynomial time for submodular flow (where n is the number of nodes, h is the time for computing an exchange capacity, and C is the maximum absolute value of arc costs). The second algorithm generalizes Goldberg's cycle canceling algorithm for min cost flow to submodular flow to also get a running time of O(n 4 h log(nC)). We show how to modify these algorithms to make them strongly polynomial, with running times of O(n 6 h log n), which matches the fastest strongly polynomial time bound for submodular flow. We also show how to extend both algorithms to solve submodular flow with separable convex objectives.

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Section: Submodular Flowmentioning

confidence: 95%

Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*

2003

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“…It is known that p type can be computed in polynomial time (Frank and Tardos 1988;Naitoh and Fujishige 1992). However, as the algorithm is cumbersome to describe and implement, we further simplify the problem using the properties of supermodular cooperative games.…”

Section: Characterizations Of Least Coresmentioning

confidence: 99%

Computing Least Cores of Supermodular Cooperative Games

Hatano

Yoshida

2017

AAAI

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One of the goals of a cooperative game is to compute a valuedivision to the players from which they have no incentive todeviate. This concept is formalized as the notion of the core.To obtain a value division that motivates players to cooperate to a greater extent or that is more robust under noise, the notions of the strong least core and the weak least core have been considered. In this paper, we characterize the strong and the weak least cores of supermodular cooperative games using the theory of minimizing crossing submodular functions. We then apply our characterizations to two representative supermodular cooperative games, namely, the induced subgraph game generalized to hypergraphs and the airport game. For these games, we derive explicit forms of the strong and weak least core values, and provide polynomial-time algorithms that compute value divisions in the strong and weak least cores.

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“…With the bi-truncation algorithm [12,16] Remark. With the bi-truncation algorithm one can get a doublepartition that minimizes (3).…”

Section: Algorithmic Aspectmentioning

confidence: 99%

Tree-compositions and orientations

Frank¹,

Király²

2013

Operations Research Letters

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a b s t r a c tA tree-composition is a tree-like family that serves to describe the obstacles to k-edge-connected orientability of mixed graphs. Here we derive a structural result on tree-compositions that gives rise to a simple algorithm for computing an obstacle when the orientation does not exist.As another application, we show a min-max theorem on the minimal in-degree of a given node set in a k-edge-connected orientation of an undirected graph. This min-max formula can be simplified in the special case of k = 1.

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A note on the Frank—Tardos bi-truncation algorithm for crossing-submodular functions

Cited by 7 publications

References 2 publications

Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*

Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*

Computing Least Cores of Supermodular Cooperative Games

Tree-compositions and orientations

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