The principal lattice of partitions of a submodular function

Narayanan, Hariharan

doi:10.1016/0024-3795(91)90070-d

Cited by 44 publications

(62 citation statements)

References 10 publications

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“…The result reveals the relation among the minimizers of subpartitions eorresponding to different number K. Because the procedure of deriving the following theorem is parallel to what Narayanan did in [6], we omit the proof.…”

Section: (Ii) + ?(Fin) _> ?(Ii V Fin) + ](Ii a Iinmentioning

confidence: 89%

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Structures of subpartitions related to a submodular function minimization

Naitoh

Nakayama

1997

Japan J. Indust. Appl. Math.

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Let f be a submodular function on 2 E for a nonempty finite set E and~ be a parameter that takes on values between -~ and ~. In this paper, we show that the set of the subpartitions T/of E on which ~x~17(f-)~)(X) attains the minimum has a structure similar to that of the intersecting family. Moreover, we apply this result to the minimum augmentation problem with respect to the edge-connectivity. We reveal that when a given graph G ----(V, A) is not K-edge-connected, the set of the subpartitions /7 of the vertex set except {V} on which ~XEH(d --K)(X) attains the minimum has a structure like the cointersecting family, where d(X) denotes the number of edges between X and V -X.

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Section: (Ii) + ?(Fin) _> ?(Ii V Fin) + ](Ii a Iinmentioning

confidence: 89%

“…For the purpose of solving problems in electrical network theory, H. Narayanan [6] studied the structure of the set of the partitions…”

Section: Introductionmentioning

confidence: 99%

Structures of subpartitions related to a submodular function minimization

Naitoh

Nakayama

1997

Japan J. Indust. Appl. Math.

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“…This running time was improved to nT F by Narayanan [2], Barahona [3] and d'Auriac et al [4], who gave three different algorithms for the attack problem.…”

Section: Introductionmentioning

confidence: 96%

“…The current best known method due to Narayanan [2] has complexity n 2 T F , or O(n) calls to the network attack problem.…”

Section: Introductionmentioning

confidence: 99%

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A Faster Algorithm for Computing the Principal Sequence of Partitions of a Graph

Kolmogorov

2008

Algorithmica

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We consider the following problem: given an undirected weighted graph G = (V , E, c) with nonnegative weights, minimize function c(δ( )) − λ| | for all values of parameter λ. Here is a partition of the set of nodes, the first term is the cost of edges whose endpoints belong to different components of the partition, and | | is the number of components. The current best known algorithm for this problem has complexity O(|V | 2 ) maximum flow computations. We improve it to |V | parametric maximum flow computations. We observe that the complexity can be improved further for families of graphs which admit a good separator, e.g. for planar graphs.

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