An Algorithm to Enumerate All Cutsets of a Graph in Linear Time per Cutset

Tsukiyama, Shuji; Shirakawa, Isao; Ozaki, Hiroshi; Ariyoshi, Hiromu

doi:10.1145/322217.322220

Cited by 83 publications

(37 citation statements)

References 7 publications

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“…In this section we present a technique which is a variant of the so called supergraph approach that has been used in the literature for instance, to generate all minimal feedback vertex and arc sets [13], minimal s-t cuts [16], minimal spanning trees [14], and minimal blockers of perfect matchings in bipartite graphs [3]. To explain the method briefly, a supergraph is a strongly connected directed graph G whose vertices are the objects that we would like to generate.…”

Section: Generating Minimal Transversals Of a Hypergraphmentioning

confidence: 99%

“…Given a graph G = (V , E) and two vertices s, t ∈ V , the two-terminal cut generation problem calls for listing all minimal subsets of edges whose removal disconnects s and t. This problem is known to be solvable in O(Nm + m + n) time and O(n + m) space [16], where n and m are the numbers of vertices and edges in the input graph, and N is the total number of cuts. In this paper, we study the following natural extension of this problem: Note that for i = j , s i and s j , or s i and t j , or t i and t j may coincide.…”

Section: Introductionmentioning

confidence: 99%

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Generating Cut Conjunctions in Graphs and Related Problems

et al. 2007

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Let G = (V , E) be an undirected graph, and let B ⊆ V × V be a collection of vertex pairs. We give an incremental polynomial time algorithm to generate all minimal edge sets X ⊆ E such that every pair (s, t) ∈ B of vertices is disconnected in (V , E X), generalizing well-known efficient algorithms for generating all minimal s-t cuts, for a given pair s, t of vertices. We also present an incremental polynomial time algorithm for generating all minimal subsets X ⊆ E such that no (s, t) ∈ B is This research was partially supported by the National Science Foundation (Grant IIS-0118635), by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science, and by the Scientific Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan. Our friend and colleague, Leonid Khachiyan passed away with tragic suddenness while we were preparing this manuscript. (V , X ∪ B). Both above problems are special cases of a more general problem that we call generating cut conjunctions for matroids: given a matroid M on ground set S = E ∪ B, generate all minimal subsets X ⊆ E such that no element b ∈ B is spanned by E X. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V , E ∪ B), the more general problem of generating cut conjunctions for vectorial matroids turns out to be NP-hard.

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Section: Generating Minimal Transversals Of a Hypergraphmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generating Cut Conjunctions in Graphs and Related Problems

et al. 2007

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“…In contrast, we seek to provide a more direct means to determine where perturbation of the probability of transitions between states leads to large system performance degradations. To do this, we leverage previous work on minimal s-t cut set identification [32][33][34][35] described below. Minimal cut set identification methods have long been used for analysis of VLSI designs, network systems, and design of various other distributed systems.…”

Section: Previous Workmentioning

confidence: 99%

“…By reducing the transition probability values of these common transitions to 0, the flow of tasks to the Tasks Completed state is also reduced to 0. In discrete mathematics, a set of one or more edges, which if removed, disconnects all paths between two vertices s and t is often referred to as an s-t cut set, as for example [32]. An s-t cut set is defined to be a minimal s-t cut set if removal of any edge from the cut set causes s and t to be reconnected.…”

Section: Finding State Transition That Disconnect Paths To Absorbing mentioning

confidence: 99%

Improving efficiency of markov chain analysis of complex distributed systems

Dabrowski¹,

Hunt²,

Morrison³

2010

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Abstract:In large-scale distributed systems, the interactions of many independent components may lead to emergent global behaviors with unforeseen, often detrimental, outcomes. The increasing importance of distributed systems such as clouds and computing grids will require analytical tools to understand and predict, complex system behavior to ensure system reliability. In previous work, we described how a piecewise homogeneous Discrete Time Markov chain representation of a computing grid can be systematically perturbed to predict situations that lead to performance degradations. While the execution time of this approach compared favorably with detailed large-scale simulation, a sizable number of perturbations of the model were needed to identify scenarios in which system performance degraded. Here, we evolve our original approach and describe two novel methods for quickly identifying portions of the Markov chain that are sensitive to perturbation. The first method involves finding minimal s-t cut sets, consisting of state transitions that disconnect all paths in a Markov chain from the initial to a desired end state. By perturbing state transitions in the cut set, it is possible to quickly identify scenarios in which system performance is adversely affected. We show this method can be applied to larger Markov models than the approach described in our earlier work. We then present a second method, in which the Spectral Expansion Theorem is used to analyze the eigensystem of a set of Markov transition probability matrices to predict which state transitions, if perturbed, are likely to adversely affect system performance. Results are presented for both methods to show that they can be used to identify the same failure scenarios presented in our earlier paper (as well as additional scenarios, using the first method), while reducing the number of perturbations needed. We argue that these methods provide a basis for creating practical tools for analysis of complex systems behavior in distributed systems.

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“…This problem, for example, has an application in network reliability, see Colbourn [4]. Abel and Bicker [1], Bellmore and Jensen [3] and Tsukiyama et al [6] In this paper, we propose a generalized framework and discuss its application.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Framework for Listing Cuts and Graphs

Yaguchi

2014

JORSJ

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An Algorithm to Enumerate All Cutsets of a Graph in Linear Time per Cutset

Cited by 83 publications

References 7 publications

Generating Cut Conjunctions in Graphs and Related Problems

Generating Cut Conjunctions in Graphs and Related Problems

Improving efficiency of markov chain analysis of complex distributed systems

A Generalized Framework for Listing Cuts and Graphs

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