On enumerating all minimal solutions of feedback problems

Schwikowski, Benno; Speckenmeyer, Ewald

doi:10.1016/s0166-218x(00)00339-5

Cited by 68 publications

(55 citation statements)

References 18 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%

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%

Generating Cut Conjunctions in Graphs and Related Problems

et al. 2007

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“…It is also known [19] that all minimal feedback arc sets for a directed graph G can be listed with polynomial delay. Theorem 2 states that we can also list, in incremental polynomial time, all minimal strongly connected subgraphs of G, while Theorem 1 states that such a result cannot hold for the family of minimal dicuts unless P = NP .…”

Section: Some Related Geometric Problems and Concluding Remarksmentioning

confidence: 99%

On Enumerating Minimal Dicuts and Strongly Connected Subgraphs

Khachiyan¹,

Boros

Elbassioni

et al. 2007

Algorithmica

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We consider the problems of enumerating all minimal strongly connected subgraphs and all minimal dicuts of a given strongly connected directed graph G = (V , E). We show that the first of these problems can be solved in incremental polynomial time, while the second problem is NP-hard: given a collection of minimal dicuts for G, it is NP-hard to tell whether it can be extended. The latter result implies, in particular, that for a given set of points A ⊆ R n , it is NP-hard to generate all maximal subsets of A contained in a closed half-space through the origin. We also discuss the enumeration of all minimal subsets of A whose convex hull contains the origin as an interior point, and show that this problem includes as a special case the well-known hypergraph transversal problem.

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“…It is also known [20] that all minimal feedback arc sets for a directed graph G can be listed with polynomial delay. Theorem 2 states that we can also list, in incremental polynomial time, all minimal strongly connected subgraphs of G, while Theorem 1 states that such a result is cannot hold for the family of minimal dicuts unless P=NP.…”

Section: Some Related Geometric Problemsmentioning

confidence: 99%

“…Furthermore, if π(X) is the property that the subgraph (V, X) of a directed graph G = (V, E) contains a directed cycle, then F π is the family of minimal directed circuits of G, while F d π consists of all minimal feedback arc sets of G. Both of these families can be generated with polynomial delay per output element, see e.g. [20].…”

Section: Introductionmentioning

confidence: 99%

Enumerating Minimal Dicuts and Strongly Connected Subgraphs and Related Geometric Problems

Boros

Elbassioni

Gurvich

et al. 2004

Integer Programming and Combinatorial Optimization

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Abstract.We consider the problems of enumerating all minimal strongly connected subgraphs and all minimal dicuts of a given directed graph G = (V, E). We show that the first of these problems can be solved in incremental polynomial time, while the second problem is NP-hard: given a collection of minimal dicuts for G, it is NP-complete to tell whether it can be extended. The latter result implies, in particular, that for a given set of points A ⊆ R n , it is NP-hard to generate all maximal subsets of A contained in a closed half-space through the origin. We also discuss the enumeration of all minimal subsets of A whose convex hull contains the origin as an interior point, and show that this problem includes as a special case the well-known hypergraph transversal problem.

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On enumerating all minimal solutions of feedback problems

Cited by 68 publications

References 18 publications

Generating Cut Conjunctions in Graphs and Related Problems

Generating Cut Conjunctions in Graphs and Related Problems

On Enumerating Minimal Dicuts and Strongly Connected Subgraphs

Enumerating Minimal Dicuts and Strongly Connected Subgraphs and Related Geometric Problems

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