On generating all maximal independent sets

Johnson, David S.; Yannakakis, Mihalis; Papadimitriou, Christos H.

doi:10.1016/0020-0190(88)90065-8

Cited by 624 publications

(462 citation statements)

References 6 publications

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“…Because S can be found and G ′ can be constructed in polynomial time, the minimal connected vertex covers of G can be enumerated in time O * (3 n/3 ) using the algorithms of e.g., [26,21] as mentioned in the preliminaries.…”

Section: Theoremmentioning

confidence: 99%

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Golovach

Heggernes

Kratsch

2016

Lecture Notes in Computer Science

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Abstract. Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson [16]. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the maximum number of minimal connected vertex covers of a graph is at most 1.8668 n , and these can be enumerated in time O(1.8668 n ). For graphs of chordality at most 5, we are able to give a better upper bound, and for chordal graphs and distance-hereditary graphs we are able to give tight bounds on the maximum number of minimal connected vertex covers.

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Section: Theoremmentioning

confidence: 99%

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Golovach

Heggernes

Kratsch

2016

Lecture Notes in Computer Science

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“…On the other hand, it can be much harder to get an enumeration algorithm with polynomial delay than an output polynomial algorithm. For an overview of various notions of enumeration complexity (see Johnson, Yannakakis, & Papadimitriou, 1988).…”

Section: Introductionmentioning

confidence: 99%

Generating Models of a Matched Formula With a Polynomial Delay

Savický

Kučera

2016

jair

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A matched formula is a CNF formula whose incidence graph admits a matching which matches a distinct variable to every clause. Such a formula is always satisfiable. Matched formulas are used, for example, in the area of parametrized complexity. We prove that the problem of counting the number of the models (satisfying assignments) of a matched formula is #P-complete. On the other hand, we define a class of formulas generalizing the matched formulas and prove that for a formula in this class one can choose in polynomial time a variable suitable for splitting the tree for the search of the models of the formula. As a consequence, the models of a formula from this class, in particular of any matched formula, can be generated sequentially with a delay polynomial in the size of the input. On the other hand, we prove that this task cannot be performed efficiently for linearly satisfiable formulas, which is a generalization of matched formulas containing the class considered above.

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“…Regarding existing dominating clique, trivial O * (2 |V | ) bound has been initially broken by [6] down to O * (3 |V |/3 ) = O * (1.443 |V | ) using a result by [8], namely that the number of maximal (for inclusion) independent sets in a graph is at most 3 |V |/3 . Recently, [7] have proposed a branching algorithm that, according to a measure and conquer analysis [2], solves min dominating clique with polynomial space and running time O * (1.3387 |V | ), and another one that requires O * (1.3234 |V | ) time and space.…”

Section: Introductionmentioning

confidence: 99%

“…Consider first the case f (v) 0.10686. From [8,6], it is possible to compute any maximal independent set, thus any maximal clique, with running time:…”

Section: Introductionmentioning

confidence: 99%

Exact Algorithms for Dominating Clique Problems

Bourgeois

Croce

Escoffier

et al. 2009

Algorithms and Computation

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We handle in this paper three dominating clique problems, namely, the decision problem itself when one asks if there exists a dominating clique in a graph G and two optimization versions where one asks for a maximum-and a minimum-size dominating clique, if any. For the three problems we propose optimal algorithms with provably worst-case upper bounds improving existing ones by (D. Kratsch and M. Liedloff, An exact algorithm for the minimum dominating clique problem, Theoretical Computer Science 385(1-3), pp. [226][227][228][229][230][231][232][233][234][235][236][237][238][239][240] 2007). We then settle all the three problems in sparse and dense graphs also providing improved upper running time bounds.

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On generating all maximal independent sets

Cited by 624 publications

References 6 publications

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Generating Models of a Matched Formula With a Polynomial Delay

Exact Algorithms for Dominating Clique Problems

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