## New Results on Monotone Dualization and Generating Hypergraph Transversals (2002)

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Venue: | SIAM JOURNAL ON COMPUTING |

Citations: | 41 - 13 self |

### BibTeX

@INPROCEEDINGS{Eiter02newresults,

author = {Thomas Eiter and Georg Gottlob and Kazuhisa Makino},

title = {New Results on Monotone Dualization and Generating Hypergraph Transversals},

booktitle = {SIAM JOURNAL ON COMPUTING},

year = {2002},

pages = {14--22},

publisher = {}

}

### Years of Citing Articles

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### Abstract

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time resp. output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log² n/log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.

### Citations

932 | A theory of diagnosis from first principles
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Citation Context ...15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., =-=[21, 28, 29, 44]-=-), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output CNF ψ can be exponential in the size of ϕ, it is currently not known whether ψ can ... |

371 |
Graph minors. II. Algorithmic aspects of tree-width
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Citation Context ...ard as the general case.) In this paper, we give a positive answer and show that for α-acyclic (prime) ϕ, DUALIZATION is solvable with polynomial delay. • Formulas of Bounded Treewidth. The treewidth =-=[45]-=- of a graph expresses its degree of cyclicity. Treewidth is an extremely general notion, and bounded treewidth generalizes almost all other notions of near-acyclicity. Following [13], we define the tr... |

180 | How to assign votes in distributed system - Garcia-Molina, Barbara - 1985 |

160 |
On generating all maximal independent sets
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- 1988
(Show Context)
Citation Context ...ld significantly advance the state of the art of several problems in the above application areas. Similarly, the complexity of DUAL (equivalently, TRANS-HYP) is open since more than 20 years now (cf. =-=[3, 15, 30, 31, 33]-=-). Note that DUALIZATION is solvable in polynomial total time on a class C of hypergraphs iff DUAL is in PTIME for all pairs (H, G), where H ∈ C [3]. DUAL is known to be in co-NP and the best currentl... |

149 |
On the complexity of dualization of monotone disjunctive normal forms
- Fredman, Khachiyan
- 1996
(Show Context)
Citation Context ... total time on a class C of hypergraphs iff DUAL is in PTIME for all pairs (H, G), where H ∈ C [3]. DUAL is known to be in co-NP and the best currently known upper time-bound is quasi-polynomial time =-=[17, 19, 47]-=-. Determining the complexities of DUALIZATION and DUAL, and of equivalent problems such as the transversal problems, is a prominent open problem. This is witnessed by the fact that these problems are ... |

133 | Hypertree decompositions and tractable queries
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(Show Context)
Citation Context ...otions of treewidth for hypergraphs resp. monotone CNFs. For example, tree decomposition of type II of CNF ϕ = ∧ c∈C c is defined as type-I tree decomposition of its incident 2-CNF (i.e., graph) G(ϕ) =-=[13, 24]-=-. That is, for each clause c ∈ ϕ, we introduce a new variable yc and construct G(ϕ) = ∧ xi∈c∈ϕ (xi ∨ yc) (here, xi ∈ c denotes that xi appears in c). Let Tw2(ϕ) denote the type-II treewidth of ϕ. Prop... |

132 | Identifying the minimal transversals of a hypergraph and related problems
- Eiter, Gottlob
- 1995
(Show Context)
Citation Context ...N is deciding, given hypergraphs H and G, whether G = Tr(H). DUALIZATION and several problems which are like transversal computation known to be computationally equivalent to problem DUALIZATION (see =-=[15]-=-) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21... |

98 | Conjunctive query containment revisited
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(Show Context)
Citation Context ...otions of treewidth for hypergraphs resp. monotone CNFs. For example, tree decomposition of type II of CNF ϕ = ∧ c∈C c is defined as type-I tree decomposition of its incident 2-CNF (i.e., graph) G(ϕ) =-=[13, 24]-=-. That is, for each clause c ∈ ϕ, we introduce a new variable yc and construct G(ϕ) = ∧ xi∈c∈ϕ (xi ∨ yc) (here, xi ∈ c denotes that xi appears in c). Let Tw2(ϕ) denote the type-II treewidth of ϕ. Prop... |

78 | Degrees of acyclicity for hypergraphs and relational database schemes
- Fagin
- 1983
(Show Context)
Citation Context ... Like in graphs, acyclicity is appealing in hypergraphs resp. monotone CNFs from a theoretical as well as a practical point of view. However, there are many notions of acyclicity for hypergraphs (cf. =-=[16]-=-), since different generalizations from graphs are possible. We refer to α-, β-,γ-, and Berge-acyclicity as stated in [16], for which the following proper inclusion hierarchy is known: Berge-acyclic ⊆... |

69 | Data mining, hypergraph transversals and machine learning
- Gunopulos, Khardon, et al.
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(Show Context)
Citation Context ... computation known to be computationally equivalent to problem DUALIZATION (see [15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., =-=[6, 7, 12, 22]-=-), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output ... |

68 |
Complexity of identification and dualization of positive Boolean functions
- Bioch, Ibaraki
- 1995
(Show Context)
Citation Context ...ld significantly advance the state of the art of several problems in the above application areas. Similarly, the complexity of DUAL (equivalently, TRANS-HYP) is open since more than 20 years now (cf. =-=[2, 13, 26, 27, 29]-=-). Note that DUALIZATION is solvable in polynomial total time on a class C of hypergraphs iff DUAL is in PTIME for all pairs (H, G), where H ∈ C [2]. DUAL is known to be in co-NP and the best currentl... |

56 |
Design by example: An application of Armstrong relations
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- 1986
(Show Context)
Citation Context ...ION and several problems which are like transversal computation known to be computationally equivalent to problem DUALIZATION (see [15]) are of interest in various areas such as database theory (e.g. =-=[38, 49]-=-), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed sys... |

55 |
Simple linear time algorithms to test chordality of graphs, test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs
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- 1984
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Citation Context ...lauses c and c ′ satisfy V (c) ⊆ V (c′), remove c from ϕ. to 0 (i.e., the empty clause). Note that α-acyclicity of a monotone CNF ϕ can be checked, and a suitable GYO-reduction output, in O(‖ϕ‖) time =-=[48]-=-. A monotone CNF ϕ is β-acyclic iff every CNF consisting of clauses in ϕ is α-acyclic. As shown in [15], the prime implicants of a monotone f represented by a β-acyclic CNF ϕ can be enumerated (and th... |

42 | On the complexity of generating maximal frequent and minimal infrequent sets
- Boros, Gurvich, et al.
(Show Context)
Citation Context ... computation known to be computationally equivalent to problem DUALIZATION (see [15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., =-=[6, 7, 12, 22]-=-), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output ... |

35 | Sideri “On Horn Envelopes and Hypergraph Transversals
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- 1993
(Show Context)
Citation Context ...15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., =-=[21, 28, 29, 44]-=-), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output CNF ψ can be exponential in the size of ϕ, it is currently not known whether ψ can ... |

31 | NP-completeness: A retrospective
- Papadimitriou
- 1997
(Show Context)
Citation Context ...oblem. This is witnessed by the fact that these problems are cited in a rapidly growing body of literature and have been referenced in various survey papers and complexity theory retrospectives, e.g. =-=[30, 34, 40]-=-. Given the importance of monotone dualization and equivalent problems for many application areas, and given the long standing failure to settle the complexity of these problems, emphasis was put on f... |

29 |
T: A theory of coteries: mutual exclusion in distributed systems
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(Show Context)
Citation Context ...earning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed systems (e.g., =-=[18, 27]-=-) to mention a few. While the output CNF ψ can be exponential in the size of ϕ, it is currently not known whether ψ can be computed in output-polynomial (or polynomial total) time, i.e., in time polyn... |

26 | Dual subimplicants of positive Boolean functions
- Boros, Gurvich, et al.
- 1998
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Citation Context ..., emphasis was put on finding tractable cases of DUAL and corresponding polynomial total-time cases of DUALIZATION. In fact, several relevant tractable classes were found by various authors; see e.g. =-=[4, 8, 9, 10, 12, 14, 15, 20, 35, 36, 39, 41]-=- and references therein. Moreover, classes of formulas were identified on which DUALIZATION is not just polynomial total-time, but where the conjuncts of the dual formula can be enumerated with increm... |

25 |
Refining nondeterminism in relativized polynomial-time bounded computations
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- 1980
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Citation Context ...many guessed bits by a polynomial-time non-deterministic Turing machine. This result might bring new insight towards settling the complexity of the problem. We adopt Kintala and Fischer’s terminology =-=[32]-=- and write g(n)-P for the class of sets accepted by a nondeterministic Turing machine in polynomial time making at most g(n) nondeterministic steps on every input of length n. For every integer k ≥ 1,... |

25 |
The maximum latency and identification of positive Boolean functions
- Makino, Ibaraki
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Citation Context ..., emphasis was put on finding tractable cases of DUAL and corresponding polynomial total-time cases of DUALIZATION. In fact, several relevant tractable classes were found by various authors; see e.g. =-=[4, 8, 9, 10, 12, 14, 15, 20, 35, 36, 39, 41]-=- and references therein. Moreover, classes of formulas were identified on which DUALIZATION is not just polynomial total-time, but where the conjuncts of the dual formula can be enumerated with increm... |

24 | Translating between Horn representations and their characteristic models
- Khardon
- 1995
(Show Context)
Citation Context ...15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., =-=[21, 28, 29, 44]-=-), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output CNF ψ can be exponential in the size of ϕ, it is currently not known whether ψ can ... |

23 |
Coherent Structures and Simple Games
- Ramamurthy
- 1990
(Show Context)
Citation Context ...nally equivalent to problem DUALIZATION (see [15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. =-=[26, 42, 43]-=-), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output CNF ψ can be exponential in the s... |

22 | Efficient read-restricted monotone CNF/DNF dualization by learning with membership queries
- Domingo, Mishra, et al.
- 1999
(Show Context)
Citation Context ... computation known to be computationally equivalent to problem DUALIZATION (see [15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., =-=[6, 7, 12, 22]-=-), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output ... |

22 | Incremental recompilation of knowledge
- Gogic, Papadimitriou, et al.
- 1998
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Citation Context |

22 |
Generating all maximal independent sets: Np-hardness and polynomial-time algorithms
- Lawler, Kan
- 1980
(Show Context)
Citation Context ...ld significantly advance the state of the art of several problems in the above application areas. Similarly, the complexity of DUAL (equivalently, TRANS-HYP) is open since more than 20 years now (cf. =-=[3, 15, 30, 31, 33]-=-). Note that DUALIZATION is solvable in polynomial total time on a class C of hypergraphs iff DUAL is in PTIME for all pairs (H, G), where H ∈ C [3]. DUAL is known to be in co-NP and the best currentl... |

22 |
Ozsoyoglu, An algorithm for tree-query membership of a distributed query
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- 1979
(Show Context)
Citation Context ...s known: Berge-acyclic ⊆ γ-acyclic ⊆ β-acyclic ⊆ α-acyclic. The notion of α-acyclicity came up in relational database theory. A monotone CNF ϕ is α-acyclic iff ϕ = 1 or reducible by the GYO-reduction =-=[25, 51]-=-, i.e., repeated application of one of the two rules: (1) If variable xi occurs in only one clause c, remove xi from c. (2) If distinct clauses c and c ′ satisfy V (c) ⊆ V (c′), remove c from ϕ. to 0 ... |

21 |
On the universal relation
- Graham
- 1980
(Show Context)
Citation Context ...s known: Berge-acyclic ⊆ γ-acyclic ⊆ β-acyclic ⊆ α-acyclic. The notion of α-acyclicity came up in relational database theory. A monotone CNF ϕ is α-acyclic iff ϕ = 1 or reducible by the GYO-reduction =-=[25, 51]-=-, i.e., repeated application of one of the two rules: (1) If variable xi occurs in only one clause c, remove xi from c. (2) If distinct clauses c and c ′ satisfy V (c) ⊆ V (c′), remove c from ϕ. to 0 ... |

19 |
A fast and simple algorithm for identifying 2-monotonic positive Boolean functions
- Makino, Ibaraki
- 1998
(Show Context)
Citation Context ..., emphasis was put on finding tractable cases of DUAL and corresponding polynomial total-time cases of DUALIZATION. In fact, several relevant tractable classes were found by various authors; see e.g. =-=[4, 8, 9, 10, 12, 14, 15, 20, 35, 36, 39, 41]-=- and references therein. Moreover, classes of formulas were identified on which DUALIZATION is not just polynomial total-time, but where the conjuncts of the dual formula can be enumerated with increm... |

17 | An efficient incremental algorithm for generating all maximal independent sets for hypergraphs of bounded dimension
- Boros, Elbassioni, et al.
- 2000
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17 | Exact transversal hypergraphs and application to Boolean µ-functions
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- 1994
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16 |
Dualization of regular Boolean functions
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15 | Dual-bounded generating problem: partial and multiple transversals of a hypergraph
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14 | Molecular computing, bounded nondeterminism, and efficient recursion
- Beigel, Fu
- 1999
(Show Context)
Citation Context ...s) can be found in polynomial time with O(χ(n) · log n) nondeterministic steps. 5.3 Application of Beigel and Fu’s results While our independently developed methods substantially differ from those in =-=[1, 2]-=-, membership of problem DUAL in co-β2P may also be obtained by exploiting Beigel and Fu’s Theorem 8 in [1] (or, equivalently, Theorem 11 in [2]). They show how to convert certain recursive algorithms ... |

14 |
Nash-solvability of games in pure strategies
- Gurvich
- 1975
(Show Context)
Citation Context ...nally equivalent to problem DUALIZATION (see [15]) are of interest in various areas such as database theory (e.g. [38, 49]), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. =-=[26, 42, 43]-=-), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed systems (e.g., [18, 27]) to mention a few. While the output CNF ψ can be exponential in the s... |

14 |
An O(nm)-time algorithm for computing the dual of a regular Boolean function, Discrete Applied Mathematics 49
- Peled, Simeone
- 1994
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13 | Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle
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- 1997
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13 |
Combinatorial optimization: some problems and trends
- Lovász
- 1992
(Show Context)
Citation Context ...oblem. This is witnessed by the fact that these problems are cited in a rapidly growing body of literature and have been referenced in various survey papers and complexity theory retrospectives, e.g. =-=[30, 34, 40]-=-. Given the importance of monotone dualization and equivalent problems for many application areas, and given the long standing failure to settle the complexity of these problems, emphasis was put on f... |

12 |
Every one a winner, or how to avoid isomorphism when cataloging combinatorial configurations
- Read
- 1978
(Show Context)
Citation Context ...nally equivalent to problem DUALIZATION (see [13]) are of interest in various areas such as database theory (e.g. [34, 43]), machine learning and data mining (e.g., [4, 5, 10, 18]), game theory (e.g. =-=[22, 38, 39]-=-), artificial intelligence (e.g., [17, 24, 25, 40]), mathematical programming (e.g., [3]), and distributed systems (e.g., [16, 23]) to mention a few. While the output CNF ψ can be exponential in the s... |

10 | Generating all maximal independent sets of bounded-degree hypergraphs
- Mishra, Pitt
- 1997
(Show Context)
Citation Context |

9 |
Efficient dualization of O(log n)-term monotone disjunctive normal forms
- Makino
(Show Context)
Citation Context ...∩V (c) = {xj}). Thus Theorem 3.4 implies the following. Theorem 4.1 For k-degenerate CNFs ϕ, DUALIZATION is solvable with O(‖ϕ‖ · n k+1 ) polynomial delay if k ≥ 1 is constant. Applying the result of =-=[37]-=- that log-clause CNF is dualizable in incremental polynomial time, we obtain a polynomiality result also for non-constant degeneracy: Theorem 4.2 For O(log ‖ϕ‖)-degenerate CNFs ϕ, problem DUALIZATION ... |

9 |
Space-efficient enumeration of minimal transversals of a hypergraph
- Tamaki
- 2000
(Show Context)
Citation Context ... total time on a class C of hypergraphs iff DUAL is in PTIME for all pairs (H, G), where H ∈ C [3]. DUAL is known to be in co-NP and the best currently known upper time-bound is quasi-polynomial time =-=[17, 19, 47]-=-. Determining the complexities of DUALIZATION and DUAL, and of equivalent problems such as the transversal problems, is a prominent open problem. This is witnessed by the fact that these problems are ... |

9 |
stable sets and perfect graphs
- Colouring
- 1995
(Show Context)
Citation Context ...of ψ with polynomial delay or incremental polynomial delay. In particular, we show positive results for the following input classes: • Degenerate CNFs. We generalize the notion of k-degenerate graphs =-=[50]-=- to hypergraphs and define k-degenerate monotone CNFs resp. hypergraphs. We prove that for any constant k, DUALIZE works with polynomial delay on k-degenerate CNFs. Moreover, it works in output-polyno... |

6 |
Open and closed problems in NP-completeness, Lecture given at the
- Johnson
- 1991
(Show Context)
Citation Context ...ld significantly advance the state of the art of several problems in the above application areas. Similarly, the complexity of DUAL (equivalently, TRANS-HYP) is open since more than 20 years now (cf. =-=[3, 15, 30, 31, 33]-=-). Note that DUALIZATION is solvable in polynomial total time on a class C of hypergraphs iff DUAL is in PTIME for all pairs (H, G), where H ∈ C [3]. DUAL is known to be in co-NP and the best currentl... |

6 |
On the sequential method for listing minimal hitting sets
- Takata
- 2002
(Show Context)
Citation Context ...] pointed out that a well-known sequential algorithm, in which the clauses ci of a CNF ϕ = c1 ∧ · · · ∧ cm are processed in order i = 1,... ,m, is not polynomial-total time in general. Most recently, =-=[46]-=- showed that this holds even if an optimal ordering of the clauses is assumed (i.e., they may be arbitrarily arranged for free). Main Goal. The main goal of this paper is to present important new poly... |

6 |
Minimal keys and antikeys
- Thi
- 1986
(Show Context)
Citation Context ...ION and several problems which are like transversal computation known to be computationally equivalent to problem DUALIZATION (see [15]) are of interest in various areas such as database theory (e.g. =-=[38, 49]-=-), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., [5]), and distributed sys... |

5 | On generating all minimal integer solutions to a monotone system of linear inequalities
- Boros, Elbassioni, et al.
- 2001
(Show Context)
Citation Context ...ase theory (e.g. [38, 49]), machine learning and data mining (e.g., [6, 7, 12, 22]), game theory (e.g. [26, 42, 43]), artificial intelligence (e.g., [21, 28, 29, 44]), mathematical programming (e.g., =-=[5]-=-), and distributed systems (e.g., [18, 27]) to mention a few. While the output CNF ψ can be exponential in the size of ϕ, it is currently not known whether ψ can be computed in output-polynomial (or p... |

5 |
Self-duality of bounded monotone Boolean functions and related problems
- Gaur, Krishnamurti
- 1968
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Citation Context |

3 |
Satisfiability and self-duality of monotone Boolean functions
- Gaur
- 1999
(Show Context)
Citation Context ... total time on a class C of hypergraphs iff DUAL is in PTIME for all pairs (H, G), where H ∈ C [3]. DUAL is known to be in co-NP and the best currently known upper time-bound is quasi-polynomial time =-=[17, 19, 47]-=-. Determining the complexities of DUALIZATION and DUAL, and of equivalent problems such as the transversal problems, is a prominent open problem. This is witnessed by the fact that these problems are ... |

2 |
Private communication
- Domingo
(Show Context)
Citation Context ...nd, is not polynomial delay, and outputs the clauses of ψ in no specific order. The result for k = O(log ‖ϕ‖) is a non-trivial generalization of the result in [12], which was posed as an open problem =-=[11]-=-. • Acyclic CNFs. There are several notions of hypergraph resp. monotone CNF acyclicity [16], where the most general and well-known is α-acyclicity. As shown in [15], DUALIZATION is polynomial total t... |