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

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Venue: | SIAM JOURNAL ON COMPUTING |

Citations: | 38 - 12 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

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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 ... |

366 |
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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... |

177 | How to assign votes in a distributed system - Garcia-Molina, Barbará - 1985 |

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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... |

143 |
On the complexity of dualization of monotone disjunctive normal forms
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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 ... |

128 | Hypertree decompositions and tractable queries
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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... |

125 | Identifying the minimal transversals of a hypergraph and related problems
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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... |

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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... |

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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 ⊆... |

67 |
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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... |

65 | Data mining, hypergraph transversals, and machine learning
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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 ... |

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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... |

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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... |

39 | On the complexity of generating maximal frequent and minimal infrequent sets
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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 ... |

34 | On Horn envelopes and hypergraph transversals
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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 ... |

31 | NP-completeness: A retrospective
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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 |
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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... |

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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,... |

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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
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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 ... |

24 |
The maximum latency and identification of positive boolean functions
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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... |

22 | Incremental recompilation of knowledge
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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 |
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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 |
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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 | Efficient read-restricted monotone cnf/dnf duaiization by learning with memebership queries. Submitted to a journal, also available from www.lsi.upc.es/~carios/papers. html
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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 ... |

21 |
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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 ... |

18 |
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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... |

17 | Exact transversal hypergraphs and application to Boolean µ-functions
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16 | An efficient incremental algorithm for generating all maximal independent sets in hypergraphs of bounded dimension
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15 | Dual-bounded generating problems: partial and multiple transversals of a hypergraph
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14 | Molecular computing, bounded nondeterminism, and efficient recursion
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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 |
An O(nm)-time algorithm for computing the dual of a regular Boolean function. Discrete Applied Mathematics 49:309–323
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13 | Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle
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Nash-solvability of games in pure strategies
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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... |

13 |
Combinatorial optimization: some problems and trends
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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... |

11 |
Every one a winner, or how to avoid isomorphism when cataloging combinatorial configurations. Annals of Discrete Mathematics 2:107–120
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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
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- 1997
(Show Context)
Citation Context |

10 |
stable sets and perfect graphs
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(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... |

9 |
Efficient dualization of O(log n)-term monotone disjunctive normal forms
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(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 ... |

8 |
Space-efficient enumeration of minimal transversals of a hypergraph, IPSJAL 75
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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 ... |

6 |
Open and closed problems in NP-completeness, Lecture given at the
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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
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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 |
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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... |

4 |
On generating all minimal integer solutions for a monotone system of linear inequalities
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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... |

4 |
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2 |
Private communication
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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... |

2 |
Satisfiability and self-duality of monotone Boolean functions
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(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 ... |