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36
New Results on Monotone Dualization and Generating Hypergraph Transversals
 SIAM JOURNAL ON COMPUTING
, 2002
"... 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 NPcompleteness. We present a number of new polynomial time resp. outputpolynomial time results for significant ..."
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Cited by 51 (12 self)
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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 NPcompleteness. We present a number of new polynomial time resp. outputpolynomial 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.
Generating All Maximal Models of a Boolean Expression
, 1999
"... We examine the computational problem of generating all maximal models of a Boolean expression in CNF. We give a resolutionlike method that reduces the unnegated variables of an expression while preserving its set of maximal models. We present an outputpolynomial algorithm for the 2CNF case and we ..."
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Cited by 28 (7 self)
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We examine the computational problem of generating all maximal models of a Boolean expression in CNF. We give a resolutionlike method that reduces the unnegated variables of an expression while preserving its set of maximal models. We present an outputpolynomial algorithm for the 2CNF case and we show that the problem cannot be solved in outputpolynomial time in the case of Horn expressions, unless P=NP, despite an affinity of this case to the recently subexponentially solved transversal hypergraph problem. The problem is of course trivial for 1valid and antiHorn expressions, and open for exclusiveors; it is NPhard in all other cases.
Evaluation of an Algorithm for the Transversal Hypergraph Problem
 In Algorithm Engineering
, 1999
"... The Transversal Hypergraph Problem is the problem of computing, given a hypergraph, the set of its minimal transversals, i.e. the hypergraph whose hyperedges are all minimal hitting sets of the given one. This problem turns out to be central in various fields of Computer Science. We present and ..."
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Cited by 18 (5 self)
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The Transversal Hypergraph Problem is the problem of computing, given a hypergraph, the set of its minimal transversals, i.e. the hypergraph whose hyperedges are all minimal hitting sets of the given one. This problem turns out to be central in various fields of Computer Science. We present and experimentally evaluate a heuristic algorithm for the problem, which seems able to handle large instances and also possesses some nice features especially desirable in problems with large output such as the Transversal Hypergraph Problem.
An efficient algorithm for the transversal hypergraph generation
 Journal of Graph Algorithms and Applications
"... The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving t ..."
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Cited by 18 (0 self)
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The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving this problem. We show that the proposed algorithm operates in a way that rules out regeneration and, thus, its memory requirements are polynomially bounded to the size of the input hypergraph. Although no time bound for the algorithm is given, experimental evaluation and comparison with other approaches have shown that it behaves well in practice and it can successfully handle large problem instances.
Computational aspects of mining maximal frequent patterns
 Theoretical Computer Science
, 2006
"... In this paper we study the complexitytheoretic aspects of mining maximal frequent patterns, from the perspective of counting the number of all distinct solutions. We present the first formal proof that the problem of counting the number of maximal frequent itemsets in a database of transactions, gi ..."
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In this paper we study the complexitytheoretic aspects of mining maximal frequent patterns, from the perspective of counting the number of all distinct solutions. We present the first formal proof that the problem of counting the number of maximal frequent itemsets in a database of transactions, given an arbitrary support threshold, is #Pcomplete, thereby providing theoretical evidence that the problem of mining maximal frequent itemsets is NPhard. We also extend our complexity analysis to other similar data mining problems that deal with complex data structures, such as sequences, trees, and graphs. We investigate several variants of these mining problems in which the patterns of interest are subsequences, subtrees, or subgraphs, and show that the associated problems of counting the number of maximal frequent patterns are all either #Pcomplete or #Phard. Key words: data mining, complexity, maximal frequent patterns, #Pcomplete 1
Abduction and the Dualization Problem
, 2003
"... Computing abductive explanations is an important problem, which has been studied extensively in Artificial Intelligence (AI) and related disciplines. ..."
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Cited by 8 (0 self)
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Computing abductive explanations is an important problem, which has been studied extensively in Artificial Intelligence (AI) and related disciplines.
Efficiently ordering subgoals with access constraints
"... In this paper, we study the problem of ordering subgoals under binding pattern restrictions for queries posed as nonrecursive Datalog programs. We prove that despite their limited expressive power, the problem is computationally hard — PSPACEcomplete in the size of the nonrecursive Datalog program ..."
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In this paper, we study the problem of ordering subgoals under binding pattern restrictions for queries posed as nonrecursive Datalog programs. We prove that despite their limited expressive power, the problem is computationally hard — PSPACEcomplete in the size of the nonrecursive Datalog program even for fairly restricted cases. As a practical solution to this problem, we develop an asymptotically optimal algorithm that runs in time linear in the size of the query plan. We also study extensions of our algorithm that efficiently solve other query planning problems under binding pattern restrictions. These problems include conjunctive queries with nested grouping constraints, distributed conjunctive queries, and firstorder queries.
Lower bounds for three algorithms for transversal hypergraph generation
 Discrete Appl. Math
"... Abstract. The computation of all minimal transversals of a given hypergraph in outputpolynomial time is a long standing open question known as the transversal hypergraph generation. One of the first attempts on this problem—the sequential method [Ber89]—is not outputpolynomial as was shown by Takat ..."
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Cited by 6 (2 self)
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Abstract. The computation of all minimal transversals of a given hypergraph in outputpolynomial time is a long standing open question known as the transversal hypergraph generation. One of the first attempts on this problem—the sequential method [Ber89]—is not outputpolynomial as was shown by Takata [Tak02]. Recently, three new algorithms improving the sequential method were published and experimentally shown to perform very well in practice [BMR03, DL05, KS05]. Nevertheless, a theoretical worstcase analysis has been pending. We close this gap by proving lower bounds for all three algorithms. Thereby, we show that none of them is outputpolynomial. 1
Algorithms for compact letter displays: Comparison and evaluation
, 2006
"... Multiple pairwise comparisons are one of the most frequent tasks in applied statistics. In this context, letter displays may be used for a compact presentation of results of multiple comparisons. A heuristic previously proposed for this task is compared with two new algorithmic approaches. The latte ..."
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Cited by 4 (3 self)
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Multiple pairwise comparisons are one of the most frequent tasks in applied statistics. In this context, letter displays may be used for a compact presentation of results of multiple comparisons. A heuristic previously proposed for this task is compared with two new algorithmic approaches. The latter rely on the equivalence of computing compact letter displays to computing clique covers in graphs, a problem that is wellstudied in theoretical computer science. A thorough discussion of the three approaches aims to give a comparison of the algorithms’ advantages and disadvantages. The three algorithms are compared in a series of experiments on simulated and real data, e.g., using data from wheat and triticale yield trials.
Complexity of DNF and Isomorphism of Monotone Formulas
"... Abstract. We investigate the complexity of finding prime implicants and minimal equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case strongly differs from the arbitrary case. We show that ..."
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Abstract. We investigate the complexity of finding prime implicants and minimal equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case strongly differs from the arbitrary case. We show that it is DPcomplete to check whether a monomial is a prime implicant for an arbitrary formula, but checking prime implicants for monotone formulas is in L. We show PPcompleteness of checking whether the minimum size of a DNF for a monotone formula is at most k. For k in unary, we show the complexity of the problem to drop to coNP. In [Uma01] a similar problem for arbitrary formulas was shown to be Σ p 2complete. We show that calculating the minimal DNF for a monotone formula is possible in outputpolynomial time if and only if P = NP. Finally, we disprove a conjecture from [Rei03] by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas. 1