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50
New Algorithms for Enumerating All Maximal Cliques
, 2004
"... Abstract. In this paper, we consider the problems of generating all maximal (bipartite) cliques in a given (bipartite) graph G = (V, E) with n vertices and m edges. We propose two algorithms for enumerating all maximal cliques. One runs with O(M(n)) time delay and in O(n 2) space and the other runs ..."
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Cited by 58 (1 self)
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Abstract. In this paper, we consider the problems of generating all maximal (bipartite) cliques in a given (bipartite) graph G = (V, E) with n vertices and m edges. We propose two algorithms for enumerating all maximal cliques. One runs with O(M(n)) time delay and in O(n 2) space and the other runs with O( ∆ 4) time delay and in O(n + m) space, where ∆ denotes the maximum degree of G, M(n) denotes the time needed to multiply two n × n matrices, and the latter one requires O(nm) time as a preprocessing. For a given bipartite graph G, we propose three algorithms for enumerating all maximal bipartite cliques. The first algorithm runs with O(M(n)) time delay and in O(n 2) space, which immediately follows from the algorithm for the nonbipartite case. The second one runs with O( ∆ 3) time delay and in O(n + m) space, and the last one runs with O( ∆ 2) time delay and in O(n + m + N∆) space, where N denotes the number of all maximal bipartite cliques in G and both algorithms require O(nm) time as a preprocessing. Our algorithms improve upon all the existing algorithms, when G is either dense or sparse. Furthermore, computational experiments show that our algorithms for sparse graphs have significantly good performance for graphs which are generated randomly and appear in realworld problems. 1
On Computing All Abductive Explanations
 INSTITUT FUR INFORMATIONSSYSTEME
, 2002
"... We consider the computation of all respectively a polynomial subset of the explanations of an abductive query from a Horn theory, and pay particular attention to whether the query is a positive or negative letter, the explanation is based on literals from an assumption set, and the Horn theory is ..."
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Cited by 34 (5 self)
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We consider the computation of all respectively a polynomial subset of the explanations of an abductive query from a Horn theory, and pay particular attention to whether the query is a positive or negative letter, the explanation is based on literals from an assumption set, and the Horn theory is represented in terms of formulas or characteristic models. We derive
Hypergraph Transversal Computation and Related Problems in Logic and AI
 of LNCS
, 2002
"... Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science. In the present paper, we address this problem and its decisional variant, i.e., the recognition of the transversal hypergraph for another hypergraph. ..."
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Cited by 32 (2 self)
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Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science. In the present paper, we address this problem and its decisional variant, i.e., the recognition of the transversal hypergraph for another hypergraph.
Firstorder queries on structures of bounded degree are computable with constant delay
 ACM TRANS. ON COMPUTATIONAL LOGIC
, 2007
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Local and global methods in data mining: basic techniques and open problems
 Lecture Notes In Computer Science; Vol. 2380; ISSN 03029743 In Proceedings of the 29th International Colloquium on Automata, Languages and Programming
, 2002
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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.
Generating maximal independent sets for hypergraphs with bounded edgeintersections
 6th Latin American Theoretical Informatics Conference (LATIN 2004), (Martin FarachColton, ed.) Lecture Notes in Computer Science 2461
"... Abstract. Given a finite set V, and integers k ≥ 1 and r ≥ 0, denote by A(k, r) the class of hypergraphs A ⊆ 2V with (k, r)bounded intersections, i.e. in which the intersection of any k distinct hyperedges has size at most r. We consider the problem MIS(A, I): given a hypergraph A and a subfamily ..."
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Cited by 14 (12 self)
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Abstract. Given a finite set V, and integers k ≥ 1 and r ≥ 0, denote by A(k, r) the class of hypergraphs A ⊆ 2V with (k, r)bounded intersections, i.e. in which the intersection of any k distinct hyperedges has size at most r. We consider the problem MIS(A, I): given a hypergraph A and a subfamily I ⊆ I(A), of its maximal independent sets (MIS) I(A), either extend this subfamily by constructing a new MIS I ∈ I(A) \ I or prove that there are no more MIS, that is I = I(A). We show that for hypergraphs A ∈ A(k, r) with k + r ≤ const, problem MIS(A, I) is NCreducible to problem MIS(A′, ∅) of generating a single MIS for a partial subhypergraph A ′ of A. In particular, for this class of hypergraphs, we get an incremental polynomial algorithm for generating all MIS. Furthermore, combining this result with the currently known algorithms for finding a single maximal independent set of a hypergraph, we obtain efficient parallel algorithms for incrementally generating all MIS for hypergraphs in the classes A(1, c), A(c, 0), and A(2, 1), where c is a constant. We also show that, for A ∈ A(k, r), where k + r ≤ const, the problem of generating all MIS of A can be solved in incremental polynomialtime with space polynomial only in the size of A. 1
Version spaces and the consistency problem
 Artificial Intelligence
, 2004
"... A version space is a collection of concepts consistent with a given set of positive and negative examples. Mitchell [Mit82] proposed representing a version space by its boundary sets: the maximally general (G) and maximally specific consistent concepts (S). For many simple concept classes, the size ..."
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Cited by 12 (0 self)
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A version space is a collection of concepts consistent with a given set of positive and negative examples. Mitchell [Mit82] proposed representing a version space by its boundary sets: the maximally general (G) and maximally specific consistent concepts (S). For many simple concept classes, the size of G and S is known to grow exponentially in the number of positive and negative examples. This paper argues that previous work on alternative representations of version spaces has disguised the real question underlying version space reasoning. We instead show that tractable reasoning with version spaces turns out to depend on the consistency problem, i.e., determining if there is any concept consistent with a set of positive and negative examples. Indeed, we show that tractable version space reasoning is possible if and only if there is an efficient algorithm for the consistency problem. Our observations give rise to new concept classes for which tractable version space reasoning is now possible, e.g., 1decision lists, monotone depth two formulas, and halfspaces. 1 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.
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 7 (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