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23
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 31 (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 29 (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
Local and Global Methods in Data Mining: Basic Techniques and Open Problems
 In ICALP 2002, 29th International Colloquium on Automata, Languages, and Programming, Malaga
, 2002
"... Data mining has in recent years emerged as an interesting area in the boundary between algorithms, probabilistic modeling, statistics, and databases. Data mining research can be divided into global approaches, which try to model the whole data, and local methods, which try to find useful patterns oc ..."
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Cited by 23 (2 self)
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Data mining has in recent years emerged as an interesting area in the boundary between algorithms, probabilistic modeling, statistics, and databases. Data mining research can be divided into global approaches, which try to model the whole data, and local methods, which try to find useful patterns occurring in the data. We discuss briefly some simple local and global techniques, review two attempts at combining the approaches, and list open problems with an algorithmic flavor.
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 22 (1 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.
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 11 (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.
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 10 (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 5 (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 the Transversal Hypergraph Generation
"... 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 of Berge [Ber89]—is not outputpolynomial as shown by ..."
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Cited by 4 (1 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 of Berge [Ber89]—is not outputpolynomial as shown by Takata [Tak02]. In recent years, three new algorithms designed to improve the sequential method were published and experimentally shown to perform well on practically relevant cases [BMR03, DL05, KS05]. Nevertheless, a theoretical analysis was still pending. We close this gap by proving lower bounds for all three algorithms. Our lower bounds show that none of the algorithms is outputpolynomial. 1
Tractable Database Design through Bounded Treewidth
, 2006
"... Given that most elementary problems in database design are NPhard, the currently used database design algorithms produce suboptimal results. For example, the current 3NF decomposition algorithms may continue further decomposing a relation even though it is already in 3NF. In this paper we study dat ..."
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Cited by 4 (3 self)
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Given that most elementary problems in database design are NPhard, the currently used database design algorithms produce suboptimal results. For example, the current 3NF decomposition algorithms may continue further decomposing a relation even though it is already in 3NF. In this paper we study database design problems whose sets of functional dependencies have bounded treewidth. For such sets, which frequently occur in practice, we develop polynomialtime and highly parallelizable algorithms for a number of central database design problems such as: • primality of an attribute • 3NFtest for a relational schema or subschema • BCNFtest for a subschema. For establishing these results, we propose a new characterization for keys and for the primality of a single attribute. In order to define the treewidth of a relational schema, we shall associate a hypergraph with it. Note that there are two main possibilities of defining the treewidth of a hypergraph H: One is via the primal graph of H and one is via the incidence graph of H. Our algorithms apply to the case where the primal graph is considered. However, we also show that the tractability results still hold when the incidence graph is considered instead.