T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278 1304, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....= c(Y ) Let t and t denote the family of all the maximal frequent sets and minimal infrequent sets respectively. It is proved that, if #= #, then M ( T t 1) I t and, hence, that the complexity of generating t is equivalent to that of the transversal hypergraph problem (see [10] for the definition of this problem) As the latter problem is known to be solvable in incremental quasi polynomial time [12] then the same result holds for the joint generation of maximal frequent and minimal infrequent sets: for each k # M k sets belonging to t can be generated in ....

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278--1304, December 1995.

....an expression may have a number of models that grows exponentially in its size, one cannot hope to solve this problem in polynomial time, but instead in time polynomial in the size of the input and the output. There has been a recent surge of interest in such algorithms, called output polynomial [9, 4, 5]. The most notable result in this area is an output sub exponential algorithm for generating all the minimal transversals of a hypergraph (see [5] University of Patras, Department of Mathematics, GR 265 00 Patras, Greece. E mail: djk math.upatras.gr Athens University of Economics and ....

....86 [95 E Delta] A hypergraph H is a pair (V; E) of a finite set V of vertices and a family E of subsets of V, called hyperedges. A transversal of H is a subset of H that intersects every hyperedge of H. For more information about hypergraphs, related problems, and complexity results, see [2, 4]. The importance of this problem comes from the fact that it is equivalent to a number of other important problems in different fields of Computer Science, like Logic, Database Theory, Distributed Systems and Artificial Intelligence. In Database Theory for example, the problem of finding all ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Computing, 24(6):1278--1304, December, 1995.

....models of an expression having all its variables positive. Complexity questions related to minimal or maximal models have been discussed in [3,4,1] more recent results can be found in [13] An encyclopedic exposition of the applications of the transversal hypergraph problem can be found in [6,7]. We briefly state from there certain problems in the design of relational databases [15,16] in distributed databases [8] and in model based diagnosis [5] Another interesting connection was pointed out between the transversal hypergraph problem and the rapidly growing field of knowledge ....

....with output and performance criteria see [11,12,19] The precise complexity of the transversal hypergraph problem is still unknown. The brute force algorithm given by Berge [2] needs time exponential in both the input and the output. However, several special cases can be solved in polynomial time [6]. Recently, an output subexponential algorithm was given by Fredman and Khachiyan in [7] There, it was shown that the duality of two monotoneBooleanexpressionsinDNFcanbecheckedintimeO(n ) where n is the combined size of the input and the output. It is not hard to see that this problem is ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Computing, 24(6):1278--1304, December, 1995. D.J. Kavvadias, E.C. Stavropoulos

.... prime implicants of a monotone DNF is equivalent to the generation of the minimal transversals of a simple hypergraph [2,6,12] and to the generation of the maximal models of a Boolean expression in conjunctive normal form [11] These problems are central in various fields of Computer Science (see [5,6,8] for an exposition of applications of these problems) The algorithm of Fredman and Khachiyan gives an upper bound for the time complexity of MBD and implies that the problem can not be co NP hard, unless any co NP complete problem can be solved in quasipolynomial time. In this paper we present a ....

T. Eiter, G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM J. Comput. 24 (6) (1995) 1278--1304.

....not always polynomial (see, also [17, 16] Hence, the transversal hypergraph problem plays a central role in many problems were all minimal or maximal structures of some kind have to be generated. An encyclopedic exposition of the applications of the transversal hypergraph problem can be found in [7, 8]. We briefly state from there certain problems in the design of relational databases [18, 19] in distributed databases [9] and in model based diagnosis [6] Another interesting connection was pointed out between the transversal hypergraph problem and the rapidly growing field of Knowledge ....

....Discovery in Databases, or Data Mining [11, 20] The precise complexity of the transversal hypergraph problem is still unknown. The brute force algorithm given by Berge [2] needs time exponential in both the input and the output. However, several special cases can be solved in polynomial time [7]. An output subexponential algorithm was given by Fredman and Khachiyan in [8] In this paper, it was shown that the duality of two monotone Boolean expressions in DNF can be checked in time O(n log n ) where n is the combined size of the input and the output. It is not hard to see that this ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Computing, 24(6):1278--1304, December, 1995.

.... prime implicants of a monotone DNF is equivalent to the generation of the minimal transversals of a simple hypergraph [1, 4, 9] and to the generation of the maximal models of a Boolean expression in conjunctive normal form [8] These problems are central in various fields of Computer Science (see [3, 4, 5] for an exposition of applications of these problems) The algorithm of Fredman and Khachiyan gives an upper bound for the time complexity of mbd and implies that the problem can not be co NP hard, unless any co NP complete problem can be solved in quasi polynomial time. In this paper we ....

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Computing, 24(6):1278--1304, December, 1995.

....with important applications in many fields of Computer Science. The problem falls in the category of problems that require the generation of objects which fulfill certain criteria. By far the most known generation problem is the problem of generating all minimal transversals of a hypergraph [4]. A hypergraph H is a pair (V; E) of a finite set V of vertices and a family E of subsets of V, called hyperedges. A transversal of H is a subset of H that intersects every hyperedge of H. For more information about hypergraphs, see [2] As the number of transversals of a hypergraph may be ....

....a finite set V of vertices and a family E of subsets of V, called hyperedges. A transversal of H is a subset of H that intersects every hyperedge of H. For more information about hypergraphs, see [2] As the number of transversals of a hypergraph may be exponentially larger than the input (see [4]) the question is whether this problem can be solved in time polynomial in both its input size and output size. Thus, more refined performance and complexity measures may be defined for this problem and for any generation problem, too (see [9] The exact complexity of the problem of generating ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Computing, 24(6):1278--1304, December, 1995.

....can be solved e#ciently for many classes of hypergraphs. For example, for hypergraphs of bounded dimension, i.e. when the sizes of all the edges of are limited by a constant dim(H) max H # H # k, the dualization problem can be executed in incremental polynomial time (see e.g. [8, 26]) and even stronger, all minimal transversals of can be enumerated in lexicographic order [27] For the quadratic case (k = 2) there are even more e#cient algorithms that enumerate all edges of in lexicographic order with polynomial delay, i.e. in poly( V , H ) time per each generated ....

....DUAL(H,X ) can be solved by an NC algorithm for dim(H) 3 and by a randomized NC algorithm for dim(H) 4, 5. E#cient algorithms also exist for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs of unbounded dimension (see e.g. [5, 19, 22, 24, 26, 27, 46, 49, 50, 58, 59]) 2 Even though no incremental polynomial time algorithm for the dualization of arbitrary hypergraphs is known, the dualization problem for hypergraphs of unbounded dimension is very unlikely to be NP hard since it can be solved in incremental quasi polynomial time (see [30] or [36] ....

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T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM J. Comput., 24 (1995) 1278-1304.

....is a minimal set of vertices that intersects all edges. The following problem has lots of interesting applications also outside data mining. Problem 13 Find a polynomial time algorithm (in the size of the input and output) for enumerating all transversals. See [21] for recent results and [20] for a list of equivalent problems. In data mining the problem arises also in connection with nding functional dependencies. 5.2 Other topics Dimensionality reduction Many of the discrete datasets encountered in data mining have very high dimensionality: document databases can have hundreds of ....

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278 - 1304, Dec. 1995.

....be computed using Algorithm E from [45] The following proposition shows that for ground conjunctive probabilistic logic l f m f 1 o= l f mm denotes the cardinality of a set . This shows that the index set can be computed in output polynomial total time (see especially [13]) ouxfeh =m l f m f w1 o= l f mm . The following theorem shows that POSITIVE PROBABILITY can be reduced to the solvability of a system of linear constraints similar to the one in Theorem 7.3. This result follows immediately from Theorem 6.3 and Lemmas 7.1 and 7.2. has a ....

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. on Computing, 24(6):1278--1304, 1995.

....The most well known problem of this type is the hypergraph dualization problem which calls for generating all minimal transversals for an explicitly given hypergrraph. The hypergraph dualization problem has applications in combinatorics [30] graph theory [20, 21, 31, 33] artificial intelligence [14], game theory [16, 17, 29] convex programming [18] reliability theory [11, 29] database theory [1, 26, 34] and learning theory [2] Given a finite set V of n = points, and a hypergraph (set family) a subset B V is called a transversal of the family if B #= # for all sets # ....

....problem DUAL(A,B) can be solved in time polynomial in V , A B . The dualization problem can be e#ciently solved for many classes of hypergraphs. For example, if the sizes of all the hyperedges of are limited by a constant c, then problem DUAL(A,B) can be solved in polynomial time (see [7, 14]) moreover, it can be e#ciently solved in parallel (see [4] In addition, for c = 2 there are dualization algorithms that run with polynomial delay, i.e. in poly( V , A ) time for a specific sequence # # B # B . see e.g. 20, 21, 33] E#cient algorithms exist also for the ....

T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM Journal on Computing, 24 (1995) 1278-1304.

....problem can be solved e#ciently for many classes of hypergraphs. For example, for hypergraphs of bounded dimension, i.e. when the sizes of all the edges of are limited by a constant dim(H) max H k, the dualization problem can be executed in incremental polynomial time (see e.g. [8, 26]) and even stronger, all minimal transversals of in lexicographic order [27] For the quadratic case (k = 2) there are even more e#cient algorithms that enumerate all edges of in lexicographic order with polynomial delay, i.e. in poly( V , H ) time per each generated minimal ....

....DUAL(H,X ) can be solved by an NC algorithm for dim(H) 3 and by a randomized NC algorithm for dim(H) 4, 5. E#cient algorithms also exist for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs of unbounded dimension (see e.g. [5, 19, 22, 24, 26, 27, 47, 50, 51, 60, 61]) The hypergraph dualization problem is equivalent to the enumeration of all minimal covers of the set of edges of by subfamilies of the n sets H # H H# i #= # , V . Some applications and algorithms for enumerating minimal set covers are discussed in [31] and [72] Even though no ....

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T. Eiter and G. Gottlob (1995). Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput., 24, 1278--1304.

....t frequent. Before proceeding further, let us mention some algorithmic implications of (1) It follows from the results of [5, 16, 17] that the incremental complexity of generating t is equivalent with that of the transversal hypergraph problem (for definitions and related results see e.g. [11]) The latter problem is known to be solvable in incremental quasi polynomial time [14] implying thus the same for the joint generation of maximal frequent and minimal infrequent sets. Specifically, it follows from [14] that for each k # M we can generate k sets in t in poly(n, m) ....

T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM Journal on Computing, 24 (1995) 1278-1304.

....t frequent. Before proceeding further, let us mention some algorithmic implications of (1) It follows from the results of [5, 16, 17] that the incremental complexity of generating t is equivalent with that of the transversal hypergraph problem (for definitions and related results see e.g. [11]) The latter problem is known to be solvable in incremental quasi polynomial time [14] implying thus the same for the joint generation of maximal frequent and minimal infrequent sets. Specifically, it follows from [14] that for each # M we can generate k sets in t in poly(n, m) k ....

T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM Journal on Computing, 24 (1995) 1278-1304.

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278--1304, December 1995.

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. Technical Report CD-TR 91/16, Christian Doppler Laboratory for Expert Systems, TU Vienna, Austria, January 1991.

....and TRANS ENUM have a large number of applications in many areas of Computer Science, including Distributed Systems, Databases, Boolean Circuits and Artificial Intelligence. There, they have important applications in Diagnosis, Machine Learning, Data Mining, and Explanation Finding, see e.g. [11, 13, 24, 28, 32, 33, 36] and the references therein. Let us call a decision problem # TRANS HYP hard, if problem TRANS HYP can be reduced to it by a standard polynomial time transformation. Furthermore, # is TRANS HYP complete, if # is TRANS HYP hard and, moreover, # can be polynomially transformed into TRANS HYP; that ....

....for I (if not, trivial reductions may exist) The rest of this paper is organized as follows. In the next two sections, we illustrate some of the applications of TRANS HYP and TRANS ENUM in Logic and in Artificial Intelligence. Some of the results have been established already some time ago [10, 11], and were announced in [11] but remained yet unpublished. After that, Section 4 is dedicated to a review of recent developments on complexity of TRANS HYP, and a new result is contributed (Theorem 11) The final Section 5 presents some open issues. We close this section with some terminology. ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278--1304, 1995.

....Boolean functions. Analogously, the decision problem Trans Hyp associated with Transversal Computation is deciding, given hypergraphs H G, whether G Tr(H) Dualization and several problems which are like transversal computation known to be computationally equivalent to 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] ....

....in time polynomial in the combined size of # and #. Any such algorithm for Dualization (or Transversal Computation) would significantly advance the state of the art of many problems in the application areas. Similarly, the complexity of Dual and 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 i# Dual is in PTIME for all pairs (H, G) where H # C [2] Dual is known to be in co NP and the best currently known upper time bound is n [15] Determining the complexities of Dualization and Dual, ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput., 24(6):1278--1304, 1995.

....exists is at least as hard as the positive duality problem [4] i.e. given two positive DNFs ; decide whether represents the dual of the function represented by . The positive duality problem and equivalent problems have been tackled by many researchers, but no polynomial algorithm is known [23, 4, 15, 11, 24]. This strongly supports that a polynomial time algorithm for the unique bidual Horn extension problem, i.e. deciding whether a pdBf (T ; F ) implicitly defines a total bidual Horn function is difficult to find. We study transformation problems between different representations for bidual Horn ....

....i.e. given a Horn DNF of f , the characteristic set of f is constructible in polynomial time and vice versa. Furthermore, we show that several transformations between representations of f and its dual f are polynomial time equivalent to the well known problem of dualizing a positive function [4, 11, 15]. Namely, the transformation between (i) the characteristic set of f and a Horn DNF of f ; ii)the characteristic set of f and the characteristic set of f , and (iii) a Horn DNF of f and a Horn DNF of f , i.e. dualization of a bidual Horn function. This can be seen as a positive result, ....

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T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM J. on Computing, 24 (1995) 1278-1304.

.... polynomially equivalent to many other important problems encountered in various fields such as hypergraph theory, operations research, artificial intelligence, database theory, and reliability theory; for example, computing an Armstrong relation for a given set of functional dependencies (see e.g. [7]) or computing a prime implicant cover for a set of clauses in knowledge compilation (cf. 5] As well known, the size of the output DNF can be exponentially larger than the size of the input CNF , and in general, the output DNF is not uniquely defined. In such cases, efficient computation is ....

....shows that there exists no polynomial total time algorithm for the dualization problem of general Boolean functions unless P=NP. Therefore, research has been focused on important restricted classes of Boolean functions, and in particular on positive (also called monotone) and Horn CNFs (e.g. [3, 7, 12, 16, 17]) Recall that a CNF is positive (resp. Horn) if each clause contains only positive literals (resp. at most one positive literal) It may happen that a CNF is neither positive This work was supported by the Austrian Science Fund (FWF) Project Z29 INF and by Grants in Aid for Scientific ....

[Article contains additional citation context not shown here]

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278--1304, December 1995.

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278 1304, 1995.

No context found.

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278--1304, 1995.

No context found.

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Computing, 24(6):1278--1304, December 1995.

No context found.

T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278-1304, 1995.

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAMG Journal on Computing, 24(6):1278-1304, 1995.

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