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.

....proof of Theorem 5.5 imply the following result, which shows that the complexity of generating all strong spanned patterns of a pdBf is not easier than the dualization of a monotone non decreasing Boolean function. Various results about the complexity of this dualization problem can be found in [1, 17, 19]. Corollary 5.6 The dualization of a monotone non decreasing Boolean function can be reduced in quadratic time to the generation of all strong spanned patterns of a pdBf. Corollary 5.7 For every integer n # 2, there exists a pdBF (T, F ) depending on 2n variables, such that T = 2n, F ....

....such that T = 2n, F = n, and the number of strong spanned patterns of (T, F ) is 2 n . Proof Let us consider the following Boolean function g(z 1 , z 2n ) n i=1 z i z n i , and construct the pdBf (T, F ) as in the proof of Theorem 5.5. It is well know (see, e.g. [17]) that g(z 1 , z 2n ) has 2 n maximal false points, and therefore, by Corollary 5.3, T, F ) has 2 n strong spanned patterns. 2 It is clear from the above corollary that the generation of all strong spanned patterns can require time which is exponential in the size of the pdBf. If the ....

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

....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 brie y 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 eld of knowledge ....

....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 2 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 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 ....

[Article contains additional citation context not shown here]

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

....set B of ground atoms in inductive logic programming terminology. Then, we can say that it is a problem to determine whether or not e is provable from C over B, i.e. C # B # e. Since database schemes in relational database theory can be viewed as hypergraphs , many researchers such as [2, 4, 13, 14, 17, 35] have widely investigated the properties of database schemes or hypergraphs, together with the acyclicity of them. A hypergraph is called acyclic if it reduces to an empty hypergraph by GYO reduction (See Section 2 bellow 1 ) It is known that the acyclicity frequently makes intractable problems ....

....replacing each variable x i in A with t i . If # and # are substitutions, we will use A## to denote (A#)#. Next, we formulate the concept of acyclicity. A hypergraph H = V, E) consists of a set V of vertices and a set E # 2 V of hyperedges. For a hypergraph H = V, E) the GYO reduct GYO(H) [2, 13, 14, 17] of H is the hypergraph obtained from H by repeatedly applying the following rules as long as possible: 1. Remove hyperedges that are empty or contained in other hyperedges. 2 A recursive clause in this paper is sometimes called an ambivalent clause [16] 3 y1 y2 y3 z1 z2 z3 x1 x2 x3 y1 y2 y3 ....

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

....of H is a minimal hitting set of H. We let tr(H) be the set of all transversals of H. It turns out that tr(tr(tr(H) tr(H) Be] Generating in output polynomial time tr(H) given H, is a well studied algorithmic problem, first proposed in [JYP] which has been open. In a recent paper, [EG] point out that this problem is equivalent to a number of other important problems in Boolean Logic, database theory, switching theory, distributed systems, and artificial intelligence. In this paper we present a rather unexpected connection between the transversal problem and the problem of ....

T. Eiter, G. Gottlob "Identifying the minimal transversals of a hypergraph and related problems," SIAM J. Comp., to appear.

.... There has been a recent surge of interest in such algorithms, called output polynomial algorithms [13,17] The most notable result in this area is an incrementally output subexponential algorithm for generating all minimal transversals of a hypergraph, a central and long unresolved question [8,12,6]. One is often interested not in all models of a Boolean expression, but in the most representative or natural ones. For example, concepts of propositional circumscription [3] and minimal diagnosis [5] restrict interest in the set of models of an expression that are minimal. Moreover, nding ....

....now that no other purely negative CNF expression with fewer clauses exists. This follows from Berge s theorem stating that if a hypergraph H is simple, then tr(tr(H) H [2] A straightforward result is that, if H and G are simple hypergraphs, then tr(H) tr(G) if and only if H = G (see, also, [6]) As already mentioned, a purely negative CNF expression corresponds to a hypergraph; accordingly, a purely negative CNF expression where no clause is subsumed by another one corresponds to a simple hypergraph. Since the above expression contains no subsumed clauses, no other purely negative CNF ....

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

.... There are many instances of such enumeration problems in the literature, for which no output polynomial algorithm is known (despite the fact that, in contrast to NP complete problems, it is trivial to output the first solution) The most famous one is to compute all transversals of a hypergraph [7] (see the Appendix for a definition and discussion of this problem) As was pointed out in [7] many enumeration problems arising in AI, databases, distributed computation, and other areas of Computer Science, turn out to be what we call in this paper TRANSVERSAL hard, in the sense that, if they ....

.... algorithm is known (despite the fact that, in contrast to NP complete problems, it is trivial to output the first solution) The most famous one is to compute all transversals of a hypergraph [7] see the Appendix for a definition and discussion of this problem) As was pointed out in [7], many enumeration problems arising in AI, databases, distributed computation, and other areas of Computer Science, turn out to be what we call in this paper TRANSVERSAL hard, in the sense that, if they are solvable in output polynomial time, then the transversal problem is likewise solvable. It ....

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

....determining which natural subcases of the general problem have efficient solutions. For example, in the event that each clause of the CNF has at most two variables, efficient solutions have been given under various definitions of efficiency [42, 30, 29, 22] Extending this work, Eiter and Gottlob [12] give an efficient algorithm for the case in which the size of each clause is bounded by some constant. Finally, Makino and Ibaraki have also shown that an efficient solution exists for the class of monotone formulas with constant maximum latency [31, 32] The restriction considered in this ....

....is a subset of vertices V 0 V that intersects each edge of the hypergraph. The minimal vertex covers are precisely the complements of the maximal independent sets. In the literature, generating all minimal vertex covers of a hypergraph is also referred to as the hypergraph transversal problem [12]. The read restriction we consider here in the CNF DNF setting is equivalent to the natural restriction of limiting the degree of each vertex in the hypergraph in both the hypergraph transversal and independent set problems. Our result complements output polynomial time algorithms for versions of ....

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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.

....a finite set B of ground unit clauses in inductive logic programming. Then, we can say that it is a problem to determine whether or not e is provable from C over B, i.e. C # B # e. Since database schemes in relational database theory can be viewed as hypergraphs, many researchers such as [2, 4, 13, 14, 16, 34] have widely investigated the properties of database schemes or hypergraphs, together with the acyclicity of them 1 . It is known that the acyclicity frequently makes intractable problems in cyclic cases tractable. The conjunctive query problem is such an example: While the conjunctive query ....

....in this paper, we call a nonrecursive definite clause containing no constant symbols a conjunctive query. Next, we formulate the concept of acyclicity. A hypergraph H = V, E) consists of a set V of vertices and a set E # 2 V of hyperedges. For a hypergraph H = V, E) the GYO reduct GYO(H) [2, 13, 14, 16] of H is the hypergraph obtained from H by repeatedly applying the following rules as long as possible: 1. Remove hyperedges that are empty or contained in other hyperedges; 2. Remove vertices that appear in # 1 hyperedges. Definition 1. A hypergraph H is called acyclic if GYO(H) is the empty ....

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

....which can perform the above tasks in time which is polynomial in both the input size and the output size. These are called output polynomial algorithms. A related problem we discuss is that of translating between the CNF and DNF representations of monotone functions, a well known problem [FK94, EG94, KPS93, Rei80] for which no polynomial algorithm is known. We call this problem the duality 1 problem and denote it by DME (for Dualization of Monotone Expressions) 1 The duality problem, DME, has a lot of equivalent manifestations which appear in various branches of computer science, and is known as the ....

....is known. We call this problem the duality 1 problem and denote it by DME (for Dualization of Monotone Expressions) 1 The duality problem, DME, has a lot of equivalent manifestations which appear in various branches of computer science, and is known as the Hypergraph Transversal problem [EG94] and the hitting set problem [Rei80] A comprehensive study of these problems is given in [EG94] 3 We first show that the problems CCM,SID are at least as hard as DME. By that we mean that if there is an output polynomial algorithm for CCM (or for SID) then there is an output polynomial ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal of Computing, 1994. To appear.

....paper [KMR95] there is a correspondence between Horn expressions and Functional Dependencies, and a correspondence between characteristic models and an Armstrong relation. The equivalent question of translating between functional dependencies and Armstrong relations has been studied before [BDFS84, MR86, EG91, GL90] and is relevant for the design of relational databases [MR86] While this paper does not discuss the problems in the database domain, some of the results presented here can be alternatively derived from previous results in database theory using the above mentioned equivalence. We identify those ....

....subset. It is shown that in this special case, SID, CCM, and DME are equivalent under polynomial reductions. Therefore, the algorithm presented by Fredman and Khachiyan [FK94] can be used to solve CCM, and SID in time n O(logn) This result can be alternatively derived from the discussion in [EG91] on functional dependencies in MAK form. We show however that our argument generalizes to the larger family of k quasi Horn expressions. The second relaxation is the problem of computing all the prime implicants for a given Horn expression. This is a relaxation of CCM since using the prime ....

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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.

....we consider some problems involving the generation of all subsets of a finite set satisfying certain conditions. The most well known problem of this type, the generation of all minimal transversals, has applications in combinatorics [29] graph theory [18, 20, 30, 32] artificial intelligence [10], game theory [13, 14, 28] reliability theory [8, 28] database theory [1, 24, 33] and learning theory [2] # The research of the first two authors was supported in part by the O#ce of Naval Research (Grant N00014 92 J 1375) the National Science Foundation (Grant DMS 98 06389) and DIMACS. The ....

....intriguing problems, the true complexity of which is not yet known. For many special classes of hypergraphs it can be solved e#ciently. For example, if the sizes of all the hyperedges of A are limited by a constant r, then dualization can be executed in incremental polynomial time, see e.g. [5, 10]) In the quadratic case, i.e. when r = 2, there are even more e#cient dualization algorithms that run with polynomial delay, i.e. in poly( V , A ) time, where B is systematically enlarged from B = # during the generation process of A d (see e.g. 18, 20, 32] E#cient algorithms ....

T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM Journal on Computing, 24 (1995) 1278-1304. 12

....we consider some problems involving the generation of all subsets of a finite set satisfying certain conditions. The most well known problem of this type, the generation of all minimal transversals, has applications in combinatorics [31] graph theory [20, 22, 32, 34] artificial intelligence [12], game theory [15, 16, 30] reliability theory [9, 30] database theory [1, 26, 35] and learning theory [2, 11] Given a finite set V of n = V points, and a hypergraph (set family) A # 2 V , a subset B # V is called a transversal of the family A if A # B #= # for all sets A # ....

....can be solved in time polynomial in V , A and B . The dualization problem can be e#ciently solved for many classes of hypergraphs. For example, if the sizes of all the hyperedges of A are limited by a constant r, then dualization can be executed in incremental polynomial time, see e.g. [6, 12]) In the quadratic case, i.e. when r = 2, there are even more e#cient dualization algorithms that run with polynomial delay, i.e. in poly( V , A ) time, where B is systematically enlarged from B = # during the generation process of A d (see e.g. 20, 22, 34] E#cient algorithms ....

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

....problem of identifying a general positive function by membership queries is equivalent to many other problems, in the sense that the former is solvable in polynomial time in n and m if and only if the latter problems are solvable in polynomial time. Among the many problems of this type (see e.g. [4, 12]) we mention the dualization of positive Boolean functions, the recognition of self dual positive functions, the recognition of saturated simple hypergraphs, and so forth. Although the exact complexity of these problems is still open, the recent result by Fredman and Khachiyan [13] shows that ....

T. Eiter and B. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, Technical Report CD-TR 91/16, Christial Doppler Labor fur Expertensysteme, Technische Universitat Wien, 1991.

....paper we consider some problems involving the generation of all subsets of a nite set satisfying certain conditions. The most well known problem of this type, the generation of all minimal transversals, has applications in combinatorics [33] graph theory [21, 23, 34, 36] arti cial intelligence [14], game theory [16, 17, 32] reliability theory [11, 32] database theory [1, 28, 37] and learning theory [2] Given a nite set V of n = jV j points, and a hypergraph (set family) A 2 V , a subset B V is called a transversal of the family A if B A 6= for all sets A 2 A; it is called a ....

....DUAL(A;B) can be solved in time polynomial in jV j; jAj and jBj. The dualization problem can be eciently solved for many classes of hypergraphs. For example, if the sizes of all the hyperedges of A are limited by a constant c, then problem DUAL(A;B) can be solved in polynomial time (see e.g. [7, 14]) In addition, for c = 2 there are dualization algorithms that run with polynomial delay, i.e. in poly(jV j; jAj) time for a speci c sequence ; B 1 B 2 : A d , see e.g. 21, 23, 36] Ecient algorithms exist also for the dualization of 2 monotonic, threshold, matroid, read bounded, ....

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

....DUAL(A;B) Given a hypergraph A and a collection B A d of minimal transversals to A, either nd a new minimal transversal B 2 A d n B or show that B = A d . The hypergraph dualization problem has applications in combinatorics [29] graph theory [19, 24, 30, 31] arti cial intelligence [13], game theory [17, 18, 28] reliability theory [10, 28] database theory [2, 6, 7, 27, 32] integer programming [6, 7] and learning theory [3] It is an open question whether problem DUAL(A;B) or equivalently MIS(A; I) can be solved in polynomial time for arbitrary hypergraphs. The fastest ....

....MIS(A; I) can be solved in polynomial time for arbitrary hypergraphs. The fastest currently known algorithm [14] for DUAL(A;B) is quasi polynomial and runs in time O(nm) m o(log m) 1 In fact, our reduction is in AC 0 2 where n = jV j and m = jAj jBj: However, as shown in [13, 5], for hypergraphs of bounded dimension problem DUAL(A;B) can be solved in polynomial time. Theorem 1 strengthens this result by implying that DUAL(A;B) 2 NC for dim(A) 3 and DUAL(A;B) 2 RNC for dim(A) 4; 5; As mentioned above, Theorem 1 is a corollary of Theorem 2 and the results of ....

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

....threshold can be listed with polynomial delay in terms of the size of the matrix, appears to be an important open problem that could also be of interest from the point of view of practical applications. The definitions of different efficiency criteria for listing algorithms are given in, e.g. [5, 7, 14]. In the rest of this section we mention some related listing problems, which are known to be easy or hard in some sense. Perhaps the central problem in the area is that of listing the maximal 0 vectors of a monotone disjunctive normal form, which is equivalent to finding the minimal hitting ....

....which are known to be easy or hard in some sense. Perhaps the central problem in the area is that of listing the maximal 0 vectors of a monotone disjunctive normal form, which is equivalent to finding the minimal hitting sets (or transversals) of a hypergraph and many other listing problems [3, 5, 21]. A recent important result of Fredman and Khachiyan [6] shows that this can be done with incremental quasipolynomial time. Listing the maximal 0 vectors of a monotone CNF can be done trivially in polynomial delay, as these are the complements of the minimal characteristic vectors of the clauses. ....

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

....paper we consider some problems involving the generation of all subsets of a nite set satisfying certain conditions. The most well known problem of this type, the generation of all minimal transversals, has applications in combinatorics [29] graph theory [18, 20, 30, 32] arti cial intelligence [10], game theory [13, 14, 28] reliability theory [8, 28] database theory [1, 24, 33] and learning theory [2] Given a nite set V of n = jV j points, and a hypergraph (set family) A 2 V , a subset B V is called a transversal of the family A if A B 6= for all sets A 2 A; it is called a ....

....can be solved in time polynomial in jV j; jAj and jBj. The dualization problem can be eciently solved for many classes of hypergraphs. For example, if the sizes of all the hyperedges of A are limited by a constant r, then dualization can be executed in incremental polynomial time, see e.g. [6, 10]) In the quadratic case, i.e. when r = 2, there are even more ecient dualization algorithms that run with polynomial delay, i.e. in poly(jV j; jAj) time, where B is systematically enlarged from B = during the generation process of A d (see e.g. 18, 20, 32] Ecient algorithms exist also for ....

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

....Unfortunately, even this possibility can be ruled out (conditionally) Many enumeration problems arising in Artificial Intelligence and Computer Science have the property that no output polynomial algorithm is known for them. A prototypical example is the hypergraph transversal problem [ Eiter and Gottlob, 1995 ] As it turns out, a considerable number of these enumeration problems can be classified as being TRANSVERSAL hard, which means that any output polynomial algorithm for those problems would lead to an outputpolynomial problem of the hypergraph transversal problem [ Eiter and Gottlob, 1995 ] ....

.... [ Eiter and Gottlob, 1995 ] As it turns out, a considerable number of these enumeration problems can be classified as being TRANSVERSAL hard, which means that any output polynomial algorithm for those problems would lead to an outputpolynomial problem of the hypergraph transversal problem [ Eiter and Gottlob, 1995 ] Now, generating a revised base under the full meet base revision scheme for Horn logic is one of those problems. Theorem 8.5 ( Gogic et al. 1994] Generating a revised base under the full meet base revision scheme for Horn logic is TRANSVERSAL hard. As a positive result, Gogic et al. ....

Thomas Eiter and Georg Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):421--457, 1995.

....according to a certain criterion, such as availability and load (equivalently, construct an optimal positive self dual function) and (iii) generate all ND coteries (equivalently, all positive self dual functions) systematically. Unfortunately, the complexity of question (i) is still unknown [8, 16, 22], although a result by M. Fredman and L. Khachiyan [17] suggests that it is unlikely that the problem is NP hard. References [8, 16] give a 1 The ith component of the characteristic vector is 1 (0) if i 2 U is (not) contained in the subset. 2 This de nition was motivated by the de nition of ....

....and (iii) generate all ND coteries (equivalently, all positive self dual functions) systematically. Unfortunately, the complexity of question (i) is still unknown [8, 16, 22] although a result by M. Fredman and L. Khachiyan [17] suggests that it is unlikely that the problem is NP hard. References [8, 16] give a 1 The ith component of the characteristic vector is 1 (0) if i 2 U is (not) contained in the subset. 2 This de nition was motivated by the de nition of regular Boolean functions. See Section 2.3. 2 number of interesting equivalent problems, which arise in various elds of ....

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

.... Sigmag. 25 Theorem 5.11 There exists a polynomial total time algorithm for the FD Inference problem when a Horn theory is represented as the Horn envelope of a set of models if and only if there exists a polynomial total time algorithm for dualizing a positive theory. Proof. It is known [13] that the problem of dualizing a positive theory is equivalent to the problem of generating all minimal solutions y subject to Ay e; and y j 2 f0; 1g for all j = 1; 2; m; 14) where A denotes a 0 1 l Theta m matrix (i.e. a ij 2 f0; 1g for all i and j) and e is the l dimensional column ....

....This proves the if part. 26 The problem of dualizing a positive theory has been well studied. Although it is still unknown whether a polynomial total time algorithm exists for this problem, a wide variety of computationally equivalent problems have been discovered (see e.g. [3, 13]) and a quasipolynomial algorithm has been developed (see [16] 6 Condensation of Horn Theories The procedure of condensation introduced in Subsection 2.3 aims at simplifying a given theory by eliminating variables that are functionally dependent on other variables in the theory. In the case of ....

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

....integer ff bounded by a constant, then there is a polynomial total time algorithm for dualizing a positive Boolean function. Here, the problem of dualizing a positive Boolean function asks to output maxF (f) from minT (f) of a given positive function f . This problem has been intensively studied [4, 9, 12, 13], and there is an O(m o(log m) time algorithm, where m = j minT (f)j j maxF (f)j, found by M. L. Fredman and L. Khachiyan [10] In this paper we rst strengthen the result (IV) and show that, even if ff is bounded by a constant, there is no polynomial total time algorithm for problem ....

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

....of O(n 2 ) where n is the number of items being clustered. The use of hypergraphs in data mining has been studied recently. For example in [GKMT97] it is shown that the problem of finding maximal elements in a lattice of patterns is closely related to the hypergraph transversal problem [EG95] The clustering or grouping of association rules have been proposed in [TKR 95] LSW97] and [KA96] In [TKR 95] and [LSW97] the focus is on finding clusters of association rules that have the same right hand side rather than on finding item clusters. In [KA96] a scheme is proposed to ....

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

....which it describes. An immediate corollary from the equivalence shown here, and recent results on characteristic models [Kha95] is that the complexity of the two translation problems is equivalent under polynomial reductions. While the complexity of these problem is an open problem [MR86, EG94, FK94, Kha95] we show that results from the theory of reasoning with models [KR94b] and computational learning theory [Bsh93] can be used to derive some positive results. In particular, we show that a closed form (although not in the form of functional dependencies) for a given relation can be ....

....The task of the design tool is therefore to translate from relations to dependencies and vice versa. Unfortunately, no polynomial time algorithm for these tasks have been found, even for the restricted case of functional dependencies. In fact, the complexity of the problem has been studied [MR86, EG94, FK94, Kha95] but is still an open problem. In general, these problems are at least as hard as the hypergraph transversal problem. In [EG94, Kha95] it is shown that certain special cases are equivalent to the latter, and therefore using the algorithm in [FK94] these can be solved in ....

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T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal of Computing, 1994. To appear.

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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, ....

[Article contains additional citation context not shown here]

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.

....functions. Analogously, the decision problem TRANS HYP associated with TRANSVERSAL COMPUTATION 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, 28, 29, 44] mathematical programming (e.g. 5] and distributed systems (e.g. 18, 27] to ....

....combined size of and . Any such algorithm for DUALIZATION (or for TRANSVERSAL COMPUTATION) would 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 2 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 ....

[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.

....related problems and applications in different fields of computer science. For space reasons the exposition is rather succinct. In particular, we omit several formal definitions and all proofs. A full discussion, exact definitions, all proofs, and more material can be found in the extended report [15]; other interesting problems related to TRANS HYP have recently been studied in [32] MINIMAL TRANSVERSALS OF A HYPERGRAPH 1299 6.1. Clause satisfiability. We have identified the following restrictions of the wellknown SATISFIABILITY problem which are p m equivalent to TRANS HYP (resp. ....

....prime implicants from a function table or a logical expression has been topic of research over decades. It is well known that every monotone Boolean function has a unique minimal conjunctive normal form consisting of a conjunction of disjunctions of atoms where no conjunct subsumes any other. In [15] we show the following results. Given a monotone Boolean expression E and a set P of prime implicants of E, it is NP complete to determine whether there exists an additional prime implicant of E. However, if E is in minimal conjunctive normal form, then the same problem is p m equivalent to ....

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

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 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.

No context found.

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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