E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In H. Alt and A. Ferreira, editors, STACS 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....calculations (e.g. #AC ) furthermore, these classes enclose problems with highly parallelizable algorithmic structure. For further details, see [25] 9 3 Related work and contributions As far as we know, some computational complexity results pertaining to association rules were presented in [15, 19, 20, 26, 27, 8]. We briefly survey the results presented in these works and then pinpoint relationships with this paper. In [15] is stated the NP completeness of the problem T , sup, k, s# on boolean databases, therein called 0 1 relations. This is done through reducing the Balanced Bipartite Clique problem ....

....of a bipartite graph. In [19] and [20] an NP hardness result is stated regarding the induction of boolean association rules (or, in general, of conditions) having an optimal entropy or chi square, although entropy and chi square are indices that we do not consider in this work. The authors of [8] dealt with the complexity of computing all the maximal frequent sets and all the minimal infrequent sets in a boolean database. Given a set of boolean attributes I , a database T on I , and a threshold t (1 # T ) a subset X of I is said to be frequent, if T C(X) # t, while is said to ....

E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In STACS: Annual Symposium on Theoretical Aspects of Computer Science, pages 134-- 141. Springer, 2002.

....X Y where Xand Y are subsets of R. The most popular interestingness measure for an association rule X Y is its accuracy (or confidence) which is defined as acc(X Y, d) fr(X, d) Also several other classes of patterns and measures of interestingness have been studied (see e.g. [4, 6, 9, 13, 14, 15, 27, 32, 33, 36, 37, 43, 44, 47, 48]) It is not always easy to define an interestingness measure # in such a way that there would be a threshold value # such that #(p) # for almost all interesting patterns p and for only very few uninteresting ones. One way to augment the interestingness measure is to define additional ....

....sets is achieved with one discretization point. Then the closed frequent sets are the frequent sets that have no frequent supersets. They determine the set of frequent sets and are called maximal frequent sets. Because of their structural importance, the maximal sets have been studied extensively [3, 6, 21, 22, 23, 36]. The maximum error is obviously minimized by choosing the value of the one discretization point to be the average of the minimum and the maximum frequencies. We computed the discretizations for several di#erent minimum frequency thresholds and in all cases the number of closed frequent sets ....

E. Boros, V. Gurvich, L. Khachiyan, and K. Makino, On the complexity of generating maximal frequent and minimal infrequent sets, in STACS 2002.

....rules, correlations, and other tasks. Of particular interest are the maximal frequent sets M t F t and the minimal infrequent sets I t F t , since they mark the boundary of frequent sets (both maximal and minimal under set inclusion) The following result has been recently proved in [6]. Theorem 6. The problem of computing, given a 0 1 matrix A and a threshold t, the sets M t and I t is TRANS ENUM complete. A related but di#erent task in data mining is dependency inference. Here the problem is, given a (not necessarily 0 1) m n matrix A, to find the dependencies C i 1 C i ....

E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In Proc. STACS-02, LNCS 2285, pp. 133--141, 2002.

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E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In Proc. 19th Annual Symposium on Theoretical Aspects of Computer Science (STACS-02), LNCS 2285, pp. 133--141, 2002.

....turn out to be uniformly dual bounded. In the next section we survey several inequalities proving uniformly dual boundedness for a number of hypergraphs (see [10, 12, 14, 17] Finally, in Section 3 we discuss several applications of the cited results and as well as some generalizations (see [10, 12, 13, 15, 16, 17, 28, 29]) 2 Classes of Uniformly Dual Bounded Hypergraphs All monotone properties considered in this paper, can be described in terms a system of monotone inequalities t i , i = 1, r, 2) over the subsets X V (or more generally, over elements of products of partially ordered sets) where ....

....it follows that X , F , where max is the hypergraph with largest size. As it was shown in [17] the bound of Theorem 4 is sharp within a polylogarithmic factor. A similar reverse inequality does not hold, in general, and the incremental generation of is NP hard [16]. 3 Applications In this section we give some specific examples of the four classes of functions described in the previous section. We then conclude by an application in data mining, and a generalization of it over products of partially ordered sets. 3.1 Matroids Several examples of ....

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E. Boros, V. Gurvich, L. Khachiyan and K. Makino, On the complexity of generating maximal frequent and minimal infrequent sets, Proc. 19th International Symposium on Theoretical Aspects of Computer Science, STACS 2002.

....solutions of the polymatroid inequality f(x) jDj t 1. Consequently, 13 for products of lattices with bounded width, this set can be generated in incremental quasipolynomial time by Theorem 5. The special case of the above result for databases D of binary attributes can be found in [5, 6]. We remark nally that many other examples of polymatroid functions and systems in the Boolean case can be found in [11] and [4] ....

E. Boros, V. Gurvich, L. Khachiyan and K. Makino, On the complexity of generating maximal frequent and minimal infrequent sets, in 19th Int. Symp. on Theoretical Aspects of Computer Science, (STACS), March 2002, LNCS 2285, pp. 133-141.

.... 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] to mention a few. While the output CNF # can be exponential in the size of #, it is currently not known whether # can be computed ....

E. Boros, V. Gurvich, L. Khachiyan and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. to appear in Proceedings of 19th International Symposium on Theoretical Aspects of Computer Science (STACS 2002).

....feasible solutions of the polymatroid inequality f(x) D t 1. Consequently, for products of lattices with bounded width, this set can be generated in incremental quasi polynomial time by Theorem 5. The special case of the above result for databases of binary attributes can be found in [5, 6]. # We remark finally that many other examples of polymatroid functions and systems in the Boolean case can be found in [11] and [4] 14 ....

E. Boros, V. Gurvich, L. Khachiyan and K. Makino, On the complexity of generating maximal frequent and minimal infrequent sets, in 19th Int. Symp. on Theoretical Aspects of Computer Science, (STACS), March 2002, LNCS 2285, pp. 133--141.

....maximal frequent sets can be bounded by a quasi polynomial in the number of minimal infrequent sets and the sizes of V, and that the minimal infrequent sets can be generated in quasi polynomial time. In fact, the bound of Theorem 4 can be strengthened to a sharp linear bound in this case, see [7]. Connectivity ensuring collections of subgraphs: Let R be a finite set of r vertices and let E 1 , E n R R be a collection of n graphs on R. Given a . n define k(X) to be the number of connected components in the graph (R, i#X E i ) Then k(X) is an anti monotone ....

....size of can be bounded by a log t degree polynomial in n and t (f) and thus all sets in t (f) can be enumerated in incremental quasi polynomial time. It is worth mentioning that in all of the above examples, generating all maximal infeasible sets for (1) turns out to be NP hard, see [7, 11, 15]. 4 Proof of Theorem 2 In this Section we show that Theorem 2 follows from Theorem 4. be the set of minimal feasible sets for (1) Clearly, we can incrementally generate all sets in by initializing and then iteratively solving problem GEN(B,H) a number of B 1 times. It is easy ....

E. Boros, V. Gurvich, L. Khachiyan and K. Makino, On the complexity of generating maximal frequent and minimal infrequent sets, in STACS 2002, LNCS 2285, pp. 133-141.

.... 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 mention a few. While the output CNF can be exponential in the size of , it is currently not known whether can be computed in ....

E. Boros, V. Gurvich, L. Khachiyan and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In: Proc. 19th International Symposium on retical Aspects of Computer Science (STACS), pp. 133--141, Springer LNCS 2285, 2002.

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E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In H. Alt and A. Ferreira, editors, STACS 2002.

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Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of Lecture Notes in Computer Science., Springer-Verlag (2002) 133--141

No context found.

E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In H. Alt and A. Ferreira, editors, STACS 2002.

No context found.

Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of Lecture Notes in Computer Science., Springer-Verlag (2002) 133--141

No context found.

Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of LNCS., Springer-Verlag (2002) 133--141

No context found.

Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of LNCS., Springer-Verlag (2002) 133--141

No context found.

Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of LNCS., Springer-Verlag (2002) 133--141

No context found.

Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of LNCS., Springer-Verlag (2002) 133--141

No context found.

Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of Lecture Notes in Computer Science., Springer-Verlag (2002) 133--141

No context found.

Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In Alt, H., Ferreira, A., eds.: STACS 2002. Volume 2285 of Lecture Notes in Computer Science., Springer-Verlag (2002) 133--141

No context found.

E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In H. Alt and A. Ferreira, editors, STACS 2002.

No context found.

E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In H. Alt and A. Ferreira, editors, STACS 2002.

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