The development of novel platforms and techniques for emerging "Big Data" applications requires the availability of real-life datasets for data-driven experiments, which are however out of reach for academic research in most cases as they are typically proprietary. A possible solution is to use synthesized datasets that reflect patterns of real ones in order to ensure high quality experimental findings. A first step in this direction is to use inverse mining techniques such as inverse frequent itemset mining (IFM) that consists of generating a transactional database satisfying given support constraints on the itemsets in an input set, that are typically the frequent ones. This paper introduces an extension of IFM, called many-sorted IFM, where the schemes for the datasets to be generated are those typical of Big Tables as required in emerging big data applications, e.g., social network analytics.
A simple solution to exclude unexpected frequent itemset from a feasible solution is the formulation proposed in @cite_13 , which is called @math : only itemsets in @math can be included as transactions in @math . The decision complexity of this problem is @math -complete as stated in @cite_13 and proved in the Appendix. The version of with infrequency support constraint ( @math for short), has been recently proposed in @cite_8 and its decision complexity is @math -complete as proven in @cite_15 .
@cite_15 @cite_13 @cite_8
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We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Inverse frequent set mining (IFM) is the problem of computing a transaction database D satisfying given support constraints for some itemsets, which are typically the frequent ones. This article proposes a new formulation of IFM, called IFMI (IFM with infrequency constraints), where the itemsets that are not listed as frequent are constrained to be infrequent; that is, they must have a support less than or equal to a specified unique threshold. An instance of IFMI can be seen as an instance of the original IFM by making explicit the infrequency constraints for the minimal infrequent itemsets, corresponding to the so-called negative generator border defined in the literature. The complexity increase from PSPACE (complexity of IFM) to NEXP (complexity of IFMI) is caused by the cardinality of the negative generator border, which can be exponential in the original input size. Therefore, the article introduces a specific problem parameter  that computes an upper bound to this cardinality using a hypergraph interpretation for which minimal infrequent itemsets correspond to minimal transversals. By fixing a constant k, the article formulates a k-bounded definition of the problem, called k-IFMI, that collects all instances for which the value of the parameter  is less than or equal to kits complexity is in PSPACE as for IFM. The bounded problem is encoded as an integer linear program with a large number of variables (actually exponential w.r.t. the number of constraints), which is thereafter approximated by relaxing integer constraintsthe decision problem of solving the linear program is proven to be in NP. In order to solve the linear program, a column generation technique is used that is A complete set of filters @math for the optimal-depth @math -input sorting network problem is such that if there exists an @math -input sorting network of depth @math then there exists one of the form @math for some @math . Previous work on the topic presents a method for finding complete set of filters @math and @math that consists only of networks of depths one and two respectively, whose outputs are minimal and representative up to permutation and reflection. We present a novel practical approach for finding a complete set of filters @math containing only networks of depth three whose outputs are minimal and representative up to permutation and reflection. In previous work, we have developed a highly efficient algorithm for finding extremal sets ( i.e. outputs of comparator networks; itemsets; ) up to permutation. In this paper we present a modification to this algorithm that identifies the representative itemsets up to permutation and reflection. Hence, the here presented practical approach is the successful combination of known theory and practice that we apply to the domain of sorting networks. For all @math , we empirically compute the complete set of filters @math of the representative minimal up to permutation and reflection @math -input networks of depth three.