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 reformulation of in terms of frequencies instead of supports has been introduced in @cite_3 @cite_6 with the name @math . The two problems are equivalent and have been shown to be in @math and @math -hard. The basic version of the frequency formulation, called , does not fix the number @math of transaction in a feasible database -- the corresponding decision problem has been proved to be @math -complete. A further variant of the problem has been introduced in @cite_3 @cite_6 with the name @math : all itemsets may occur as transactions in @math at most a fixed number of times ( @math ). This problem is in @math and @math -hard.
@cite_6 @cite_3
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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 . We consider the @math -Center problem and some generalizations. For @math -Center a set of @math center vertices needs to be found in a graph @math with edge lengths, such that the distance from any vertex of @math to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any @math computing a @math -approximation is W[2]-hard for parameter @math and NP-hard for graphs with highway dimension @math . The latter does not rule out fixed-parameter @math -approximations for the highway dimension parameter @math , but implies that such an algorithm must have at least doubly exponential running time in @math if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of @math by combining the parameters @math and @math : we develop a fixed-parameter @math -approximation with running time @math . Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter @math -approximations for only the parameter @math . We also provide similar fixed-parameter approximations for the weighted @math -Center and @math -Partition problems, which generalize @math -Center.