Association Rule Hiding Based on Intersection Lattice

Association rule hiding has been playing a vital role in sensitive knowledge preservation when sharing data between enterprises. The aim of association rule hiding is to remove sensitive association rules from the released database such that side effects are reduced as low as possible. This research proposes an efficient algorithm for hiding a specified set of sensitive association rules based on intersection lattice of frequent itemsets. In this research, we begin by analyzing the theory of the intersection lattice of frequent itemsets and the applicability of this theory into association rule hiding problem. We then formulate two heuristics in order to (a) specify the victim items based on the characteristics of the intersection lattice of frequent itemsets and (b) identify transactions for data sanitization based on the weight of transactions. Next, we propose a new algorithm for hiding a specific set of sensitive association rules with minimum side effects and low complexity. Finally, experiments were carried out to clarify the efficiency of the proposed approach. Our results showed that the proposed algorithm, AARHIL, achieved minimum side effects and CPU-Time when compared to current similar state of the art approaches in the context of hiding a specified set of sensitive association rules.

The Face Semigroup Algebra of a Hyperplane Arrangement

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete systemof primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposablemodules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

Geometrically Constructed Bases for Homology of Non-Crossing Partition Lattices

For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Björner and Wachs using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$.

Affine and toric arrangements

International audience We extend the Billera―Ehrenborg―Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky's fundamental results on the number of regions. Nous étendons l'opérateur de Billera―Ehrenborg―Readdy entre le trellis d'intersection et la treillis de faces d'un arrangement hyperplans centraux aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.

The Dimension of the Cohomology Groups of the Orlik–Solomon Algebras

The Orlik–Solomon algebra is a graded algebra defined by the partially ordered set of subspace intersections of the hyperplanes in an arrangement. Define the cohomology of an Orlik–Solomon algebra as that of the complex formed by its homogeneous components with the differential defined via multiplication by an element of degree one. The dimension of the cohomology of the Orlik–Solomon algebra in dimension one has been determined by Libgöber and Yuzvinsky. Using similar techniques, we study the dimension of the cohomology groups of the Orlik–Solomon algebra in higher dimensions under the special case where the element of degree one which defines the multiplication is concentrated under an element of the intersection lattice of codimension two. We provide computational methods for the dimension of the second cohomology group.