Privacy-preserving data splitting is a technique that aims to protect data privacy by storing different fragments of data in different locations. In this work we give a new combinatorial formulation to the data splitting problem. We see the data splitting problem as a purely combinatorial problem, in which we have to split data attributes into different fragments in a way that satisfies certain combinatorial properties derived from processing and privacy constraints. Using this formulation, we develop new combinatorial and algebraic techniques to obtain solutions to the data splitting problem. We present an algebraic method which builds an optimal data splitting solution by using Gr " o bner bases. Since this method is not efficient in general, we also develop a greedy algorithm for finding solutions that are not necessarily minimal sized.
The optimal decomposition problem described in @cite_8 is hard to solve even if vertex deletion is not allowed. In fact, @cite_13 proved that it is NP-hard to color a 2-colorable, 4-uniform hypergraph using only @math colors for any constant @math . This means that, in the case that all 4-tuples of attributes are sensitive, it is NP-hard to find a partition of attributes into two sets that satisfies all privacy constraints, even knowing that it exists. Because of the hardness of this problem, in @cite_8 they present three different heuristics to solve it.
@cite_13 @cite_8
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We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCPs). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the proof or not, based on the given random string. We define the covering complexity of such a verifier, on a given input, to be the minimum number of proofs needed to "satisfy" the verifier on every random string; i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems and, in particular, (hyper)graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple not-all-equal check on the four bits it reads. This enables us to prove that, for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors and also yields a superconstant inapproximability result under a stronger hardness assumption. Recent trends towards database outsourcing, as well as concerns and laws governing data privacy, have led to great interest in enabling secure database services. Previous approaches to enabling such a service have been based on data encryption, causing a large overhead in query processing. We propose a new, distributed architecture that allows an organization to outsource its data management to two untrusted servers while preserving data privacy. We show how the presence of two servers enables efficient partitioning of data so that the contents at any one server are guaranteed not to breach data privacy. We show how to optimize and execute queries in this architecture, and discuss new challenges that emerge in designing the database schema.