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 continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree. We take a graph theoretic view of deletion correcting codes. The problem of finding an n-bit s-deletion correcting code is equivalent to finding an independent set in a particular graph. We discuss the relationship between codes and colorings and demonstrate that the VT codes are optimal in a coloring sense. We describe a method of partitioning the set of bit strings by Hamming weight and finding codes within each partition. In the single deletion case, we find an optimal coloring of the constant Hamming weight induced subgraphs. We show that the resulting code is asymptotically optimal. We also prove a lower bound on size of codes constructed using these partitions for any number of deletions. Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer @math and arbitrary small @math , the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor @math even on graphs where the vertices can be almost partitioned into @math solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the "It Ain't Over Till It's Over" theorem) than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is NP-hard to approximate within any constant assuming the Unique Games Conjecture. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations. We study deterministic, distributed algorithms for two weak variants of the standard graph coloring problem. We consider defective colorings, i.e., colorings where nodes of a color class may induce a graph of maximum degree d for some parameter d>0. We also look at colorings where a minimum number of multi-chromatic edges is required. For an integer k>0, we call a coloring k-partially proper if every node v has at least min k,deg(v) neighbors with a different color. We show that for all d∈ 1,...,Δ , it is possible to compute a O(Δ2 d2)-coloring with defect d in time O(log*n) where Δ is the largest degree of the network graph. Similarly, for all k∈ 1,...,Δ , a k-partially proper O(k2)-coloring can be computed in O(log*n) rounds. As an application of our weak defective coloring algorithm, we obtain a faster deterministic algorithm for the standard vertex coloring problem on graphs with moderate degrees. We show that in time O(Δ+log*n), a (Δ+1)-coloring can be computed, a task for which the best previous algorithm required time O(Δ*log(Δ) + log*n). The same result holds for the problem of computing a maximal independent set.