Parameterized Inapproximability of Degree Anonymization

The Degree Anonymity problem arises in the context ofcombinatorial graph anonymization. It asks, given a graph G and twointegers k and s, whether G can be made k-anonymous with at most smodifications. Here, a graph is k-anonymous if the graph contains for everyvertex at least k − 1 other vertices of the same degree. Complementingrecent investigations on its computational complexity, we show that thisproblem is very hard when studied from the viewpoints of approximationas well as parameterized approximation. In particular, for the optimizationvariant where one wants to minimize the number of either edge or vertexdeletions there is no factor-n1−ε approximation running in polynomialtime unless P = NP, for any constant 0 < ε ≤ 1. For the variant where onewants to maximize k and the number s of either edge or vertex deletions isgiven, there is no factor-n1/2−ε approximation running in time f(s) ·nunless W[1] = FPT, for any constant 0 < ε ≤ 1/2 and any function f .On the positive side, we classify the general decision version as fixed-parameter tractable with respect to the combined parameter solutionsize s and maximum degree.