Parameterized Inapproximability of Degree Anonymization

Abstract

The Degree Anonymity  problem arises in the context of combinatorial graph anonymization. It asks, given a graph $G$ and two integers $k$ and $s$, whether $G$ can be made k-anonymous with at most $s$ modifications. Here, a graph is k-anonymous if the graph contains for every vertex at least $k-1$ other vertices of the same degree. Complementing recent investigations on its computational complexity, we show that this problem is very hard when studied from the viewpoints of approximation as well as parameterized approximation. In particular, for the optimization variant where one wants to minimize the number of either edge or vertex deletions there is no factor-$n^{1-\varepsilon }$ approximation running in polynomial time unless P = NP, for any constant $0 < \varepsilon \le 1$. For the variant where one wants to maximize $k$ and the number $s$ of either edge or vertex deletions is given, there is no factor-$n^{{1}/{2}-\varepsilon }$ approximation running in time $f(s) \cdot n^{O(1)}$ unless W[1] = FPT, for any constant $0 < \varepsilon \le {1}/{2}$ and any function $f$. On the positive side, we classify the general decision version as fixed-parameter tractable with respect to the combined parameter solution size $s$ and maximum degree.