Finding large degree-anonymous subgraphs is hard

A graph is said to be k-anonymous for an integer k, if for every vertex in the graph there are at least k - 1 other vertices with the same degree. We examine the computational complexity of making a given undirected graph k-anonymous either through at most s vertex deletions or through at most s edge deletions; the corresponding problem variants are denoted by ANONYM V-DEL and ANONYM E-DEL. We present a variety of hardness results, most of them hold for both problems. The two variants are intractable from the parameterized as well as from the approximation point of view. In particular, we show that both variants remain NP-hard on very restricted graph classes like trees even if k = 2. We further prove that both variants are W[1]-hard with respect to the combined parameter solutions size s and anonymity level k. With respect to approximability, we obtain hardness results showing that neither variant can be approximated in polynomial time within a factor better than n(1/2) (unless P = NP). Furthermore, for the optimization variants where the solution size s is given and the task is to maximize the anonymity level k, this inapproximability result even holds if we allow a running time of f(s) . n(O(1)) for any computable function f. On the positive side, we classify both problem variants as fixed-parameter tractable with respect to the combined parameter solution size s and maximum degree Delta. (C) 2016 Elsevier B.V. All rights reserved.