Finding large degree-anonymous subgraphs is hard

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 n12 (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)⋅nO(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 Δ.