A Note on the Computational Complexity of Unsmoothened Vertex Attack Tolerance

We have previously introduced vertex attack tolerance (VAT) and unsmoothened VAT (UVAT), denoted respectively as τ(G) = minS⊂V |S| |V−S−Cmax(V−S)|+1 and τ̂(G) = minS⊂V |S| |V−S−Cmax(V−S)| , where Cmax(V − S) is the largest connected component in V − S, as appropriate mathematical measures of resilience in the face of targeted node attacks for arbitrary degree networks. Here we prove the hardness of approximating τ̂ under various plausible computational complexity hypotheses. 1. Definitions and Preliminaries Given a connected, undirected graph G = (V,E), the Vertex Attack Tolerance of G is denoted by τ(G) defined as follows:[3, 7, 2] τ(G) = min S⊂V,S 6=∅ { |S| |V − S −Cmax(V − S)|+ 1 } where Cmax(V −S) is the largest connected component in V −S. As in [2], we refer to connected, undirected graphs G = (V,E) with more than one node (|V | ≥ 2) as non-trivial. Remark 1.1. [2] For nontrivial G = (V,E), 0 < τ(G) ≤ 1. VAT was originally introduced as τ̂ (UVAT for “unsmoothened VAT”), of which τ is a smoothened variation, defined as follows[3, 7]: τ̂ = min S⊂V,S 6=∅ { |S| |V − S − Cmax(V − S)| } where Cmax(V −S) is the largest connected component in V −S. Note that for any graph G = (V,E) such that G is not a clique, the pair of nodes u, v Preprint submitted to arXiv March 29, 2016