Tight Inapproximability of Target Set Reconfiguration

Given a graph G with a vertex threshold function τ, consider a dynamic process in which any inactive vertex v becomes activated whenever at least τ(v) of its neighbors are activated. A vertex set S is called a target set if all vertices of G would be activated when initially activating vertices of S. In the Minmax Target Set Reconfiguration problem, for a graph G and its two target sets X and Y, we wish to transform X into Y by repeatedly adding or removing a single vertex, using only target sets of G, so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of 2-o(1/polylog n), where n is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a 2-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016).