Given a graph $G$ with a vertex threshold function $\tau$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $\tau(v)$ 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 $2-o\left(\frac{1}{\operatorname{polylog} n}\right)$, 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).

翻译：给定一个具有顶点阈值函数$\tau$的图$G$，考虑一个动态过程：当任意非活跃顶点$v$的至少$\tau(v)$个邻居被激活时，$v$变为活跃状态。顶点集$S$被称为目标集，如果初始激活$S$中的顶点后，图$G$的所有顶点最终都会被激活。在最小化最大目标集重构问题中，对于图$G$及其两个目标集$X$和$Y$，我们希望通过反复添加或删除单个顶点，仅使用$G$的目标集将$X$转换为$Y$，以最小化任何中间目标集的最大规模。我们证明，在因子$2-o\left(\frac{1}{\operatorname{polylog} n}\right)$内逼近最小化最大目标集重构是NP困难的，其中$n$是顶点数。该结果建立了最小化最大目标集重构可逼近性的紧致下界，而该问题本身存在一个2因子逼近算法。证明基于从目标集选择到最小化最大目标集重构的保间隙归约，其中前者问题的逼近NP困难性由Chen（SIAM J. Discrete Math., 2009）以及Charikar、Naamad和Wirth（APPROX/RANDOM 2016）证明。