目标集重构的紧不可近似性
Tight Inapproximability of Target Set Reconfiguration
AI总结：
该研究证明Minmax目标集重构问题难以在2-o(1/polylog n)因子内近似，确立其近似性紧下界，基于间隙保持归约，而该问题存在简单2因子近似算法。
摘要：
给定一个带有顶点阈值函数τ的图G，考虑这样一个动态过程：当非活跃顶点v的邻居中至少有τ(v)个被激活时，v就会变为活跃顶点。顶点集S被称为目标集，指的是当初始仅激活S中的顶点时，G的所有顶点最终都会被激活。在Minmax目标集重构问题中，对于图G及其一对目标集X和Y，我们希望通过反复添加或移除单个顶点，且仅使用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，在顶点数为n的情况下，要以2-o(1/polylog n)的因子近似Minmax目标集重构问题是NP难的。我们的结果确立了该问题近似性的紧下界，而该问题存在一个简单的2因子近似算法。该证明基于从目标集选择问题到Minmax目标集重构问题的间隙保持归约，其中前者问题的近似NP难性由Chen（SIDMA 2009）以及Charikar、Naamad和Wirth（APPROX/RANDOM 2016）证明。
英文摘要：
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 have been activated. A vertex set $S$ is called a target set if all vertices of $G$ would eventually be activated when initially activating exactly the vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and a pair of its 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 $\mathbf{NP}$-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 simple $2$-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where $\mathbf{NP}$-hardness of approximation for the former problem is proven by Chen (SIDMA 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
