In the Minmax Set Cover Reconfiguration problem, given a set system $\mathcal{F}$ over a universe and its two covers $\mathcal{C}^\mathsf{start}$ and $\mathcal{C}^\mathsf{goal}$ of size $k$, we wish to transform $\mathcal{C}^\mathsf{start}$ into $\mathcal{C}^\mathsf{goal}$ by repeatedly adding or removing a single set of $\mathcal{F}$ while covering the universe in any intermediate state. Then, the objective is to minimize the maximize size of any intermediate cover during transformation. We prove that Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguration are $\mathsf{PSPACE}$-hard to approximate within a factor of $2-\frac{1}{\operatorname{polyloglog} N}$, where $N$ is the size of the universe and the number of vertices in a graph, respectively, improving upon Ohsaka (SODA 2024) and Karthik C. S. and Manurangsi (2023). This is the first result that exhibits a sharp threshold for the approximation factor of any reconfiguration problem because both problems admit a $2$-factor approximation algorithm as per Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno (Theor. Comput. Sci., 2011). Our proof is based on a reconfiguration analogue of the FGLSS reduction from Probabilistically Checkable Reconfiguration Proofs of Hirahara and Ohsaka (2024). We also prove that for any constant $\varepsilon \in (0,1)$, Minmax Hypergraph Vertex Cover Reconfiguration on $\operatorname{poly}(\varepsilon^{-1})$-uniform hypergraphs is $\mathsf{PSPACE}$-hard to approximate within a factor of $2-\varepsilon$.

翻译：在最小最大集合覆盖重构问题中，给定一个集合系统$\mathcal{F}$（定义在全集上）及其两个大小为$k$的覆盖$\mathcal{C}^\mathsf{start}$和$\mathcal{C}^\mathsf{goal}$，我们希望通过反复添加或删除$\mathcal{F}$中的单个集合，将$\mathcal{C}^\mathsf{start}$转换为$\mathcal{C}^\mathsf{goal}$，且在任意中间状态下覆盖全集。此时，目标是最小化中间覆盖的最大规模。我们证明了最小最大集合覆盖重构问题与最小最大支配集重构问题是$\mathsf{PSPACE}$-难的，即使在近似因子为$2-\frac{1}{\operatorname{polyloglog} N}$的情况下仍成立，其中$N$分别为全集大小和图的顶点数，这一结果改进了Ohsaka (SODA 2024)以及Karthik C. S.和Manurangsi (2023)的研究。这是首个展示重构问题近似因子尖锐阈值的结果，因为根据Ito、Demaine、Harvey、Papadimitriou、Sideri、Uehara和Uno (Theor. Comput. Sci., 2011)的研究，这两个问题均存在$2$因子近似算法。我们的证明基于Hirahara和Ohsaka (2024)的概率可校验重构证明的FGLSS归约的推广形式。此外，我们证明：对于任意常数$\varepsilon \in (0,1)$，在$\operatorname{poly}(\varepsilon^{-1})$-均匀超图上，最小最大超图顶点覆盖重构问题在近似因子为$2-\varepsilon$时也是$\mathsf{PSPACE}$-难的。