In this paper, we demonstrate gap amplification for reconfiguration problems. In particular, we prove an explicit factor of PSPACE-hardness of approximation for three popular reconfiguration problems only assuming the Reconfiguration Inapproximability Hypothesis (RIH) due to Ohsaka (STACS 2023). Our main result is that under RIH, Maxmin Binary CSP Reconfiguration is PSPACE-hard to approximate within a factor of $0.9942$. Moreover, the same result holds even if the constraint graph is restricted to $(d,\lambda)$-expander for arbitrarily small $\frac{\lambda}{d}$. The crux of its proof is an alteration of the gap amplification technique due to Dinur (J. ACM, 2007), which amplifies the $1$ vs. $1-\epsilon$ gap for arbitrarily small $\epsilon > 0$ up to the $1$ vs. $1-0.0058$ gap. As an application of the main result, we demonstrate that Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguratio} are PSPACE-hard to approximate within a factor of $1.0029$ under RIH. Our proof is based on a gap-preserving reduction from Label Cover to Set Cover due to Lund and Yannakakis (J. ACM, 1994). However, unlike Lund--Yannakakis' reduction, the expander mixing lemma is essential to use. We highlight that all results hold unconditionally as long as "PSPACE-hard" is replaced by "NP-hard," and are the first explicit inapproximability results for reconfiguration problems without resorting to the parallel repetition theorem. We finally complement the main result by showing that it is NP-hard to approximate Maxmin Binary CSP Reconfiguration within a factor better than $\frac{3}{4}$.

翻译：在本文中，我们展示了重配置问题的间隙放大。特别地，仅假设Ohsaka（STACS 2023）提出的重配置不可近似性假设（RIH），我们证明了三个流行重配置问题的显式PSPACE-难近似因子。我们的主要结果是：在RIH下，最大最小二元CSP重配置问题在$0.9942$因子内近似是PSPACE-难的。此外，即使约束图被限制为$(d,\lambda)$-扩展图且$\frac{\lambda}{d}$任意小，相同的结果依然成立。其证明的关键在于对Dinur（J. ACM, 2007）间隙放大技术的改进，该技术将任意小的$\epsilon > 0$对应的$1$ vs. $1-\epsilon$间隙放大至$1$ vs. $1-0.0058$间隙。作为主要结果的应用，我们证明在RIH下，最小最大集合覆盖重配置问题和最小最大支配集重配置问题在$1.0029$因子内近似是PSPACE-难的。我们的证明基于Lund和Yannakakis（J. ACM, 1994）从标签覆盖到集合覆盖的间隙保持归约。然而，与Lund-Yannakakis归约不同，扩展图混合引理的使用是必要的。我们强调，只要将“PSPACE-难”替换为“NP-难”，所有结果无条件成立，并且这是重配置问题中首个无需依赖并行重复定理的显式不可近似性结果。最后，我们补充了主要结果：在优于$\frac{3}{4}$的因子内近似最大最小二元CSP重配置问题是NP-难的。