AI 中文总结
本文证明了对任意q≥2和ε>0，Maxmin q-CSP重配置问题在因子1/(2^(q-1))+ε下近似是PSPACE-难的，且完美完备情形下其1/(2^(q-1))-ε因子近似属于NP，确立了该问题近似的最优PSPACE-难度。
AI 中文摘要
在Maxmin q-CSP重配置问题中，给定一个可满足的q-CSP实例及其一对满足赋值，要求通过反复修改单个变量的赋值，将一个赋值转换为另一个，目标是找到此类转换，使转换过程中满足约束的最小分数最大化。本文证明，对任意q≥2和ε>0，Maxmin q-CSP重配置问题在因子为1/(2^(q-1))+ε的近似下是PSPACE-难的。为补充该难度结果，本文还证明，在完美完备情形下，Maxmin q-CSP重配置问题的1/(2^(q-1))-ε因子近似属于NP问题。这些结果在NP≠PSPACE的假设下，确立了对所有q≥2，Maxmin q-CSP重配置问题近似的最优PSPACE-难度。
英文摘要
In the Maxmin $q$-CSP Reconfiguration problem, given a satisfiable $q$-CSP instance and a pair of its satisfying assignments, we are asked to transform one assignment into the other by repeatedly changing the value assigned to a single variable. The objective is to find such a transformation that maximizes the minimum fraction of satisfied constraints along the transformation. In this paper, we prove that for any $q \geq 2$ and $\varepsilon > 0$, Maxmin $q$-CSP Reconfiguration is $\mathsf{PSPACE}$-hard to approximate within a factor of $\frac{1}{2^{q-1}}+\varepsilon$. To complement this hardness result, we prove that a $\bigl(\frac{1}{2^{q-1}}-\varepsilon\bigr)$-factor approximation for Maxmin $q$-CSP Reconfiguration is in $\mathsf{NP}$ in the perfect completeness case. These results establish the optimal $\mathsf{PSPACE}$-hardness of approximating Maxmin $q$-CSP Reconfiguration for every $q \geq 2$ under $\mathsf{NP} \neq \mathsf{PSPACE}$.
Comments79 pages, to appear in Proceedings of the 67th IEEE Symposium on Foundations of Computer Science (FOCS 2026)
