Optimal PSPACE-hardness of Approximating $q$-CSP Reconfiguration

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 $\mathsf{NP} \neq \mathsf{PSPACE}$.