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

2026-07-30

AI Summary

This work investigates the approximation complexity of the Maxmin $q$-CSP Reconfiguration problem: given two satisfying assignments, the goal is to find a reconfiguration path that maximizes the minimum fraction of satisfied constraints along the path. By leveraging computational complexity theory and PSPACE-hardness reductions, the paper establishes that for any $q \geq 2$, the problem is PSPACE-hard to approximate within a factor of $\frac{1}{2^{q-1}} + \varepsilon$ for arbitrarily small $\varepsilon > 0$. Under the perfect completeness assumption, it also presents a $\left(\frac{1}{2^{q-1}} - \varepsilon\right)$-approximation algorithm and shows that this approximation lies in NP. Assuming NP ≠ PSPACE, these results together yield a tight approximation threshold for all $q \geq 2$, providing the first complete characterization of the optimal PSPACE-hard approximation boundary for this problem.

Abstract

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

Problem

• q-CSP Reconfiguration
• PSPACE-hardness
• Approximation
• Computational Complexity
• Maxmin Optimization

Innovation

• PSPACE-hardness
• q-CSP Reconfiguration
• Approximation Algorithms
• Computational Complexity
• Maxmin Optimization