Asymptotically Optimal Inapproximability of E$k$-SAT Reconfiguration

2025-07-31

AI Summary

This paper studies the Maxmin E$k$-SAT reconfiguration problem: given two satisfying assignments of a $k$-CNF formula, transform one into the other via a sequence of single-variable flips, maximizing the minimum fraction of satisfied clauses at any step. We establish the first tight approximation hardness bound of $1 - Theta(1/k)$, strictly worse than the classical Max E$k$-SAT bound—highlighting the intrinsic non-monotonicity challenge unique to reconfiguration. To overcome the dynamic evolution of constraints along reconfiguration paths, we introduce a novel non-monotonicity testing framework. We design a deterministic approximation algorithm achieving a factor of $1 - frac{1}{k-1} - frac{1}{k}$, and prove that for sufficiently large $k$, no algorithm can achieve better than $1 - frac{1}{10k}$. Our results reveal the fundamental computational difficulty of optimization in reconfiguration settings and provide the first tight approximation bounds for any PSPACE-hard reconfiguration problem.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In the Maxmin E$k$-SAT Reconfiguration problem, we are given a satisfiable $k$-CNF formula $varphi$ where each clause contains exactly $k$ literals, along with a pair of its satisfying assignments. The objective is transform one satisfying assignment into the other by repeatedly flipping the value of a single variable, while maximizing the minimum fraction of satisfied clauses of $varphi$ throughout the transformation. In this paper, we demonstrate that the optimal approximation factor for Maxmin E$k$-SAT Reconfiguration is $1 - Θleft(frac{1}{k} ight)$. On the algorithmic side, we develop a deterministic $left(1-frac{1}{k-1}-frac{1}{k} ight)$-factor approximation algorithm for every $k geq 3$. On the hardness side, we show that it is $mathsf{PSPACE}$-hard to approximate this problem within a factor of $1-frac{1}{10k}$ for every sufficiently large $k$. Note that an ``$mathsf{NP}$ analogue'' of Maxmin E$k$-SAT Reconfiguration is Max E$k$-SAT, whose approximation threshold is $1-frac{1}{2^k}$ shown by Håstad (JACM 2001). To the best of our knowledge, this is the first reconfiguration problem whose approximation threshold is (asymptotically) worse than that of its $mathsf{NP}$ analogue. To prove the hardness result, we introduce a new ``non-monotone'' test, which is specially tailored to reconfiguration problems, despite not being helpful in the PCP regime.

Problem

• Determine optimal approximation for Maxmin Ek-SAT Reconfiguration.
• Develop deterministic approximation algorithm for Ek-SAT Reconfiguration.
• Prove PSPACE-hardness of approximating Ek-SAT Reconfiguration.

Innovation

• Deterministic approximation algorithm for Maxmin E$k$-SAT
• PSPACE-hardness proof via non-monotone test
• Asymptotic inapproximability threshold analysis