Gap Preserving Reductions Between Reconfiguration Problems
Naoto Ohsaka
TL;DR
This work investigates the approximability of optimization variants of combinatorial reconfiguration problems and introduces the Reconfiguration Inapproximability Hypothesis (RIH), a PSPACE-hardness conjecture for a gap version of Gap$_{1,1-\varepsilon}$ $q$-CSP Reconfiguration. By a sequence of gap-preserving reductions, starting from RIH the authors establish PSPACE-hardness of approximating Maxmin $E3$-SAT$(B)$ Reconfiguration under bounded occurrence, leveraging a novel alphabet-squaring trick and explicit near-Ramanujan expanders in a degree-reduction step. The approach systematically connects $q$-CSP Reconfiguration to BCSP Reconfiguration, to Maxmin E$3$-SAT Reconfiguration, and from there to Nondeterministic Constraint Logic, yielding conditional PSPACE-hardness for optimization variants of Independent Set, Clique, Vertex Cover, and Maxmin $2$-SAT$(B)$ Reconfiguration. The results imply that, under the plausible assumption RIH, many natural reconfiguration problems resist efficient approximation, highlighting the depth of the reconfiguration landscape and guiding future efforts to prove or refute RIH. The techniques—gap-preserving reductions, alphabet squaring, and expander-based degree reductions—provide a versatile toolkit for conditional hardness in reconfiguration contexts with potential broader impact on complexity and approximation theory.
Abstract
Combinatorial reconfiguration is a growing research field studying problems on the transformability between a pair of solutions of a search problem. We consider the approximability of optimization variants of reconfiguration problems; e.g., for a Boolean formula $\varphi$ and two satisfying truth assignments $σ_{\sf s}$ and $σ_{\sf t}$ for $\varphi$, Maxmin SAT Reconfiguration requires to maximize the minimum fraction of satisfied clauses of $\varphi$ during transformation from $σ_{\sf s}$ to $σ_{\sf t}$. Solving such optimization variants approximately, we may obtain a reconfiguration sequence comprising almost-satisfying truth assignments. In this study, we prove a series of gap-preserving reductions to give evidence that a host of reconfiguration problems are PSPACE-hard to approximate, under some plausible assumption. Our starting point is a new working hypothesis called the Reconfiguration Inapproximability Hypothesis (RIH), which asserts that a gap version of Maxmin CSP Reconfiguration is PSPACE-hard. This hypothesis may be thought of as a reconfiguration analogue of the PCP theorem. Our main result is PSPACE-hardness of approximating Maxmin $3$-SAT Reconfiguration of bounded occurrence under RIH. The crux of its proof is a gap-preserving reduction from Maxmin Binary CSP Reconfiguration to itself of bounded degree. Because a simple application of the degree reduction technique using expander graphs due to Papadimitriou and Yannakakis does not preserve the perfect completeness, we modify the alphabet as if each vertex could take a pair of values simultaneously. To accomplish the soundness requirement, we further apply an explicit family of near-Ramanujan graphs and the expander mixing lemma. As an application of the main result, we demonstrate that under RIH, optimization variants of popular reconfiguration problems are PSPACE-hard to approximate.