Gap Preserving Reductions between Reconfiguration Problems

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 $\sigma_{\sf s}$ and $\sigma_{\sf t}$ for $\varphi$, Maxmin SAT Reconfiguration requires to maximize the minimum fraction of satisfied clauses of $\varphi$ during transformation from $\sigma_{\sf s}$ to $\sigma_{\sf t}$. 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.