  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.
