Probabilistically Checkable Reconfiguration Proofs and Inapproximability of Reconfiguration Problems
Shuichi Hirahara, Naoto Ohsaka
TL;DR
This work introduces probabilistically checkable reconfiguration proofs (PCRP), a PCP-style characterization of PSPACE that encodes any PSPACE computation as an exponentially long reconfiguration sequence of proofs differing by at most one bit and verifiable with $O(\log n)$ randomness and $O(1)$ queries. Centered on Succinct Graph Reachability, the authors construct a PCRP system by encoding vertices with locally testable codes and using a PCP of Proximity to probabilistically verify adjacency constraints within the reconfiguration sequence, even allowing a special symbol $\bot$ to mark transition steps. Leveraging this PCRP framework, they prove PSPACE-completeness of approximating several reconfiguration problems (e.g., Maxmin $3$-SAT Reconfiguration), resolving open questions such as the Reconfiguration Inapproximability Hypothesis, and they obtain a first polynomial-factor inapproximability for Maxmin Clique Reconfiguration via derandomized graph products. The results imply that many natural reconfiguration problems are PSPACE-hard to approximate, highlighting the deep hardness of maintaining high-quality solutions across constrained, stepwise transformations in high-complexity regimes. Overall, the work provides a fundamental PCP-style lens for reconfiguration hardness with direct consequences for approximation limits in a broad class of combinatorial problems.
Abstract
Motivated by the inapproximability of reconfiguration problems, we present a new PCP-type characterization of PSPACE, which we call a probabilistically checkable reconfiguration proof (PCRP): Any PSPACE computation can be encoded into an exponentially long sequence of polynomially long proofs such that every adjacent pair of the proofs differs in at most one bit, and every proof can be probabilistically checked by reading a constant number of bits. Using the new characterization, we prove PSPACE-completeness of approximate versions of many reconfiguration problems, such as the Maxmin $3$-SAT Reconfiguration problem. This resolves the open problem posed by Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno (ISAAC 2008; Theor. Comput. Sci. 2011) as well as the Reconfiguration Inapproximability Hypothesis by Ohsaka (STACS 2023) affirmatively. We also present PSPACE-completeness of approximating the Maxmin Clique Reconfiguration problem to within a factor of $n^ε$ for some constant $ε> 0$.