Probabilistically Checkable Reconfiguration Proofs and Inapproximability of Reconfiguration Problems

TL;DR

This paper introduces a new PCP-like characterization of PSPACE called PCRP, enabling the proof of PSPACE-completeness and inapproximability results for various reconfiguration problems, resolving longstanding open problems.

Contribution

It presents a novel probabilistically checkable reconfiguration proof (PCRP) framework that characterizes PSPACE and applies it to establish hardness and inapproximability of reconfiguration problems.

Findings

Proves PSPACE-completeness of approximate Maxmin 3-SAT Reconfiguration.

Establishes PSPACE-completeness of Maxmin Clique Reconfiguration within certain approximation factors.

Resolves open problems on reconfiguration inapproximability from prior research.

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 $n^\epsilon$ for some constant $\epsilon > 0$.