概率可检验重构证明与重构问题的不可近似性
Probabilistically Checkable Reconfiguration Proofs and Inapproximability of Reconfiguration Problems
AI总结：
本文提出概率可检验重构证明（PCRP）这一 PSPACE 的新 PCP 型刻画，并据此证明 Maxmin 3-SAT 与 Clique 等重构问题近似版本的 PSPACE 完全性，解决相关开放猜想。
摘要：
受重构问题不可近似性的启发，我们提出了一种新的 PCP 型 PSPACE 刻画，称之为概率可检验重构证明（PCRP）：任何 PSPACE 计算都可以编码为一个指数长度的序列，该序列由多项式长度的证明组成，使得每对相邻证明至多相差一个比特，并且每个证明都可以通过读取常数个比特来进行概率检验。利用这一新刻画，我们证明了许多重构问题的近似版本具有 PSPACE 完全性，例如 Maxmin 3-SAT Reconfiguration 问题。这肯定地解决了 Ito、Demaine、Harvey、Papadimitriou、Sideri、Uehara 和 Uno（ISAAC 2008；Theor. Comput. Sci. 2011）提出的开放问题，以及 Ohsaka（STACS 2023）提出的 Reconfiguration Inapproximability Hypothesis。我们还证明了，对于某个常数 ε>0，将 Maxmin Clique Reconfiguration 问题近似到 n^ε 因子以内是 PSPACE 完全的。
英文摘要：
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 $ε> 0$.
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