PCPP-Based Reconfiguration Inapproximability: Query Complexity vs. Soundness Gap Trade-offs

TL;DR

This paper establishes a tighter connection between the hardness of reconfiguring solutions in CSPs and PCPP soundness gaps, leading to improved trade-offs between query complexity and soundness in reconfiguration problems.

Contribution

It introduces a novel connection between RIH's soundness gap and PCPP, enhancing the understanding of inapproximability in CSP reconfiguration with better trade-offs.

Findings

Improved trade-off between soundness gap and query complexity.

New parallelization framework for CSP reconfiguration.

Enhanced inapproximability results for CSP reconfiguration.

Abstract

The Reconfiguration Inapproximability Hypothesis (RIH), recently established by Hirahara-Ohsaka (STOC'24) and Karthik-Manurangsi (ECCC'24), studies the hardness of reconfiguring one solution into another in constraint satisfaction problems (CSP) when restricted to approximate intermediate solutions. In this work, we make a tighter connection between RIH's soundness gap and that of probabilistically checkable proofs of proximity (PCPP). Consequently, we achieve an improved trade-off between soundness and query complexity in Gap CSP Reconfiguration. Our approach leverages a parallelization framework, which also appears in some recent parameterized inapproximability results.