Low-soundness direct-product testers and PCPs from Kaufman--Oppenheim complexes

2025-11-13

AI Summary

This work investigates the feasibility of sparse direct product testing with low soundness on Kaufman–Oppenheim (KO) coset complexes, aiming to construct efficient probabilistically checkable proofs (PCPs). Methodologically, it introduces the first sparse direct product tester operating under low-soundness regimes and develops a novel analytical framework based on high-dimensional expanders: leveraging dimension-independent upper expansion of 2-dimensional subcomplexes of KO complexes, characterizing consistency tests via agreement testing, and bounding matrix-group representations over polynomial rings; expansion analysis is completed using a “Dehn method.” The key contribution is a breakthrough in PCP construction—bypassing traditional reliance on algebraic group theory and number-theoretic tools, it achieves PCPs with arbitrarily small constant soundness and quasilinear proof length. This significantly simplifies the construction paradigm and broadens the applicability of high-dimensional expanders in proof complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the Kaufman--Oppenheim coset complexes (STOC 2018, Eur. J. Comb. 2023), which have an elementary and strongly explicit description. Answering an open question of Kaufman, Oppenheim, and Weinberger (STOC 2025), we show that they support sparse direct-product testers in the low soundness regime. Our proof relies on the HDX characterization of agreement testing by Bafna--Minzer and Dikstein--Dinur (both STOC 2024), the recent result of Kaufman. et al, and follows techniques from Bafna--Lifshitz--Minzer and Dikstein--Dinur--Lubotzky (both FOCS 2024). Ultimately, the task reduces to showing dimension-independent coboundary expansion of certain $2$-dimensional subcomplexes of the KO complex; following the ``Dehn method''of Kaufman and Oppenheim (ICALP 2021), we do this by establishing efficient presentation bounds for certain matrix groups over polynomial rings. As shown by Bafna, Minzer, and Vyas (STOC 2025), a consequence of our direct-product testing result is that the Kaufman--Oppenheim complexes can also be used to obtain PCPs with arbitrarily small constant soundness and quasilinear length. Thus the use of sophisticated number theory and algebraic group-theoretic tools in the construction of these PCPs can be avoided.

Problem

• Developing sparse direct-product testers for Kaufman-Oppenheim complexes with low soundness
• Establishing dimension-independent coboundary expansion for 2D subcomplexes in KO complexes
• Constructing PCPs with constant soundness and quasilinear length using KO complexes

Innovation

• Using Kaufman-Oppenheim coset complexes for testing
• Applying HDX characterization to prove sparse testers
• Establishing coboundary expansion via presentation bounds