How does the coboundary expansion of these coset complexes impact the efficiency of the resulting PCP constructions compared to other approaches?
Answer:
The paper suggests that proving coboundary expansion of their coset complex construction over the symmetric group could lead to more efficient PCP (Probabilistically Checkable Proof) constructions compared to some existing methods. Here's how:
Elementary Construction: Coset complexes, particularly the variant used in the paper, offer a relatively elementary and explicit construction compared to other families of complexes exhibiting similar properties, like Ramanujan complexes. This simplicity can translate to more practical and efficient PCP constructions.
Bounded Degree and Expansion: The coset complexes presented have bounded degree, meaning each vertex is connected to a limited number of other vertices. This property is crucial for constructing efficient PCPs as it ensures the proof verification process remains manageable. The paper establishes cosystolic expansion and conjectures coboundary expansion for these complexes. Coboundary expansion, a stronger property implying cosystolic expansion, is directly linked to agreement expansion in the low soundness regime, a key ingredient for efficient PCPs.
Potential for Improvement: While the paper proves cosystolic expansion, they acknowledge that proving coboundary expansion is still an open problem for their specific construction. If proven, it would directly imply agreement expansion with parameters potentially surpassing those achievable by other approaches, leading to more efficient PCPs.
However, it's important to note:
Local Expansion Challenge: Even with coboundary expansion of the complex, achieving efficient PCPs requires the coboundary expansion constants of the links to be independent of the complex dimension. This local expansion property is still an open problem for these coset complexes.
Comparison with Other Expanders: Recent works have shown coboundary expansion in other families of complexes, like those based on [CL24]. Direct comparison of efficiency would require a deeper analysis of the resulting PCP parameters in each construction.
In summary, while coboundary expansion of these coset complexes holds the potential for more efficient PCP constructions due to their elementary nature and good expansion properties, further research is needed to confirm and quantify these efficiency gains.
Could there be alternative families of simplicial complexes that exhibit coboundary expansion over the symmetric group but do not rely on the specific properties of coset complexes?
Answer:
Yes, it's certainly possible. While the paper focuses on coset complexes, the pursuit of simplicial complexes exhibiting coboundary expansion over the symmetric group extends beyond this specific construction. Here are some potential avenues for finding alternative families:
Generalizations of Existing Constructions:
Building on [CL24]: The works of [BLM24, DDL24] demonstrate coboundary expansion in complexes derived from the work of [CL24]. Exploring generalizations or variations of these constructions could yield new families with the desired properties.
Tweaking Coset Complexes: Modifying the coset complex construction itself, such as exploring different groups or subgroup choices, might lead to new examples. The paper [OP22], for instance, generalizes the coset complex idea, offering potential starting points.
New Topological Constructions:
High-Dimensional Expanders: The field of high-dimensional expanders (HDX) is rich with diverse constructions. Investigating their coboundary expansion properties, particularly over the symmetric group, could uncover hidden gems.
Geometric and Topological Methods: Drawing inspiration from geometric and topological tools, like those used in the study of buildings, could lead to entirely new families of complexes with the desired expansion properties.
Combinatorial Approaches:
Agreement Expanders: Exploring families of complexes specifically designed to exhibit strong agreement expansion properties, even without directly aiming for coboundary expansion, might indirectly lead to the desired outcome.
Randomized Constructions: While current randomized constructions often focus on spectral expansion, adapting these techniques to target coboundary expansion over the symmetric group could be fruitful.
The key challenge in all these approaches lies in proving the global property of coboundary expansion, which, as the paper highlights, cannot be deduced solely from local considerations. New global techniques, potentially inspired by the paper's novel approach to proving vanishing cohomology, will be crucial in this endeavor.
What are the potential implications of this research for other areas of theoretical computer science beyond PCPs, such as coding theory or cryptography?
Answer:
The research on coboundary expansion over the symmetric group, particularly in the context of coset complexes, has the potential to impact various areas of theoretical computer science beyond PCPs. Here are some examples:
Coding Theory:
Locally Testable Codes (LTCs): Coboundary expansion, particularly in conjunction with spectral expansion, is closely related to the construction of LTCs. These codes allow for efficient verification of whether a given word belongs to the code by examining only a small number of its bits. Improved constructions of coboundary expanders could lead to LTCs with better parameters, such as higher rate or larger distance.
Locally Decodable Codes (LDCs): Similar to LTCs, LDCs allow for efficient retrieval of individual bits of the encoded message by querying only a small number of codeword bits. Coboundary expansion properties of the underlying complexes used in LDC constructions could translate to improved decoding efficiency and robustness.
Cryptography:
Secret Sharing Schemes: Coboundary expansion over the symmetric group could lead to new and improved secret sharing schemes. These schemes allow for distributing a secret among multiple parties such that only authorized subsets can reconstruct it. The properties of coboundary expanders could enhance the security and efficiency of these schemes.
Secure Multi-Party Computation (MPC): MPC protocols enable multiple parties to jointly compute a function over their private inputs without revealing anything beyond the output. The strong expansion properties of coboundary expanders could be leveraged to design more secure and efficient MPC protocols, particularly in the context of dishonest majority settings.
Other Areas:
Property Testing: Coboundary expansion has implications for property testing, where the goal is to efficiently determine whether a given object (e.g., a graph or a function) satisfies a certain property or is far from satisfying it. New coboundary expanders could lead to more efficient property testing algorithms for various combinatorial and algebraic properties.
Derandomization: The study of coboundary expansion and related concepts could contribute to the field of derandomization, which aims to design efficient deterministic algorithms for problems that are currently only solvable efficiently using randomness. The explicit and structured nature of coboundary expanders might offer new tools for derandomizing algorithms.
Overall, the research on coboundary expansion over the symmetric group, as explored in the context of coset complexes, has the potential to significantly impact various areas of theoretical computer science by providing new tools and techniques for designing efficient and secure algorithms and protocols.
