Quasi-Linear Size PCPs with Small Soundness from HDX (2407.12762v2)

Abstract: We construct 2-query, quasi-linear size probabilistically checkable proofs (PCPs) with arbitrarily small constant soundness, improving upon Dinur's 2-query quasi-linear size PCPs with soundness $1-\Omega(1)$. As an immediate corollary, we get that under the exponential time hypothesis, for all $\epsilon>0$ no approximation algorithm for $3$-SAT can obtain an approximation ratio of $7/8+\epsilon$ in time $2^{n/\log^C n}$, where $C$ is a constant depending on $\epsilon$. Our result builds on a recent line of independent works by Bafna, Lifshitz and Minzer, and Dikstein, Dinur and Lubotzky, that showed the existence of linear size direct product testers with small soundness. The main new ingredient in our proof is a technique that embeds a given 2-CSP into a 2-CSP on a prescribed graph, provided that the latter is a graph underlying a sufficiently good high-dimensional expander (HDX). We achieve this by establishing a novel connection between PCPs and fault-tolerant distributed computing, more precisely, to the almost-everywhere reliable transmission problem introduced by Dwork, Peleg, Pippenger and Upfal (1986). We instantiate this connection by showing that graphs underlying HDXs admit routing protocols that are tolerant to adversarial edge corruptions, also improving upon the state of the art constructions of sparse edge-fault-tolerant networks in the process. Our PCP construction requires variants of the aforementioned direct product testers with poly-logarithmic degree. The existence and constructability of these variants is shown in an appendix by Zhiwei Yun.

• The paper constructs 2-query quasi-linear size PCPs with arbitrarily small constant soundness, surpassing previous soundness bounds.
• It leverages high-dimensional expanders and fault-tolerant routing protocols to design resilient and efficient proof-checking systems.
• Under ETH, its results imply that no 3-SAT approximation can exceed a 7/8+ε ratio in subexponential time, refining NP-hardness limits.

Quasi-Linear Size PCPs with Small Soundness from HDX

The paper at hand introduces a significant contribution to the paper of probabilistically checkable proofs (PCPs). Specifically, it constructs two-query, quasi-linear sized PCPs with arbitrarily small constant soundness, thereby improving upon the previous results of Dinur that achieved soundness of $1-\Omega(1)$. This advancement has profound implications, particularly under the exponential time hypothesis (ETH), on the intractability of achieving certain approximation ratios for NP-hard problems like 3-SAT within certain time bounds.

Main Results and Contributions

The primary achievement of the paper revolves around the construction and proof of the existence of 2-query PCPs of quasi-linear size with minimal soundness. This is encapsulated in the following theorem:

Theorem: For all $\delta>0$, there exists $C = C(\delta)>0$ and a polynomial time procedure such that given an instance $\phi$ of 3-SAT of size $n$, it produces a label cover instance $\Psi$ with the following properties:

1. The size of $\Psi$ is at most $n (\log n)^{C}$ and the alphabet size of $\Psi$ is at most $O_{\delta}(1)$.
2. If $\phi$ is satisfiable, then $\text{val}(\Psi) = 1$.
3. If $\phi$ is unsatisfiable, then $\text{val}(\Psi) \leq \delta$.

This result is immediately consequential for the exponential time hypothesis (ETH). For instance, under ETH, it guarantees that for all $\epsilon>0$, no approximation algorithm for 3-SAT can achieve an approximation ratio of $7/8+\epsilon$ in time $2^{n/\log^C n}$, where $C$ depends on $\epsilon$.

Technical Innovations

The paper leverages several significant innovations and methodologies to achieve the main result. Central to the construction of the PCPs is the embedding of high-dimensional expanders (HDXs) into the PCP frameworks. HDXs, due to their robust expansion properties, serve as a pivotal tool in achieving the desired soundness properties while maintaining quasi-linear proof sizes.

Interplay with Fault-Tolerant Distributed Computing

Another compelling aspect is the utilization of techniques from fault-tolerant distributed computing, particularly drawing from the almost everywhere agreement problem literature. This connection catalyzes the embedding of a given PCP construction into a prescribed graph that is based on an HDX. The paper demonstrates that HDX graphs admit routing protocols that are resilient to adversarial edge corruptions, which is crucial for maintaining the structural integrity and robustness of the PCPs.

Direct Product Testers

Complementing these methodologies are advancements in direct product testers, which are necessary for the construction of the quasi-linear size PCPs. Specifically, the paper requires variants of these testers that possess poly-logarithmic degree, whose existence is substantiated in an appendix by Zhiwei Yun.

Implications and Future Directions

The paper’s results have far-reaching implications both in theory and practical algorithm design. Theoretical implications mostly concern the hardness of approximation results. Many hardness of approximation results initiate with the hardness of Label Cover with small soundness, and the results of this paper allow for the replication of these results with quasi-linear blow-ups. Furthermore, in distributed computing, the protocols designed for HDXs could directly influence the design of more fault-tolerant networks, enhancing reliability in distributed systems under adversarial conditions.

Realizing Future Potential in AI

As AI and complex system designs grow in importance, the principles argued in this paper could significantly impact the development of efficient verification systems and the robustness of distributed AI models. For example, in scenarios such as federated learning or decentralized AI systems, the ability to verify computational results efficiently using minimal resources could be transformative.

Conclusion

The construction of quasi-linear PCPs with minimal soundness represents a notable stride in computational complexity and PCP theory. By employing HDXs, routing protocols from fault-tolerant distributed computing, and advanced direct product testers, the paper not only achieves theoretical advancement but also lays potential groundwork for practical applications in AI and distributed systems. As future works build upon these foundations, the principles elucidated here may very well inform the next generation of PCP-based verification systems and distributed AI architectures.