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Near Optimal Hardness of Approximating 2-Prover-1-Round Games with Small Alphabet Size


Core Concepts
For all ε > 0, it is NP-hard to distinguish whether a 2-Prover-1-Round projection game with alphabet size q has value at least 1-δ or at most 1/q^(1-ε), establishing a nearly optimal alphabet-to-soundness tradeoff for 2-query PCPs.
Abstract
The content presents a new result that improves upon previous work on the alphabet-soundness tradeoff for 2-query PCPs. The key contributions are: The main technical result (Theorem 1.3) shows that for all ε, δ > 0, it is NP-hard to distinguish whether a 2-Prover-1-Round game with alphabet size q has value at least 1-δ or at most 1/q^(1-ε). This establishes a nearly optimal tradeoff between the alphabet size and soundness error of PCPs. This improved tradeoff has several applications, including: Improved hardness of approximating Quadratic Programming within a factor of (log n)^(1-o(1)) (Theorem 1.4). Improved hardness of approximating bounded degree 2-CSPs within a factor of (1/2-η)d, where d is the maximum degree (Theorem 1.5). Improved hardness results for various connectivity problems in graphs (Theorem 1.6). The technical approach involves composing an "inner PCP" based on the Grassmann graph with an "outer PCP" using smooth parallel repetition. The analysis requires new techniques in low-degree testing and list decoding over the Grassmann graph.
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Key Insights Distilled From

by Dor Minzer,K... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07441.pdf
Near Optimal Alphabet-Soundness Tradeoff PCPs

Deeper Inquiries

How can the techniques developed in this work be applied to make progress on the sliding scale conjecture for PCPs

The techniques developed in this work, particularly the composition of Inner PCP and Outer PCP, can be applied to make progress on the sliding scale conjecture for PCPs. The sliding scale conjecture is concerned with achieving soundness error that is inversely polynomial in the instance size. By refining the analysis of the covering property and improving the list decoding bounds, the techniques in this work could potentially be adapted to construct PCPs with soundness error that approaches the desired level for the sliding scale conjecture. This would involve further optimizing the parameters in the PCP construction, such as the smoothness parameter and the size of the subspaces, to achieve a tighter tradeoff between soundness error, alphabet size, and instance size.

Are there other combinatorial optimization problems beyond the ones considered here that could benefit from the improved hardness results for 2-Prover-1-Round games

Beyond the combinatorial optimization problems discussed in the paper, there are several other problems that could benefit from the improved hardness results for 2-Prover-1-Round games. For example, problems related to graph coloring, maximum matching, and network design could potentially be tackled using the near-optimal alphabet-to-soundness tradeoff established in this work. Additionally, problems in computational biology, such as sequence alignment and protein structure prediction, could also benefit from the enhanced hardness results, as these problems often involve complex optimization tasks that can be formulated as constraint satisfaction problems.

What are the limitations of the PCP construction techniques used in this work, and how could they be further improved to achieve even stronger hardness of approximation results

While the PCP construction techniques used in this work represent a significant advancement in the field of theoretical computer science, there are some limitations that could be addressed to achieve even stronger hardness of approximation results. One limitation is the dependency between the soundness error and the alphabet size in the constructed PCPs. Improving the analysis of the covering property and refining the list decoding bounds could help in achieving a more optimal tradeoff between these parameters. Additionally, further research could focus on developing new techniques for coordinating zoom-outs in the PCP composition step, as this is a critical aspect of the construction that impacts the overall hardness results. By addressing these limitations and exploring new avenues for improvement, the PCP construction techniques could be further enhanced to yield even stronger hardness results for a wider range of optimization problems.
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