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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