Near Optimal Alphabet-Soundness Tradeoff PCPs

We show a nearly optimal alphabet-soundness tradeoff for NP-hardness of 2-Prover-1-Round Games (2P1R). More specifically, we show that for all \eps > 0, for sufficiently large M, it is NP-hard to decide whether a 2P1R instance of alphabet size M has value nearly 1 or at most M^{-1+\eps}. 2P1R are equivalent to 2-Query PCPs, and are widely used in obtaining hardness of approximation results. As such, our result implies the following: 1) hardness of approximating Quadratic Programming within a factor of nearly log(n), 2) hardness of approximating d-bounded degree 2-CSP within a factor of nearly d/2, and 3) improved hardness of approximation results for various k-vertex connectivity problems. For the former two applications, our results nearly match the performance of the best known algorithms.