Near Optimal Alphabet-Soundness Tradeoff PCPs

We show that for all ɛ > 0, for sufficiently large q ∈ ℕ that is a power of 2, for all δ > 0, it is NP-hard to distinguish whether a given 2-Prover-1-Round projection game with alphabet size q has value at least 1-δ, or value at most 1/ q 1-ɛ . This establishes a nearly optimal alphabet-to-soundness tradeoff for 2-query PCPs with alphabet size q , improving upon a result of [Chan, J. ACM 2016]. Our result has the following implications: (1) Near optimal hardness for Quadratic Programming: it is NP-hard to approximate the value of a given Boolean Quadratic Program within factor (log n ) 1 - o (1) under quasi-polynomial time reductions. This improves upon a result of [Khot, Safra, ToC 2013] and nearly matches the performance of the best known algorithms due to [Megretski, IWOTA 2000], [Nemirovski, Roos, Terlaky, Mathematical Programming 1999] and [Charikar, Wirth, FOCS 2004] that achieve O (log n ) approximation ratio. (2) Bounded degree 2-CSPs: under randomized reductions, for sufficiently large d > 0, it is NP-hard to approximate the value of 2-CSPs in which each variable appears in at most d constraints to within a factor of (1- o (1)) \(\frac{d}{2}\) , improving upon a result of [Lee, Manurangsi, ITCS 2024]. (3) Improved hardness results for connectivity problems: using results of [Laekhanukit, SODA 2014] and [Manurangsi, Inf. Process. Lett., 2019], we deduce improved hardness results for the Rooted k -Connectivity Problem, the Vertex-Connectivity Survivable Network Design Problem and the Vertex-Connectivity k -Route Cut Problem.