Near Optimal Alphabet-Soundness Tradeoff PCPs

Abstract

We show that for all > 0, for su ciently large prime power ∈ N, for all > 0, it is NP-hard to distinguish whether a 2-Prover1-Round projection game with alphabet size has value at least 1 − , or value at most 1/ 1− . This establishes a nearly optimal alphabet-to-soundness tradeo for 2-query PCPs with alphabet size , improving upon a result of [Chan 2016]. Our result has the following implications: (1) Near optimal hardness for Quadratic Programming: it is NPhard to approximate the value of a given Boolean Quadratic Program within factor (log) 1− (1) under quasi-polynomial time reductions. This result improves a result of [Khot-Safra 2013] and nearly matches the performance of the best known approximation algorithm [Megrestki 2001, Nemirovski-RoosTerlaky 1999 Charikar-Wirth 2004] that achieves a factor of (log). (2) Bounded degree 2-CSP’s: under randomized reductions, for su ciently large > 0, it is NP-hard to approximate the value of 2-CSPs in which each variable appears in at most constraints within factor (1 − (1)) 2 , improving upon a recent result of [Lee-Manurangsi 2023]. (3) Improved hardness results for connectivity problems: using results of [Laekhanukit 2014] and [Manurangsi 2019], we deduce improved hardness results for the Rooted -Connectivity Problem, the Vertex-Connectivity Survivable Network Design Problem and the Vertex-Connectivity -Route Cut Problem.