Scheme-Theoretic Approach to Computational Complexity. IV. A New Perspective on Hardness of Approximation

We provide a new approach for establishing hardness of approximation results, based on the theory recently introduced by the author. It allows one to directly show that approximating a problem beyond a certain threshold requires super-polynomial time. To exhibit the framework, we revisit two famous problems in this paper. The particular results we prove are: MAX-3-SAT$(1,\frac{7}{8}+\epsilon )$ requires exponential time for any constant $\epsilon$ satisfying $\frac{1}{8}\ge \epsilon >0$. In particular, the gap exponential time hypothesis (Gap-ETH) holds. MAX-3-LIN-2$(1-\epsilon ,\frac{1}{2}+\epsilon )$ requires exponential time for any constant $\epsilon$ satisfying $\frac{1}{4}\ge \epsilon >0$.