New Upper Bounds for MAX-2-SAT and MAX-2-CSP w.r.t. the Average Variable Degree

Abstract

MAX-2-SAT and MAX-2-CSP are important NP-hard optimization problems generalizing many graph problems. Despite many efforts, the only known algorithm (due to Williams) solving them in less than 2ⁿ steps uses exponential space. Scott and Sorkin give an algorithm with $2^{n(1-\frac{2}{d+1})}$ time and polynomial space for these problems, where d is the average variable degree. We improve this bound to $O^*(2^{n(1-\frac{10/3}{d+1})})$ for MAX-2-SAT and $O^*(2^{n(1-\frac{3}{d+1})})$ for MAX-2-CSP. We also prove stronger upper bounds for d bounded from below. E.g., for d ≥ 10 the bounds improve to $O^*(2^{n(1-\frac{3.469}{d+1})})$ and $O^*(2^{n(1-\frac{3.221}{d+1})})$, respectively. As a byproduct we get a simple proof of an $O^*(2^\frac{m}{5.263})$ upper bound for MAX-2-CSP, where m is the number of constraints. This matches the best known upper bound w.r.t. m due to Gaspers and Sorkin.

Research is partially supported by Yandex, Parallels and JetBrains.