Beating a Random Assignment

Max CSP(P) is the problem of maximizing the weight of satisfied constraints, where each constraint acts over a k-tuple of literals and is evaluated using the predicate P. The approximation ratio of a random assignment is equal to the fraction of satisfying inputs to P. If it is NP-hard to achieve a better approximation ratio for Max CSP(P), then we say that P is approximation resistant. Our goal is to characterize which predicates that have this property. A general approximation algorithm for Max CSP(P) is introduced. For a multitude of different P, it is shown that the algorithm beats the random assignment algorithm, thus implying that P is not approximation resistant. In particular, over 2/3 of the predicates on four binary inputs are proved not to be approximation resistant, as well as all predicates on 2s binary inputs, that have at most 2s+1 accepting inputs. We also prove a large number of predicates to be approximation resistant. In particular, all predicates of arity 2s+s2 with less than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{s^2}$\end{document} non-accepting inputs are proved to be approximation resistant, as well as almost 1/5 of the predicates on four binary inputs.