Every 2-csp Allows Nontrivial Approximation

We use semidefinite programming to prove that any constraint satisfaction problem in two variables over any domain allows an efficient approximation algorithm that does better than picking a random assignment. Specifically we consider the case when each variable can take values in [d] and that each constraint rejects t out of the d2 possible input pairs. Then, for some universal constant c, we can, in probabilistic polynomial time, find an assignment whose objective value is, in expectation, within a factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1- \frac{t} {d^{2}} +\frac{ct} {d^{4}log d}$$\end{document} of optimal, improving on the trivial bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1- \frac{t} {d^{2}}$$\end{document}.