Beating the random assignment on constraint satisfaction problems of bounded degree

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $frac{1}{2} + Omega(1/sqrt{D})$ fraction of constraints, where $D$ is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a emph{quantum} algorithm to find an assignment satisfying a $frac{1}{2} + Omega(D^{-3/4})$ fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for triangle-free instances; i.e., an efficient algorithm that finds an assignment satisfying at least a $mu + Omega(1/sqrt{D})$ fraction of constraints, where $mu$ is the fraction that would be satisfied by a uniformly random assignment.