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 1 2 + Ω(1/ √ D) fraction of =’s 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 quantum algorithm to find an assignment satisfying a 1 2 + Ω(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 μ+Ω(1/ √ D) fraction of constraints, where μ is the fraction that would be satisfied by a uniformly random assignment. ∗Microsoft Research New England. †MIT Mathematics Department. ‡Department of Computer Science, Carnegie Mellon. §U.C.Berkeley, Department of Electrical Engineering & Computer Sciences. ¶Courant Institute of Mathematical Sciences, New York University. ‖Cornell University. ∗∗Courant Institute of Mathematical Sciences, New York University. ar X iv :1 50 5. 03 42 4v 2 [ cs .C C ] 1 1 A ug 2 01 5