Constraint Satisfaction over a Non-Boolean Domain: Approximation Algorithms and Unique-Games Hardness

We study the approximability of the MAX k-CSP problem over non-boolean domains, more specifically over {0,1,...,qi¾? 1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [19] to obtain a UGC hardness result when qis a prime. More precisely, assuming the Unique Games Conjecture, we show that it is NP-hard to approximate the problem to a ratio greater than q2k/qk. Independent of this work, Austrin and Mossel [2] obtain a more general UGC hardness result using entirely different techniques. We also obtain an approximation algorithm that achieves a ratio of C(q) ·k/qkfor some constant C(q) depending only on q, via a subroutine for approximating the value of a semidefinite quadratic form when the variables take values on the corners of the q-dimensional simplex. This generalizes an algorithm of Nesterov [16] for the ±1-valued variables. It has been pointed out to us [15] that a similar approximation ratio can be obtained by reducing the non-boolean case to a boolean CSP.