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, . . . , q−1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [18] to obtain a UGC hardness result when q is 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 qk/q. 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/q for some constant C(q) depending only on q. Except for constant factors depending on q, the algorithm and the UGC hardness result have the same dependence on the arity k. It has been pointed out to us [14] that a similar approximation ratio can be obtained by reducing the non-boolean case to a boolean CSP, and appealing to the CMM algorithm [3]. As a subroutine, we design a constant factor(depending on q) approximation algorithm for the problem of maximizing a semidefinite quadratic form, where the variables are constrained to take values on the corners of the q-dimensional simplex. This result generalizes an algorithm of Nesterov [15] for maximizing semidefinite quadratic forms where the variables take {−1, 1} values. ∗An extended abstract of this work was presented at APPROX 2008: the 11th International Workshop on Approximation, Randomization and Combinatorial Optimization. †Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213. Email: [email protected]. Work done while visiting the School of Mathematics, Institute for Advanced Study, Princeton, NJ. Research supported in part by NSF CCF-0343672 and a David and Lucile Packard Fellowship. ‡Microsoft Research New England, Cambridge, MA 02142. Email: [email protected]. Work done while visiting Princeton University. Research supported in part by NSF CCF-0343672.