The Non-approximability of Non-Boolean Predicates

• Authors: Lars Engebretsen
• Venue: APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
• Year: 2001

Constraint satisfaction programs where each constraint depends on a constant number of variables have the following property: The randomized algorithm that guesses an assignment uniformly at random satisfies an expected constant fraction of the constraints. By combining constructions from interactive proof systems with harmonic analysis over finite groups, Håstad showed that for several constraint satisfaction programs this naive algorithm is essentially the best possible unless P = NP. While most of the predicates analyzed by Håstad depend on a small number of variables, Samorodnitsky and Trevisan recently extended Håstad's result to predicates depending on an arbitrarily large, but still constant, number of Boolean variables. We combine ideas from these two constructions and prove that there exists a large class of predicates on finite non-Boolean domains such that for predicates in the class, the naive randomized algorithm that guesses a solution uniformly is essentially the best possible unless P = NP. As a corollary, we show that the k-CSP problem over domains with size D cannot be approximated within Dk-O(√k) - Ɛ, for any constant Ɛ 0, unless P = NP. This lower bound matches well with the best known upper bound, Dk-1, of Serna, Trevisan and Xhafa.