Near-Optimal UGC-hardness of Approximating Max k-CSP_R

In this paper, we prove an almost-optimal hardness for Max k-CSPR based on Khot’s Unique Games Conjecture (UGC). In Max k-CSPR, we are given a set of predicates each of which depends on exactly k variables. Each variable can take any value from 1, 2, . . . , R. The goal is to find an assignment to variables that maximizes the number of satisfied predicates. Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max k-CSPR to within factor 2 log k)(logR)k/2/Rk−1 for any k,R. To the best of our knowledge, this result improves on all the known hardness of approximation results when 3 ≤ k = o(logR/ log logR). In this case, the previous best hardness result was NP-hardness of approximating within a factor O(k/Rk−2) by Chan. When k = 2, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O’Donnell. In addition, by extending an algorithm for Max 2-CSPR by Kindler, Kolla and Trevisan, we provide an Ω(logR/Rk−1)-approximation algorithm for Max k-CSPR. This algorithm implies that our inapproximability result is tight up to a factor of 2 log k)(logR)k/2−1. In comparison, when 3 ≤ k is a constant, the previously known gap was O(R), which is significantly larger than our gap of O(polylogR). Finally, we show that we can replace the Unique Games Conjecture assumption with Khot’s d-to-1 Conjecture and still get asymptotically the same hardness of approximation.