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 2O(klogk)(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/loglogR). 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 2O(klogk)(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(polylog R).
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