1 research outputs found
Near-Optimal UGC-hardness of Approximating Max k-CSP_R
In this paper, we prove an almost-optimal hardness for Max -CSP based
on Khot's Unique Games Conjecture (UGC). In Max -CSP, we are given a set
of predicates each of which depends on exactly variables. Each variable can
take any value from . 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 -CSP to within factor for any . To the best of our knowledge, this result
improves on all the known hardness of approximation results when . In this case, the previous best hardness result was
NP-hardness of approximating within a factor by Chan. When , 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-CSP by Kindler, Kolla
and Trevisan, we provide an -approximation algorithm
for Max -CSP. This algorithm implies that our inapproximability result
is tight up to a factor of . In comparison,
when is a constant, the previously known gap was , which is
significantly larger than our gap of .
Finally, we show that we can replace the Unique Games Conjecture assumption
with Khot's -to-1 Conjecture and still get asymptotically the same hardness
of approximation