3 research outputs found
Explicit Optimal Hardness via Gaussian stability results
The results of Raghavendra (2008) show that assuming Khot's Unique Games
Conjecture (2002), for every constraint satisfaction problem there exists a
generic semi-definite program that achieves the optimal approximation factor.
This result is existential as it does not provide an explicit optimal rounding
procedure nor does it allow to calculate exactly the Unique Games hardness of
the problem.
Obtaining an explicit optimal approximation scheme and the corresponding
approximation factor is a difficult challenge for each specific approximation
problem. An approach for determining the exact approximation factor and the
corresponding optimal rounding was established in the analysis of MAX-CUT (KKMO
2004) and the use of the Invariance Principle (MOO 2005). However, this
approach crucially relies on results explicitly proving optimal partitions in
Gaussian space. Until recently, Borell's result (Borell 1985) was the only
non-trivial Gaussian partition result known.
In this paper we derive the first explicit optimal approximation algorithm
and the corresponding approximation factor using a new result on Gaussian
partitions due to Isaksson and Mossel (2012). This Gaussian result allows us to
determine exactly the Unique Games Hardness of MAX-3-EQUAL. In particular, our
results show that Zwick algorithm for this problem achieves the optimal
approximation factor and prove that the approximation achieved by the algorithm
is as conjectured by Zwick.
We further use the previously known optimal Gaussian partitions results to
obtain a new Unique Games Hardness factor for MAX-k-CSP : Using the well known
fact that jointly normal pairwise independent random variables are fully
independent, we show that the the UGC hardness of Max-k-CSP is , improving on results of Austrin and Mossel (2009)