Statistics Papers

Document Type

Journal Article

Date of this Version

11-2013

Publication Source

ACM Transactions on Computation Theory

Volume

5

Issue

4

Start Page

14:1

Last Page

14:26

DOI

10.1145/2505766

Abstract

The results of Raghavendra [2008] show that assuming Khot’s Unique Games Conjecture [2002], for every constraint satisfaction problem there exists a generic semidefinite 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. Khot et al. [2004] established a general approach for determining the exact approximation factor and the corresponding optimal rounding algorithm for any given constraint satisfaction problem. However, this approach crucially relies on results explicitly proving optimal partitions in the Gaussian space. Until recently, Borell’s result [1985] was the only nontrivial Gaussian partition result known.

In this article 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 the exact Unique Games Hardness of MAX-3-EQUAL. In particular, our results show that Zwick’s algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is ≈ 0.796 as conjectured by Zwick [1998].

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 UGC hardness of Max-k-CSP is ⌈(k+1)/2⌉/2k−1, improving on results of Austrin and Mossel [2009].

Comments

At the time of publication, author Elchanan Mossel was affiliated with the University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.

Keywords

Gaussian stability, dictatorship test, unique games conjecture

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Date Posted: 27 November 2017

This document has been peer reviewed.