### Volume 2015

#### Unique Games on the Hypercube

Naman Agarwal
Princeton University
Princeton, NJ, USA
namana AT cs DOT princeton DOT edu

Guy Kindler
Hebrew University
Jerusalem, Israel
gkindler AT cs DOT huji DOT ac DOT il

Alexandra Kolla
University of Illinois Urbana-Champaign
Urbana-Champaign, Illinois, USA
akolla AT illinois DOT edu

and

Luca Trevisan
University of California Berkeley
Berkeley, CA, USA
luca AT berkeley DOT edu

February 7, 2015

#### Abstract

In this paper, we investigate the validity of the Unique Games Conjecture when the constraint graph is the boolean hypercube. We construct an almost optimal integrality gap instance on the Hypercube for the Goemans-Williamson semidefinite program (SDP) for Max-2-LIN$(Z_2)$. We conjecture that adding triangle inequalities to the SDP provides a polynomial time algorithm to solve Unique Games on the hypercube.

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