Unique Games on the Hypercube

TL;DR

This paper explores the Unique Games Conjecture on the hypercube, constructing near-optimal integrality gap instances for SDP and proposing that adding triangle inequalities could lead to a polynomial-time solution.

Contribution

It introduces a new integrality gap instance for Max-2-LIN on the hypercube and hypothesizes that enhanced SDP relaxations might efficiently solve these problems.

Findings

Constructed an almost optimal integrality gap instance for the hypercube.

Conjecture that adding triangle inequalities to SDP could solve Unique Games efficiently.

Provides insights into the complexity of Unique Games on high-dimensional structures.

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$(\mathbb{Z}_2)$. We conjecture that adding triangle inequalities to the SDP provides a polynomial time algorithm to solve Unique Games on the hypercube.