Unique Games and Games Based on Groups

TL;DR

This paper investigates the properties of unique games, especially those with quantum-assisted values near 1, and introduces a group-based family of such games generalizing XOR games, with implications for graph labelling problems.

Contribution

It proves that unique games with high quantum-assisted value have perfect deterministic strategies and introduces a new group-based family of unique games.

Findings

Quantum-assisted value close to 1 implies a perfect deterministic strategy.

Group-based unique games generalize XOR games.

Connections to 3-labelling problems in directed graphs.

Abstract

We study unique games and estimate some of their values. We prove that if a unique game has a quantum-assisted value close to 1, then it must have a perfect deterministic strategy. We introduce a family of unique games based on groups that generalize XOR games, and show that when the group is the cyclic group of order 3, then these games correspond to a 3-labelling problem for directed graphs.