# Approximating Unique Games Using Low Diameter Graph Decomposition

## TL;DR

This paper introduces approximation algorithms for Unique Games on graphs with low diameter decompositions, improving solutions for Max2Lin and MaxCut problems in minor-free graphs and general graphs, using LP relaxation and graph decomposition techniques.

## Contribution

It presents novel approximation algorithms leveraging low diameter graph decomposition directly on the constraint graph, extending prior results to broader graph classes.

## Key findings

- Approximation ratio for Max2Lin_k in K_r-minor free graphs is 1 - O(rε).
- Improved MaxCut approximation for K_r-minor free graphs.
- Algorithms extend to graphs of genus g with similar bounds involving log g.

## Abstract

We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition. For the ${\sf Max2Lin}_k$ problem in $K_r$-minor free graphs, when there is an assignment satisfying $1-\varepsilon$ fraction of constraints, we present an algorithm that produces an assignment satisfying $1-O(r\varepsilon)$ fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the ${\sf MaxCut}$ problem for $K_r$-minor free graphs. For general Unique Games in $K_r$-minor free graphs, we provide another algorithm that produces an assignment satisfying $1-O(r \sqrt{\varepsilon})$ fraction of constraints. Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus $g$, and we get similar results with $r$ replaced by $\log g$ in the ${\sf Max2Lin}_k$ problem and by $\sqrt{\log g}$ in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the ${\sf Max2Lin}_k$ case, and the latter result generalizes the result of Trevisan for general Unique Games.