# Rounding semidefinite programs for large-domain problems via Brownian motion

## TL;DR

This paper introduces a novel rounding method for semidefinite programming relaxations in large-domain problems, leveraging Brownian motion to improve approximate solutions for angular synchronization.

## Contribution

It proposes a simple, Brownian motion-based rounding scheme for SDP relaxations, specifically applied to angular synchronization problems, with conjectured near-optimal guarantees.

## Key findings

- The rounding scheme is feasible and effective based on computational evidence.
- It achieves approximation guarantees close to the best possible under the Unique-Games Conjecture.
- The method simplifies the rounding process for large-domain SDP problems.

## Abstract

We present a new simple method for rounding a semidefinite programming relaxation of a constraint satisfaction problem. We apply it to the problem of approximate angular synchronization. Specifically, we are given directed distances on a circle (i.e., directed angles) between pairs of elements and our goal is to assign the elements to positions on a circle so as to preserve these distances as much as possible. The feasibility of our rounding scheme is based on properties of the well-known stochastic process called Brownian motion. Based on computational and other evidence, we conjecture that this rounding scheme yields an approximation guarantee that is very close to the best-possible guarantee (assuming the Unique-Games Conjecture).