Belief propagation for graph partitioning

TL;DR

This paper explores belief propagation for graph bi-partitioning, demonstrating how to fix global parameters locally, and provides a phase diagram and heuristic solver for the problem on random graphs.

Contribution

It introduces a novel method to fix magnetization locally within belief propagation, enabling full phase diagram computation and efficient heuristic solutions.

Findings

Full phase diagram of bi-partitioning on random graphs

Effective heuristic solver for graph partitioning

Method to fix global parameters locally in belief propagation

Abstract

We study the belief propagation algorithm for the graph bi-partitioning problem, i.e. the ground state of the ferromagnetic Ising model at a fixed magnetization. Application of a message passing scheme to a model with a fixed global parameter is not banal and we show that the magnetization can in fact be fixed in a local way within the belief propagation equations. Our method provides the full phase diagram of the bi-partitioning problem on random graphs, as well as an efficient heuristic solver that we anticipate to be useful in a wide range of application of the partitioning problem.