Binarizing Physics-Inspired GNNs for Combinatorial Optimization

TL;DR

This paper investigates the limitations of physics-inspired GNNs in dense combinatorial problems, analyzes their training dynamics, and proposes binarization techniques to improve their performance.

Contribution

It identifies the performance drop in PI-GNNs with dense graphs and introduces binarization methods inspired by fuzzy logic to enhance their effectiveness.

Findings

PI-GNNs' performance decreases with graph density.

Phase transition in training dynamics causes degenerate solutions.

Binarization techniques improve PI-GNNs in dense problems.

Abstract

Physics-inspired graph neural networks (PI-GNNs) have been utilized as an efficient unsupervised framework for relaxing combinatorial optimization problems encoded through a specific graph structure and loss, reflecting dependencies between the problem's variables. While the framework has yielded promising results in various combinatorial problems, we show that the performance of PI-GNNs systematically plummets with an increasing density of the combinatorial problem graphs. Our analysis reveals an interesting phase transition in the PI-GNNs' training dynamics, associated with degenerate solutions for the denser problems, highlighting a discrepancy between the relaxed, real-valued model outputs and the binary-valued problem solutions. To address the discrepancy, we propose principled alternatives to the naive strategy used in PI-GNNs by building on insights from fuzzy logic and binarized neural networks. Our experiments demonstrate that the portfolio of proposed methods significantly improves the performance of PI-GNNs in increasingly dense settings.