Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks

TL;DR

This paper introduces a graph neural network that acts as a fast, reliable proxy for SDP relaxations in Max-Cut, enabling more efficient exact solutions within branch-and-bound algorithms.

Contribution

It presents a novel Max-Cut-specific GNN that predicts feasible SDP solutions efficiently, reducing computational costs significantly without requiring labeled training data.

Findings

Reduces Max-Cut bounding cost by up to 10.6 times compared to Mosek.

Predicts primal and dual SDP solutions with complexity O(n^2 + ne).

Trained in a self-supervised manner without solved SDP labels.

Abstract

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity $\mathcal{O}(n^2 + ne)$, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to $10.6 \times$ compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.