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

2026-05-07

AI Summary

This work proposes a novel self-supervised graph neural network architecture to reduce the substantial computational cost of repeatedly solving Max-Cut semidefinite programming (SDP) relaxations within branch-and-bound algorithms. Designed as a lightweight SDP proxy solver embedded in an exact optimization framework, the model requires no ground-truth SDP solutions for supervision and directly predicts primal- and dual-feasible SDP solutions. These predictions are combined with Goemans–Williamson randomized rounding to produce high-quality cut solutions. To the best of our knowledge, this is the first graph neural network approach that achieves fully self-supervised training while preserving feasibility, significantly accelerating the solution of structured convex relaxations. Experiments demonstrate that the method reduces the boundary computation cost in exact Max-Cut solving by up to 10.6× compared to the commercial solver MOSEK.

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.

Problem

• Max-Cut
• branch-and-bound
• semidefinite programming
• combinatorial optimization
• convex relaxations

Innovation

• Graph Neural Networks
• Max-Cut
• Semidefinite Programming
• Branch-and-Bound
• Self-Supervised Learning