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

• URL: http://arxiv.org/abs/2605.07113v1
• Date: Fri, 08 May 2026 01:41:40 GMT
• Title: Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks
• Abstract summary: We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for SDP solvers.<n>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.
• Score: 20.951741111898897
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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