A Scalable Lift-and-Project Differentiable Approach For the Maximum Cut Problem

We propose a scalable framework for solving the Maximum Cut (MaxCut) problem in large graphs using projected gradient ascent on quadratic objectives. Our approach is differentiable and leverages GPUs for gradient-based optimization. It is not a machine learning method and does not require training data. Starting from a continuous relaxation of the classical quadratic binary formulation, we present a parallelized strategy that explores multiple initialization vectors in batch. We analyze the relaxed objective, showing it is convex and has fixed-points corresponding to local optima, particularly at boundary points, highlighting a key challenge in non-convex optimization. To improve exploration, we introduce a lifted quadratic formulation that over-parameterizes the solution space. We also provide a theoretical characterization of these lifted fixed-points. Finally, we propose DECO, a dimension-alternating algorithm that switches between the unlifted and lifted formulations, combined with importance-based degree initialization and a population-based evolutionary hyper-parameter search. Experiments on diverse graph families show that our methods attain comparable or superior performance relative to recent neural networks and GPU-accelerated sampling approaches.

💡 Research Summary

The paper introduces a scalable, data‑free framework for solving the Maximum Cut (MaxCut) problem on large graphs by leveraging projected gradient ascent on continuous relaxations of the classic quadratic binary formulation. Starting from the well‑known QUBO representation of MaxCut, the authors relax the binary variables z ∈ {‑1, 1}ⁿ to a continuous box x ∈