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

Ismail Alkhouri, Mian Wu, Cunxi Yu, Jia Liu, Rongrong Wang, Alvaro Velasquez

TL;DR

This work tackles scaling MaxCut on large graphs with a training-data-free, differentiable approach. It introduces two continuous relaxations, an unlifted QUCO and a lifted LUCO, and augments them with a dimension-alternating DECO algorithm, degree-based initializations, and an evolutionary hyper-parameter search. Theoretical fixed-point analyses accompany practical GPU-accelerated optimizers, and extensive experiments show competitive cuts against neural and GPU-based sampling baselines, especially on very large graphs where exact or learned methods struggle. The result is a scalable, data-less optimization framework that leverages over-parameterization and parallelism to outperform several baselines while maintaining tractability.

Abstract

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.

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

TL;DR

This work tackles scaling MaxCut on large graphs with a training-data-free, differentiable approach. It introduces two continuous relaxations, an unlifted QUCO and a lifted LUCO, and augments them with a dimension-alternating DECO algorithm, degree-based initializations, and an evolutionary hyper-parameter search. Theoretical fixed-point analyses accompany practical GPU-accelerated optimizers, and extensive experiments show competitive cuts against neural and GPU-based sampling baselines, especially on very large graphs where exact or learned methods struggle. The result is a scalable, data-less optimization framework that leverages over-parameterization and parallelism to outperform several baselines while maintaining tractability.

Abstract

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.

Paper Structure

This paper contains 26 sections, 3 theorems, 30 equations, 2 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations:
  4. Proposed Objective Functions
  5. Differentiable Quadratic Optimization
  6. Lifted Quadratic Formulation
  7. Proposed Algorithms
  8. Degree-based Initialization
  9. Degree-based via Uniform Distribution Initialization (DUI):
  10. Importance-based Degree-based Initialization (IDI):
  11. Parallelized Algorithms
  12. Dimension-Alternating Quadratic Optimization Algorithm
  13. Evolution-based Hyper-parameters Search
  14. Experimental Results
  15. Main Comparison Results
  16. ...and 11 more sections

Key Result

Lemma 1

For any graph with one connected component, $\mathbf{L}$ is positive semidefinite (PSD).

Figures (2)

  • Figure 1: Convergence plots of our method (pDECO), ReSCO, and PIGNN. The y-axis corresponds to the cut value of a graph from the small ER dataset (left), a graph from the Gset dataset (middle), and a graph from a large ER dataset (right).
  • Figure 2: Convergence plots of our method (pDECO), ReSCO, ANYCSP, and PIGNN. The y-axis corresponds to the cut value (multiplied by $10^7$) using two large scale ER graphs

Theorems & Definitions (10)

  • Definition 1: Maximum Cut (MaxCut)
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Definition 2: MaxCut fixed-points
  • Remark 2
  • Theorem 1: Lifted MaxCut fixed-points
  • Remark 3
  • Remark 4
  • proof