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Decision-focused Graph Neural Networks for Combinatorial Optimization

Yang Liu, Chuan Zhou, Peng Zhang, Shirui Pan, Zhao Li, Hongyang Chen

TL;DR

This work targets combinatorial optimization (COP) by fusing graph neural networks with decision-focused learning in an end-to-end pipeline. It introduces G-DFL4CO, comprising a graph predictive model trained in an unsupervised manner and a differentiable QUBO-based optimizer, enabling backpropagation through the entire decision process. The framework is instantiated on MaxCut, MIS, and MVC, where it outperforms a standalone GNN and classical solvers, demonstrating improved accuracy and efficiency. By integrating neural graph predictions with traditional optimization, the approach offers a scalable, precise path to solving NP-hard COPs in an inductive setting.

Abstract

In recent years, there has been notable interest in investigating combinatorial optimization (CO) problems by neural-based framework. An emerging strategy to tackle these challenging problems involves the adoption of graph neural networks (GNNs) as an alternative to traditional algorithms, a subject that has attracted considerable attention. Despite the growing popularity of GNNs and traditional algorithm solvers in the realm of CO, there is limited research on their integrated use and the correlation between them within an end-to-end framework. The primary focus of our work is to formulate a more efficient and precise framework for CO by employing decision-focused learning on graphs. Additionally, we introduce a decision-focused framework that utilizes GNNs to address CO problems with auxiliary support. To realize an end-to-end approach, we have designed two cascaded modules: (a) an unsupervised trained graph predictive model, and (b) a solver for quadratic binary unconstrained optimization. Empirical evaluations are conducted on various classical tasks, including maximum cut, maximum independent set, and minimum vertex cover. The experimental results on classical CO problems (i.e. MaxCut, MIS, and MVC) demonstrate the superiority of our method over both the standalone GNN approach and classical methods.

Decision-focused Graph Neural Networks for Combinatorial Optimization

TL;DR

This work targets combinatorial optimization (COP) by fusing graph neural networks with decision-focused learning in an end-to-end pipeline. It introduces G-DFL4CO, comprising a graph predictive model trained in an unsupervised manner and a differentiable QUBO-based optimizer, enabling backpropagation through the entire decision process. The framework is instantiated on MaxCut, MIS, and MVC, where it outperforms a standalone GNN and classical solvers, demonstrating improved accuracy and efficiency. By integrating neural graph predictions with traditional optimization, the approach offers a scalable, precise path to solving NP-hard COPs in an inductive setting.

Abstract

In recent years, there has been notable interest in investigating combinatorial optimization (CO) problems by neural-based framework. An emerging strategy to tackle these challenging problems involves the adoption of graph neural networks (GNNs) as an alternative to traditional algorithms, a subject that has attracted considerable attention. Despite the growing popularity of GNNs and traditional algorithm solvers in the realm of CO, there is limited research on their integrated use and the correlation between them within an end-to-end framework. The primary focus of our work is to formulate a more efficient and precise framework for CO by employing decision-focused learning on graphs. Additionally, we introduce a decision-focused framework that utilizes GNNs to address CO problems with auxiliary support. To realize an end-to-end approach, we have designed two cascaded modules: (a) an unsupervised trained graph predictive model, and (b) a solver for quadratic binary unconstrained optimization. Empirical evaluations are conducted on various classical tasks, including maximum cut, maximum independent set, and minimum vertex cover. The experimental results on classical CO problems (i.e. MaxCut, MIS, and MVC) demonstrate the superiority of our method over both the standalone GNN approach and classical methods.
Paper Structure (20 sections, 1 theorem, 7 equations, 6 figures, 3 tables)

This paper contains 20 sections, 1 theorem, 7 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Graph neural networks
  4. Combinatorial optimization
  5. Decision-focused learning
  6. Mathematical background
  7. Notations.
  8. Graph and graph neural networks
  9. Combinatorial optimization
  10. Our proposed framework
  11. Graph predictive model
  12. Optimizer
  13. Experiments
  14. Scenarios
  15. Datasets.
  16. ...and 5 more sections

Key Result

Theorem 1

Assume $\mathbf{x}^*$ is the local maxima of $F(\mathbf{x}, \theta)$, there exists a neighborhood $\mathcal{I}$ around $\mathbf{x}^*$ such that the maximizer of $F(\mathbf{x}, \theta)$ within $\mathcal{I} \cup \mathcal{X}$ is differentiable almost everywhere.

Figures (6)

  • Figure 1: The overarching framework of our proposed G-DFL4CO involves an end-to-end decision-focused learning process.
  • Figure 2: Flow chart illustrating the end-to-end workflow for our proposed framework using a simple example with five nodes.
  • Figure 3: Some visualization examples of the d-regular instances ($n=50,100,150$).
  • Figure 4: Comparison of MaxCut solution for d-regular graphs ($d = 3,5$) in different size graphs.
  • Figure 5: Some visualization of Minimum Vertex Cover for several d-regular graphs in the framework of G-DFL4CO.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Definition 1: Maximum cut (MaxCut)
  • Definition 2: Maximum independent set (MIS)
  • Definition 3: Minimum vertex cover (MVC)
  • Definition 4: local maxima
  • Theorem 1