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Learning for Dynamic Combinatorial Optimization without Training Data

Yiqiao Liao, Farinaz Koushanfar, Parinaz Naghizadeh

TL;DR

This work tackles Dynamic Combinatorial Optimization (DCO) by presenting DyCO-GNN, an unsupervised approach that requires no training data beyond the problem instance. Building on PI-GNN, it introduces a shrink-and-perturb (SP) initialization to enable fast convergence while maintaining robust solution quality across time-evolving graph snapshots for MaxCut, MIS, and TSP. The paper provides theoretical support based on a Goemans-Williamson-inspired analysis showing perturbations can increase the probability of reaching the optimal cut, and demonstrates substantial empirical gains (up to 3–60x speedups) under tight to moderate budgets. The approach offers practical benefits for dynamic, resource-constrained decision-making and can adapt rapidly to changing graph structures without offline training or supervision.

Abstract

We introduce DyCO-GNN, a novel unsupervised learning framework for Dynamic Combinatorial Optimization that requires no training data beyond the problem instance itself. DyCO-GNN leverages structural similarities across time-evolving graph snapshots to accelerate optimization while maintaining solution quality. We evaluate DyCO-GNN on dynamic maximum cut, maximum independent set, and the traveling salesman problem across diverse datasets of varying sizes, demonstrating its superior performance under tight and moderate time budgets. DyCO-GNN consistently outperforms the baseline methods, achieving high-quality solutions up to 3-60x faster, highlighting its practical effectiveness in rapidly evolving resource-constrained settings.

Learning for Dynamic Combinatorial Optimization without Training Data

TL;DR

This work tackles Dynamic Combinatorial Optimization (DCO) by presenting DyCO-GNN, an unsupervised approach that requires no training data beyond the problem instance. Building on PI-GNN, it introduces a shrink-and-perturb (SP) initialization to enable fast convergence while maintaining robust solution quality across time-evolving graph snapshots for MaxCut, MIS, and TSP. The paper provides theoretical support based on a Goemans-Williamson-inspired analysis showing perturbations can increase the probability of reaching the optimal cut, and demonstrates substantial empirical gains (up to 3–60x speedups) under tight to moderate budgets. The approach offers practical benefits for dynamic, resource-constrained decision-making and can adapt rapidly to changing graph structures without offline training or supervision.

Abstract

We introduce DyCO-GNN, a novel unsupervised learning framework for Dynamic Combinatorial Optimization that requires no training data beyond the problem instance itself. DyCO-GNN leverages structural similarities across time-evolving graph snapshots to accelerate optimization while maintaining solution quality. We evaluate DyCO-GNN on dynamic maximum cut, maximum independent set, and the traveling salesman problem across diverse datasets of varying sizes, demonstrating its superior performance under tight and moderate time budgets. DyCO-GNN consistently outperforms the baseline methods, achieving high-quality solutions up to 3-60x faster, highlighting its practical effectiveness in rapidly evolving resource-constrained settings.

Paper Structure

This paper contains 25 sections, 2 theorems, 7 equations, 14 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Static CO: PI-GNN and quadratic binary unconstrained optimization (QUBO)
  5. DCO on graphs
  6. Method
  7. Warm-starting PI-GNN
  8. DyCO-GNN: fast convergence and robust solution quality
  9. Theoretical support
  10. Experiments
  11. Datasets
  12. Implementation details
  13. Empirical results on DCO problems
  14. Discussion
  15. Additional related work
  16. ...and 10 more sections

Key Result

Theorem 1

Fix the solution $X_0$ of the GW SDP step, and define $X_{\lambda} := \text{Proj}_{\mathcal{X}}(X_0 + \lambda Z)$, where $\lambda\in \mathbb{R}_{\geq 0}$, $Z$ is a symmetric random matrix sampled from the Gaussian Orthogonal Ensemble, and $\text{Proj}_{\mathcal{X}}(\cdot)$ is projection onto set $\m

Figures (14)

  • Figure 1: Performance of static PI-GNN and warm-started PI-GNN on dynamic MaxCut and MIS instances. Detailed setup is explained in Section \ref{['sec:exp']}.
  • Figure 2: Snapshot-level ApRs on dynamic MaxCut instance (UC Social). All methods use GCNConv.
  • Figure 3: Snapshot-level ApRs on dynamic MIS instance (UC Social). All methods use GCNConv.
  • Figure 4: Snapshot-level ApRs on dynamic TSP instance (burma14). The best checkpoint was taken.
  • Figure 5: Sensitivity analysis using UC Social.
  • ...and 9 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1