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Exploration through Generation: Applying GFlowNets to Structured Search

Mark Phillip Matovic

TL;DR

This work investigates the applicability of Generative Flow Networks (GFlowNets) to canonical graph optimization problems—Shortest Path, Minimum Spanning Tree, and Traveling Salesperson—by learning sequential policies trained with Trajectory Balance. A unified GFlowNet framework is proposed, combining problem-specific state representations, action masking, and a TB loss to ensure generated complete solutions have probability proportional to their inverse cost, yielding optimal solutions on benchmark instances. Key contributions include demonstrating a single architecture across problems of varying complexity, effectively enforcing combinatorial constraints, and analyzing the trade-offs between upfront training and amortized inference. The findings show that GFlowNets can recover classical optima on small instances and highlight scalability challenges and the need for graph-aware architectures to extend to larger, real-world problems. Overall, the work positions GFlowNets as a promising, complementary approach to classical algorithms for scalable combinatorial optimization under sufficient computational resources.

Abstract

This work applies Generative Flow Networks (GFlowNets) to three graph optimization problems: the Traveling Salesperson Problem, Minimum Spanning Tree, and Shortest Path. GFlowNets are generative models that learn to sample solutions proportionally to a reward function. The models are trained using the Trajectory Balance loss to build solutions sequentially, selecting edges for spanning trees, nodes for paths, and cities for tours. Experiments on benchmark instances of varying sizes show that GFlowNets learn to find optimal solutions. For each problem type, multiple graph configurations with different numbers of nodes were tested. The generated solutions match those from classical algorithms (Dijkstra for shortest path, Kruskal for spanning trees, and exact solvers for TSP). Training convergence depends on problem complexity, with the number of episodes required for loss stabilization increasing as graph size grows. Once training converges, the generated solutions match known optima from classical algorithms across the tested instances. This work demonstrates that generative models can solve combinatorial optimization problems through learned policies. The main advantage of this learning-based approach is computational scalability: while classical algorithms have fixed complexity per instance, GFlowNets amortize computation through training. With sufficient computational resources, the framework could potentially scale to larger problem instances where classical exact methods become infeasible.

Exploration through Generation: Applying GFlowNets to Structured Search

TL;DR

This work investigates the applicability of Generative Flow Networks (GFlowNets) to canonical graph optimization problems—Shortest Path, Minimum Spanning Tree, and Traveling Salesperson—by learning sequential policies trained with Trajectory Balance. A unified GFlowNet framework is proposed, combining problem-specific state representations, action masking, and a TB loss to ensure generated complete solutions have probability proportional to their inverse cost, yielding optimal solutions on benchmark instances. Key contributions include demonstrating a single architecture across problems of varying complexity, effectively enforcing combinatorial constraints, and analyzing the trade-offs between upfront training and amortized inference. The findings show that GFlowNets can recover classical optima on small instances and highlight scalability challenges and the need for graph-aware architectures to extend to larger, real-world problems. Overall, the work positions GFlowNets as a promising, complementary approach to classical algorithms for scalable combinatorial optimization under sufficient computational resources.

Abstract

This work applies Generative Flow Networks (GFlowNets) to three graph optimization problems: the Traveling Salesperson Problem, Minimum Spanning Tree, and Shortest Path. GFlowNets are generative models that learn to sample solutions proportionally to a reward function. The models are trained using the Trajectory Balance loss to build solutions sequentially, selecting edges for spanning trees, nodes for paths, and cities for tours. Experiments on benchmark instances of varying sizes show that GFlowNets learn to find optimal solutions. For each problem type, multiple graph configurations with different numbers of nodes were tested. The generated solutions match those from classical algorithms (Dijkstra for shortest path, Kruskal for spanning trees, and exact solvers for TSP). Training convergence depends on problem complexity, with the number of episodes required for loss stabilization increasing as graph size grows. Once training converges, the generated solutions match known optima from classical algorithms across the tested instances. This work demonstrates that generative models can solve combinatorial optimization problems through learned policies. The main advantage of this learning-based approach is computational scalability: while classical algorithms have fixed complexity per instance, GFlowNets amortize computation through training. With sufficient computational resources, the framework could potentially scale to larger problem instances where classical exact methods become infeasible.
Paper Structure (52 sections, 10 equations, 3 figures, 4 tables, 2 algorithms)

This paper contains 52 sections, 10 equations, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Challenge of Graph Optimization
  3. Classical Algorithms: Deterministic Procedures
  4. The Opportunity: Generative Models for Optimization
  5. Computational Scalability Through Learning
  6. Research Question and Contributions
  7. The GFlowNet Framework
  8. Unified Approach
  9. Key Contributions
  10. Paper Organization
  11. Methodology
  12. Problem Formulations as Sequential MDPs
  13. Shortest Path Problem
  14. Minimum Spanning Tree (MST) Problem
  15. Traveling Salesperson Problem (TSP)
  16. ...and 37 more sections

Figures (3)

  • Figure 1: Trajectory Balance loss during training for shortest path across varying graph sizes. complexity.
  • Figure 2: Trajectory Balance loss during training for minimum spanning tree across varying graph sizes. complexity.
  • Figure 3: Trajectory Balance loss during training for travelling sales person across varying graph sizes. complexity.