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Neural Bipartite Matching

Dobrik Georgiev, Pietro Liò

TL;DR

This work investigates whether graph neural networks can neurally execute a complex graph algorithm by learning to compute a maximum bipartite matching via Ford-Fulkerson on a residual graph. It adopts an encode-process-decode framework to learn subroutines (augmenting-path discovery, bottleneck identification, and capacity augmentation) with stepwise supervision. The authors demonstrate near-perfect generalization across graph sizes and edge densities, showing that well-trained GNN processors can reproduce the sequential reasoning of Ford-Fulkerson and maintain algorithmic invariants. The results suggest a promising direction for learning complex algorithmic reasoning on graphs with potential applications in scheduling, matching, and network optimization.

Abstract

Graph neural networks (GNNs) have found application for learning in the space of algorithms. However, the algorithms chosen by existing research (sorting, Breadth-First search, shortest path finding, etc.) usually align perfectly with a standard GNN architecture. This report describes how neural execution is applied to a complex algorithm, such as finding maximum bipartite matching by reducing it to a flow problem and using Ford-Fulkerson to find the maximum flow. This is achieved via neural execution based only on features generated from a single GNN. The evaluation shows strongly generalising results with the network achieving optimal matching almost 100% of the time.

Neural Bipartite Matching

TL;DR

This work investigates whether graph neural networks can neurally execute a complex graph algorithm by learning to compute a maximum bipartite matching via Ford-Fulkerson on a residual graph. It adopts an encode-process-decode framework to learn subroutines (augmenting-path discovery, bottleneck identification, and capacity augmentation) with stepwise supervision. The authors demonstrate near-perfect generalization across graph sizes and edge densities, showing that well-trained GNN processors can reproduce the sequential reasoning of Ford-Fulkerson and maintain algorithmic invariants. The results suggest a promising direction for learning complex algorithmic reasoning on graphs with potential applications in scheduling, matching, and network optimization.

Abstract

Graph neural networks (GNNs) have found application for learning in the space of algorithms. However, the algorithms chosen by existing research (sorting, Breadth-First search, shortest path finding, etc.) usually align perfectly with a standard GNN architecture. This report describes how neural execution is applied to a complex algorithm, such as finding maximum bipartite matching by reducing it to a flow problem and using Ford-Fulkerson to find the maximum flow. This is achieved via neural execution based only on features generated from a single GNN. The evaluation shows strongly generalising results with the network achieving optimal matching almost 100% of the time.

Paper Structure

This paper contains 23 sections, 1 equation, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Ford-Fulkerson
  4. Algorithm Execution
  5. Preliminary definitions
  6. Encode-process-decode
  7. Supervising algorithm execution
  8. Neurally Executing Ford-Fulkerson
  9. Finding Augmenting Path
  10. Bottleneck Finding
  11. Augmenting Path Capacities
  12. Evaluation through simulation
  13. Finding Augmenting Path and Termination
  14. Bottleneck Finding
  15. Augmenting Path Capacities
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Neural execution of Ford-Fulkerson: The GNN takes as input a residual graph $G_f$. At each step of the algorithm, the GNN computes the augmenting path which is then reused (orange) to find the bottleneck edge on the path. The bottleneck and the augmenting path are then fed through (blue) to produce the residual graph with augmented capacities. The resulting residual graph is the input to the next step (yellow).
  • Figure 2: Maximum flow accuracy (solid) and mean absolute flow error (dashed) per epoch for PNA and MPNN architectures.
  • Figure 3: PNA on $2\times$ scale. The model shows signs of overfitting.
  • Figure :