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Towards a General Recipe for Combinatorial Optimization with Multi-Filter GNNs

Frederik Wenkel, Semih Cantürk, Stefan Horoi, Michael Perlmutter, Guy Wolf

TL;DR

GCON is a novel GNN architecture that leverages a complex filter bank and localized attention mechanisms to solve CO problems on graphs and is on par with the powerful Gurobi solver on the max-cut problem.

Abstract

Graph neural networks (GNNs) have achieved great success for a variety of tasks such as node classification, graph classification, and link prediction. However, the use of GNNs (and machine learning more generally) to solve combinatorial optimization (CO) problems is much less explored. Here, we introduce GCON, a novel GNN architecture that leverages a complex filter bank and localized attention mechanisms to solve CO problems on graphs. We show how our method differentiates itself from prior GNN-based CO solvers and how it can be effectively applied to the maximum cut, minimum dominating set, and maximum clique problems in a unsupervised learning setting. GCON is competitive across all tasks and consistently outperforms other specialized GNN-based approaches, and is on par with the powerful Gurobi solver on the max-cut problem. We provide an open-source implementation of our work at https://github.com/WenkelF/copt.

Towards a General Recipe for Combinatorial Optimization with Multi-Filter GNNs

TL;DR

GCON is a novel GNN architecture that leverages a complex filter bank and localized attention mechanisms to solve CO problems on graphs and is on par with the powerful Gurobi solver on the max-cut problem.

Abstract

Graph neural networks (GNNs) have achieved great success for a variety of tasks such as node classification, graph classification, and link prediction. However, the use of GNNs (and machine learning more generally) to solve combinatorial optimization (CO) problems is much less explored. Here, we introduce GCON, a novel GNN architecture that leverages a complex filter bank and localized attention mechanisms to solve CO problems on graphs. We show how our method differentiates itself from prior GNN-based CO solvers and how it can be effectively applied to the maximum cut, minimum dominating set, and maximum clique problems in a unsupervised learning setting. GCON is competitive across all tasks and consistently outperforms other specialized GNN-based approaches, and is on par with the powerful Gurobi solver on the max-cut problem. We provide an open-source implementation of our work at https://github.com/WenkelF/copt.
Paper Structure (23 sections, 2 theorems, 26 equations, 2 figures, 7 tables, 3 algorithms)

This paper contains 23 sections, 2 theorems, 26 equations, 2 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Problem Setup
  4. Methodology
  5. Maximum Cut
  6. Maximum Clique
  7. Minimum Dominating Set
  8. The GCON Architecture
  9. The GCON filter bank: Aggregation and Comparison operations
  10. Attention over filters
  11. Theoretical Analysis
  12. Experiments
  13. Conclusion
  14. Limitations and future work.
  15. Further Theoretical Analysis
  16. ...and 8 more sections

Key Result

Theorem 1

Consider the non-decoupled architecture and a $c$-band-dominant representation $\operatorname{\mathbf{x}} \mapsto (\bar{\operatorname{\mathbf{s}}}_f, \operatorname{\mathbf{h}}_f)_{f\in\mathcal{F}}$ at a node $v\in V$ for a non-negative input signal $\operatorname{\mathbf{x}}\geq 0$. Then, if all fil where $\epsilon>0$ can be made arbitrarily small if the scales of the filters are sufficiently larg

Figures (2)

  • Figure 1: Illustration of our general framework. Graphs are first equipped with node features derived from graph statistics and then passed through a GNN composed of GCON hybrid layer blocks (detail in Fig. \ref{['fig:new']}) with a min-max or softmax output layer. A rule-based decoder then processes the node-level outputs to determine the set of interest.
  • Figure 2: Illustration of the layer-wise update for (a) the new decoupled GCON filter bank and (b) the ScatteringClique filter bank from min2022can.

Theorems & Definitions (4)

  • Definition 1: Band-dominant representation
  • Theorem 1
  • Theorem
  • proof