Annealing Machine-assisted Learning of Graph Neural Network for Combinatorial Optimization

TL;DR

The paper tackles scaling combinatorial optimization by coupling Annealing Machines (AM) with Graph Neural Networks (GNNs), leveraging AM's precision and GNNs' scalability. It introduces a multiresolution pipeline that compresses the target graph ${\mathcal{G}}_P$ into a sequence ${\mathcal{G}}_i$ via Louvain decomposition, uses AM solutions ${\mathbf{x}}^{\rm AM}_i$ as supervision for local GNNs, and aggregates their learned representations to initialize a Main GNN that solves the original problem on ${\mathcal{G}}_P$, minimizing the QUBO objective $H_{\rm QUBO}({\mathbf{x}}^{\rm GNN})$. Empirical results on MIS, MaxCut, and Graph Partition show that AM-guided mrGNN (mrGNN+AM) can handle graphs up to $|\mathcal{V}|=150{,}000$, delivering improved solution quality for large instances and faster convergence relative to purely GNN-based baselines, albeit with some runtime trade-offs. This work demonstrates a scalable bridge between AM accuracy and GNN scalability, with future directions including end-to-end differentiable training and reuse of partial solutions across related problems.

Abstract

While Annealing Machines (AM) have shown increasing capabilities in solving complex combinatorial problems, positioning themselves as a more immediate alternative to the expected advances of future fully quantum solutions, there are still scaling limitations. In parallel, Graph Neural Networks (GNN) have been recently adapted to solve combinatorial problems, showing competitive results and potentially high scalability due to their distributed nature. We propose a merging approach that aims at retaining both the accuracy exhibited by AMs and the representational flexibility and scalability of GNNs. Our model considers a compression step, followed by a supervised interaction where partial solutions obtained from the AM are used to guide local GNNs from where node feature representations are obtained and combined to initialize an additional GNN-based solver that handles the original graph's target problem. Intuitively, the AM can solve the combinatorial problem indirectly by infusing its knowledge into the GNN. Experiments on canonical optimization problems show that the idea is feasible, effectively allowing the AM to solve size problems beyond its original limits.

Figures (2)

• Figure 1: (a) Proposed approach: Original graph ${\mathcal{G}}_P$ is compressed into a sequence of decreasing size problem graphs, which are solved by AM. A local GNN solver uses those solutions as guidance, and their resulting learned node representations are aggregated through a Mapping module to initialize a Main GNN solver that attaches to the original graph ${\mathcal{G}}_P$. (b) Guiding block (GB) diagram.
• Figure 2: Evolution of the assignment scores from for a sample of 200 nodes for the MaxCut problem on the $n=50$k, $d=3$ graph using mrGNN.