# A Note on Learning Algorithms for Quadratic Assignment with Graph Neural Networks

@article{Nowak2017ANO, title={A Note on Learning Algorithms for Quadratic Assignment with Graph Neural Networks}, author={Alex W. Nowak and Soledad Villar and Afonso S. Bandeira and Joan Bruna}, journal={ArXiv}, year={2017}, volume={abs/1706.07450} }

Inverse problems correspond to a certain type of optimization problems formulated over appropriate input distributions. Recently, there has been a growing interest in understanding the computational hardness of these optimization problems, not only in the worst case, but in an average-complexity sense under this same input distribution.
In this revised note, we are interested in studying another aspect of hardness, related to the ability to learn how to solve a problem by simply observing a…

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## References

SHOWING 1-10 OF 37 REFERENCES

### Learning Combinatorial Optimization Algorithms over Graphs

- Computer ScienceNIPS
- 2017

This paper proposes a unique combination of reinforcement learning and graph embedding that behaves like a meta-algorithm that incrementally constructs a solution, and the action is determined by the output of agraph embedding network capturing the current state of the solution.

### Graph Matching: Relax at Your Own Risk

- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2016

It is proved that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimalpermutation.

### Projected power iteration for network alignment

- Computer ScienceOptical Engineering + Applications
- 2017

This work proposes the algorithm Projected Power Alignment, which is a projected power iteration version of EigenAlign, a fast spectral method with convergence guarantees for Erdős-Renyí graphs, and describes the theory that may be used to provide performance guarantees for Projected power Alignment.

### Supervised Community Detection with Line Graph Neural Networks

- Computer ScienceICLR
- 2019

This work presents a novel family of Graph Neural Networks (GNNs) for solving community detection problems in a supervised learning setting and shows that, in a data-driven manner and without access to the underlying generative models, they can match or even surpass the performance of the belief propagation algorithm on binary and multi-class stochastic block models.

### Semidefinite programming approach for the quadratic assignment problem with a sparse graph

- Computer ScienceComput. Optim. Appl.
- 2018

A new SDP relaxation involving a number of positive semidefinite matrices of dimension O(n) produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.

### Graph matching: relax or not?

- MathematicsArXiv
- 2014

It is proved that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence and in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving 2n equality and n^2 inequality constraints.

### Spectral Alignment of Networks

- Computer ScienceArXiv
- 2016

This paper proposes a network alignment framework that uses an orthogonal relaxation of the underlying QAP in a maximum weight bipartite matching optimization, and generalizes the objective function of the network alignment problem to consider both matched and mismatched interactions in a standard QAP formulation.

### Neural Networks with Finite Intrinsic Dimension have no Spurious Valleys

- Computer ScienceArXiv
- 2018

Focusing on a class of two-layer neural networks defined by smooth activation functions, it is proved that as soon as the hidden layer size matches the intrinsic dimension of the reproducing space, defined as the linear functional space generated by the activations, no spurious valleys exist, thus allowing the existence of descent directions.

### Neural Combinatorial Optimization with Reinforcement Learning

- Computer ScienceICLR
- 2017

A framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, and Neural Combinatorial Optimization achieves close to optimal results on 2D Euclidean graphs with up to 100 nodes.

### Community Detection with Graph Neural Networks

- Computer Science
- 2017

This work embeds the resulting class of algorithms within a generic family of graph neural networks and shows that they can reach detection thresholds in a purely data-driven manner, without access to the underlying generative models and with no parameter assumptions.