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

Many inverse problems are formulated as optimization problems over certain 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 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 collection of previously solved instances. These are used to supervise the training of an appropriate predictive model that parametrizes a broad class of algorithms, with the hope that the resulting “algorithm” will provide good accuracy-complexity tradeoffs in the average sense. We illustrate this setup on the Quadratic Assignment Problem, a fundamental problem in Network Science. We observe that data-driven models based on Graph Neural Networks offer intriguingly good performance, even in regimes where standard relaxation based techniques appear to suffer.