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

TL;DR

The paper investigates learning algorithms for the Quadratic Assignment Problem by training Graph Neural Networks on solved instances drawn from a distribution, aiming for favorable average-case accuracy–complexity tradeoffs. It introduces a siamese GNN that processes graph signals via local operators and residual connections, and analyzes its optimization landscape under a simplified random-graph model, linking critical points to spectral concentrations. Empirically, the data-driven GNN can outperform spectral and SDP relaxations on Erdos–Renyi and random-regular graphs under moderate perturbations, with a computational cost of $O(n^2)$. The work highlights a data-driven paradigm for approximate combinatorial optimization and outlines open questions on generalization, operator design, and the relationship between statistical and computational hardness.

Abstract

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 collection of previously solved instances. These 'planted solutions' are used to supervise the training of an appropriate predictive model that parametrizes a broad class of algorithms, with the hope that the resulting model 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.

Figures (1)

• Figure 1: Comparison of recovery rates for the SDP peng10, LowRankAlign feizi16 and our data-driven GNN, for the Erdos-Renyi model (left) and random regular graphs (right). All graphs have $n=50$ nodes and edge density $p=0.2$. The recovery rate is measured as the average number of matched nodes from the ground truth. Experiments have been repeated 100 times for every noise level except the SDP, which have been repeated 5 times due to its high computational complexity.