Tackling the Linear Sum Assignment Problem with Graph Neural Networks

Linear Assignment Problems are fundamental combinatorial optimization problems that appear throughout domains such as logistics, robotics and telecommunications. In general, solving assignment problems to optimality is computationally infeasible even for contexts of small dimensionality, and so heuristic algorithms are often employed to find near-optimal solutions. The handcrafting of a heuristic usually requires expert-knowledge to exploit the problem structure to be addressed, however if the problem description changes slightly, a previously derived heuristic may no longer be appropriate. This work explores a more general-purpose learning approach, based on the description of the problem through a bipartite graph, and the use of a Message Passing Graph Neural Network model, to attain the correct assignment permutation. The simulation results indicate that the proposed structure allows for a significant increase in classification accuracy if compared with two different DNN approaches based on Dense Networks and Convolutional Neural Networks, furthermore, the GNN has proved to be very efficient with regard to the processing time and memory requirements, thanks to intrinsic parameter-sharing capability.