  The packing problem, also known as cutting or nesting, has diverse
applications in logistics, manufacturing, layout design, and atlas generation.
It involves arranging irregularly shaped pieces to minimize waste while
avoiding overlap. Recent advances in machine learning, particularly
reinforcement learning, have shown promise in addressing the packing problem.
In this work, we delve deeper into a novel machine learning-based approach that
formulates the packing problem as conditional generative modeling. To tackle
the challenges of irregular packing, including object validity constraints and
collision avoidance, our method employs the score-based diffusion model to
learn a series of gradient fields. These gradient fields encode the
correlations between constraint satisfaction and the spatial relationships of
polygons, learned from teacher examples. During the testing phase, packing
solutions are generated using a coarse-to-fine refinement mechanism guided by
the learned gradient fields. To enhance packing feasibility and optimality, we
introduce two key architectural designs: multi-scale feature extraction and
coarse-to-fine relation extraction. We conduct experiments on two typical
industrial packing domains, considering translations only. Empirically, our
approach demonstrates spatial utilization rates comparable to, or even
surpassing, those achieved by the teacher algorithm responsible for training
data generation. Additionally, it exhibits some level of generalization to
shape variations. We are hopeful that this method could pave the way for new
possibilities in solving the packing problem.
