Deep Learning for Unrelated-Machines Scheduling: Handling Variable Dimensions

Deep learning has been effectively applied to many discrete optimization problems. However, learning-based scheduling on unrelated parallel machines remains particularly difficult to design. Not only do the numbers of jobs and machines vary, but each job-machine pair has a unique processing time, dynamically altering feature dimensions. We propose a novel approach with a neural network tailored for offline deterministic scheduling of arbitrary sizes on unrelated machines. The goal is to minimize a complex objective function that includes the makespan and the weighted tardiness of jobs and machines. Unlike existing online approaches, which process jobs sequentially, our method generates a complete schedule considering the entire input at once. The key contribution of this work lies in the sophisticated architecture of our model. By leveraging various NLP-inspired architectures, it effectively processes any number of jobs and machines with varying feature dimensions imposed by unrelated processing times. Our approach enables supervised training on small problem instances while demonstrating strong generalization to much larger scheduling environments. Trained and tested on instances with 8 jobs and 4 machines, costs were only 2.51% above optimal. Across all tested configurations of up to 100 jobs and 10 machines, our network consistently outperformed an advanced dispatching rule, which incurred 22.22% higher costs on average. As our method allows fast retraining with simulated data and adaptation to various scheduling conditions, we believe it has the potential to become a standard approach for learning-based scheduling on unrelated machines and similar problem environments.

Fast Screening Rules for Optimal Design via Quadratic Lasso Reformulation

The problems of Lasso regression and optimal design of experiments share a critical property: their optimal solutions are typically \emph{sparse}, i.e., only a small fraction of the optimal variables are non-zero. Therefore, the identification of the support of an optimal solution reduces the dimensionality of the problem and can yield a substantial simplification of the calculations. It has recently been shown that linear regression with a \emph{squared} $\ell_1$-norm sparsity-inducing penalty is equivalent to an optimal experimental design problem. In this work, we use this equivalence to derive safe screening rules that can be used to discard inessential samples. Compared to previously existing rules, the new tests are much faster to compute, especially for problems involving a parameter space of high dimension, and can be used dynamically within any iterative solver, with negligible computational overhead. Moreover, we show how an existing homotopy algorithm to compute the regularization path of the lasso method can be reparametrized with respect to the squared $\ell_1$-penalty. This allows the computation of a Bayes $c$-optimal design in a finite number of steps and can be several orders of magnitude faster than standard first-order algorithms. The efficiency of the new screening rules and of the homotopy algorithm are demonstrated on different examples based on real data.