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 learningbased scheduling on unrelated machines and similar problem environments.

Finding a policy for stochastic scheduling problems on parallel machines is a complicated task, as the problem is already NP-hard in the deterministic case. The problem becomes even more challenging in the unrelated machines environment, where the processing time of a job depends on the machine where it is executed. This article proposes an approach based on reinforcement learning to tackle this class of problems. In this study, we apply supervised learning to train a neural network able to handle small instances, using training data obtained by solving a set of instances to optimality, by dynamic programming. Different representations of the stochastic information in the training data are investigated. Our experiments show that the policy derived from the neural network outperforms the state-of-the-art heuristics used in this field, and produce schedules with expected costs within 1% of the true optimal policy. Moreover, the resulting policy generalizes well to stochastic instances with probability distributions of job durations other than those for which it has been trained, as well as instances with more jobs and machines.

The problems of Lasso regression and optimal design of experiments share a critical property: their optimal solutions are typically 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 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.

In the Symmetric Rendezvous Search on the Line with Unknown Initial Distance, two identical agents are placed on the real line with their distance, the other's location, and their orientation unknown to them. Moving along the line at unit speed and executing the same randomized search strategy, the agents' goal is to meet up as early as possible. The expected meeting time obviously depends on the unknown initial distance and orientations. The quality of a randomized search strategy is thus measured by its competitive ratio, that is, the ratio of the expected meeting time and the earliest possible meeting time (half the initial distance). We present a class of successively refined randomized search strategies together with a rigorous mathematical analysis of their continuously improved competitive ratios. These strategies all rely on the basic idea of performing an infinite sequence of steps of geometrically increasing size in random directions, always returning to the agent's initial position before starting the next step. In addition, our more refined strategies use two novel ideas. First, remembering their past random choices, the agents randomly choose the direction of the next step in a Markov-chain-like manner. Second, choosing the next few random directions in advance, each agent may combine consecutive steps in the same direction into one longer step. As our main result, we show that this combination of looking into the past as well as into the future leads to a substantially improved competitive ratio of 13.93 compared to the previously best known bound of 24.85 (Ozsoyeller et al. 2013).

We consider the stochastic scheduling problem of minimizing the expected makespan on $m$ parallel identical machines. While the (adaptive) list scheduling policy achieves an approximation ratio of $2$, any (non-adaptive) fixed assignment policy has performance guarantee $Ω\left(\frac{\log m}{\log \log m}\right)$. Although the performance of the latter class of policies are worse, there are applications in which non-adaptive policies are desired. In this work, we introduce the two classes of $δ$-delay and $τ$-shift policies whose degree of adaptivity can be controlled by a parameter. We present a policy - belonging to both classes - which is an $\mathcal{O}(\log \log m)$-approximation for reasonably bounded parameters. In other words, an exponential improvement on the performance of any fixed assignment policy can be achieved when allowing a small degree of adaptivity. Moreover, we provide a matching lower bound for any $δ$-delay and $τ$-shift policy when both parameters, respectively, are in the order of the expected makespan of an optimal non-anticipatory policy.

Let be a directed acyclic graph with arcs, a source and a sink . We introduce the cone of flow matrices, which is a polyhedral cone generated by the matrices , where is the incidence vector of the path . We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc‐representation of a flow, reduce to a linear optimization problem over . This cone is intractable: we prove that the membership problem associated to is NP‐complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial‐time the decomposition of a matrix as a linear combination of some 's. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of . We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc‐capacities.

We introduce the class of spot‐checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. In an SC game, the pure strategies of network users correspond to paths in a graph, and the pure strategies of the inspectors are subset of arcs to be controlled. Although SC games are not zero‐sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium (SSE) is more relevant for this problem and we give a mixed integer programming (MIP) formulation for this problem. We show that the computation of such an equilibrium is NP‐hard. More generally, we prove that it is NP‐hard to compute a SSE in a polymatrix game, even if the game is pairwise zero‐sum. Then, we give some bounds on the price of spite, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we report computational experiments on instances constructed from real data, for an application to the enforcement of a truck toll in Germany. These numerical results show the efficiency of the proposed methods, as well as the quality of the bounds derived in this article. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(4), 312–328 2015

We propose a new approach to optimize the deployment and the sampling rates of network monitoring tools, such as Netflow, on a large IP network. It reduces to solving a stochastic sequence of Second Order Cone Programs. We validate our approach with experiments relying on real data from a commercial network.