Consider a system with N identical single-server queues and M(N ) task types, where each server is able to process only a small subset of possible task types. Arriving tasks select d ≥ 2 random compatible servers, and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph G N between the servers and the task types. When G N is complete bipartite, the meanfield approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the meanfield approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server-degree almost by a factor N /ln(N ) compared to the complete bipartite compatibility graph.

We examine the problem of smoothed online optimization, where a decision maker must sequentially choose points in a normed vector space to minimize the sum of per-round, non-convex hitting costs and the costs of switching decisions between rounds. The decision maker has access to a black-box oracle, such as a machine learning model, that provides untrusted and potentially inaccurate predictions of the optimal decision in each round. The goal of the decision maker is to exploit the predictions if they are accurate, while guaranteeing performance that is not much worse than the hindsight optimal sequence of decisions, even when predictions are inaccurate. We impose the standard assumption that hitting costs are globally α-polyhedral. We propose a novel algorithm, Adaptive Online Switching (AOS), and prove that, for a large set of feasible δ > 0, it is (1+δ)-competitive if predictions are perfect, while also maintaining a uniformly bounded competitive ratio of 2~O (1/(α δ)) even when predictions are adversarial. Further, we prove that this trade-off is necessary and nearly optimal in the sense that any deterministic algorithm which is (1+δ)-competitive if predictions are perfect must be at least 2~Ω (1/(α δ)) -competitive when predictions are inaccurate. In fact, we observe a unique threshold-type behavior in this trade-off: if δ is not in the set of feasible options, then no algorithm is simultaneously (1 + δ)-competitive if predictions are perfect and ζ-competitive when predictions are inaccurate for any ζ < ∞. Furthermore, we discuss that memory is crucial in AOS by proving that any algorithm that does not use memory cannot benefit from predictions. We complement our theoretical results by a numerical study on a microgrid application.

Consider a system with N identical single-server queues and a number of task types, where each server is able to process only a small subset of possible task types. Arriving tasks select [Formula: see text] random compatible servers and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph between the servers and the task types. When the graph is complete bipartite, the mean-field approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the mean-field approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server degree almost by a factor [Formula: see text] compared with the complete bipartite compatibility graph.

Consider a system with N identical single-server queues and M(N) task types, where each server is able to process only a small subset of possible task types. Arriving tasks select d≥2 random compatible servers, and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph GN between the servers and the task types. When GN is complete bipartite, the meanfield approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the meanfield approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server-degree almost by a factor N/ln(N) compared to the complete bipartite compatibility graph.

Modern data centers suffer from immense power consumption. As a result, data center operators have heavily invested in capacity scaling solutions, which dynamically deactivate servers if the demand is low and activate them again when the workload increases. We analyze a continuoustime model for capacity scaling, where the goal is to minimize the weighted sum of flow-time, switching cost, and power consumption in an online fashion. We propose a novel algorithm, called the Adaptive Balanced Capacity Scaling (ABCS) algorithm, that has access to black-box machine learning predictions. ABCS aims to adapt to the predictions and is also robust against unpredictable surges in the workload. In particular, we prove that the ABCS algorithm is (1 + ")-competitive if the predictions are accurate, and yet, it has a uniformly bounded competitive ratio even if the predictions are completely inaccurate.