Improved Online Load Balancing in the Two-Norm

2025-11-05

AI Summary

This paper studies online load balancing on unrelated machines with the objective of minimizing the squared ℓ₂-norm of machine loads. We propose a novel primal-dual analysis framework based on semidefinite programming (SDP) relaxation, integrating online correlated randomized rounding with independent rounding to uniformly characterize and optimize fractional assignment policies. Our approach breaks the long-standing competitive ratio barrier of 5, achieving a competitive ratio of 4.9843—the current best-known result. We rigorously prove that this fractional algorithm is theoretically optimal within the class of independent rounding schemes and establish a matching lower bound. Moreover, our framework provides concise, unified optimality proofs for several classical algorithms, demonstrating its broad applicability and conceptual unification power in online scheduling theory.

Technology Category

• Machine Learning: Online Learning & Bandits
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the online load balancing problem on unrelated machines, with the objective of minimizing the square of the $ell_2$ norm of the loads on the machines. The greedy algorithm of Awerbuch et al. (STOC'95) is optimal for deterministic algorithms and achieves a competitive ratio of $3 + 2 sqrt{2} approx 5.828$, and an improved $5$-competitive randomized algorithm based on independent rounding has been shown by Caragiannis (SODA'08). In this work, we present the first algorithm breaking the barrier of $5$ on the competitive ratio, achieving a bound of $4.9843$. To obtain this result, we use a new primal-dual framework to analyze this problem based on a natural semidefinite programming relaxation, together with an online implementation of a correlated randomized rounding procedure of Im and Shadloo (SODA'20). This novel primal-dual framework also yields new, simple and unified proofs of the competitive ratio of the $(3 + 2 sqrt{2})$-competitive greedy algorithm, the $5$-competitive randomized independent rounding algorithm, and that of a new $4$-competitive optimal fractional algorithm. We also provide lower bounds showing that the previous best randomized algorithm is optimal among independent rounding algorithms, that our new fractional algorithm is optimal, and that a simple greedy algorithm is optimal for the closely related online scheduling problem $R || sum w_j C_j$.

Problem

• Minimizing squared L2 norm of machine loads in online load balancing
• Breaking competitive ratio barrier of 5 for unrelated machines
• Developing primal-dual framework with semidefinite programming relaxation

Innovation

• Novel primal-dual framework using SDP relaxation
• Online correlated randomized rounding implementation
• Unified analysis providing competitive ratio proofs