Online Selfish Load Balancing

Abstract

In selfish load balancing, there is a set of machines and jobs to be scheduled, where each machine is owned by a selfish agent. Agents hold the processing times of jobs as private information and may strategically misreport them to maximize their utilities. The goal is to design a truthful mechanism that minimizes the makespan. This selfish-machine model was first proposed by Nisan and Ronen (STOC~1999), who presented an $m$-approximation algorithm for unrelated machines in the offline scenario, which was later shown to be tight by Christodoulou, Koutsoupias, and Kovács (STOC 2023). The study of offline selfish load balancing on related machines was initiated by Archer and Tardos (FOCS 2001). The best-known results for this problem are two PTAS mechanisms, due to Christodoulou and Kovács (SICOMP~2013) and Epstein et al. (MOR 2016). However, there is little literature on selfish scheduling in the online scenario, which is precisely what arises in real-world applications (e.g., in cloud platforms, jobs often arrive online). In this paper, we aim to address this gap. For unrelated machines, we observe that the existing $m$-approximation algorithm can also be implemented in an online scenario, implying that $m$ remains the best possible competitive ratio. For related machines, we design the first nontrivial online mechanism that is truthful in expectation and achieves a competitive ratio of $O(\log m)$. Moreover, we extend our mechanism to also guarantee job-side truthfulness (in expectation), ensuring that jobs arriving online report their true sizes. This notion was first studied by Feldman, Fiat, and Roytman (EC 2017), but without combining it with the classic machine-side truthfulness. Finally, we generalize our two-sided truthful mechanism to the $\ell_q$-norm variant of load balancing, achieving a competitive ratio of $\tilde{O}\left(m^{\frac{1}{q}(1-\frac{1}{q})}\right)$.