In this paper, we study the classic optimization problem of Related Machine Online Load Balancing under the conditions of selfish machines and selfish jobs. We have $m$ related machines with varying speeds and $n$ jobs arriving online with different sizes. Our objective is to design an online truthful algorithm that minimizes the makespan while ensuring that jobs and machines report their true sizes and speeds. Previous studies in the online scenario have primarily focused on selfish jobs, beginning with the work of Aspnes et al. (JACM 1997). An $O(1)$-competitive online mechanism for selfish jobs was discovered by Feldman, Fiat, and Roytman (EC 2017). For selfish machines, truthful mechanisms have only been explored in offline settings, starting with Archer and Tardos (FOCS 2001). The best-known results are two PTAS mechanisms by Christodoulou and Kov\'{a}cs (SICOMP 2013) and Epstein et al. (MOR 2016). We design an online mechanism that is truthful for both machines and jobs, achieving a competitive ratio of $O(\log m)$. This is the first non-trivial two-sided truthful mechanism for online load balancing and also the first non-trivial machine-side truthful mechanism. Furthermore, we extend our mechanism to the $\ell_q$ norm variant of load balancing, maintaining two-sided truthfulness with a competitive ratio of $\tilde{O}(m^{\frac{1}{q}(1-\frac{1}{q})})$.

翻译：本文研究在机器与任务均具自利性条件下的经典优化问题——相关机器在线负载均衡。我们拥有$m$台运行速度各异的相关机器，以及$n个在线到达且规模不同的任务。我们的目标是设计一种在线的真实算法，在确保任务和机器如实报告其规模与速度的前提下，最小化完工时间。先前针对在线场景的研究主要集中于自利性任务，始于Aspnes等人（JACM 1997）的工作。Feldman、Fiat与Roytman（EC 2017）发现了一种针对自利性任务的$O(1)$竞争性在线机制。对于自利性机器，真实机制的研究仅在线下场景中展开，始于Archer与Tardos（FOCS 2001）。目前最著名的成果是Christodoulou与Kovács（SICOMP 2013）以及Epstein等人（MOR 2016）提出的两种PTAS机制。我们设计了一种对机器和任务均保持真实性的在线机制，其竞争比为$O(\log m)$。这是首个非平凡的双边真实在线负载均衡机制，同时也是首个非平凡的机器侧真实机制。此外，我们将该机制扩展至负载均衡的$\ell_q$范数变体，在保持双边真实性的同时，实现了$\tilde{O}(m^{\frac{1}{q}(1-\frac{1}{q})})$的竞争比。