In the weighted load balancing problem, the input is an $n$-vertex bipartite graph between a set of clients and a set of servers, and each client comes with some nonnegative real weight. The output is an assignment that maps each client to one of its adjacent servers, and the load of a server is then the sum of the weights of the clients assigned to it. The goal is to find an assignment that is well-balanced, typically captured by (approximately) minimizing either the $\ell_\infty$- or $\ell_2$-norm of the server loads. Generalizing both of these objectives, the all-norm load balancing problem asks for an assignment that approximately minimizes all $\ell_p$-norm objectives for $p \ge 1$, including $p = \infty$, simultaneously. Our main result is a deterministic $O(\log{n})$-pass $O(1)$-approximation semi-streaming algorithm for the all-norm load balancing problem. Prior to our work, only an $O(\log{n})$-pass $O(\log{n})$-approximation algorithm for the $\ell_\infty$-norm objective was known in the semi-streaming setting. Our algorithm uses a novel application of the multiplicative weights update method to a mixed covering/packing convex program for the all-norm load balancing problem involving an infinite number of constraints.
翻译:在加权负载平衡问题中,输入是一组客户和一组服务器之间的一个 $ $- offex 双向平衡图, 每个客户都有一些非负真实重量。 输出是将每个客户映射到相邻服务器的任务, 然后服务器的负荷是分配给它客户的重量的总和。 目标是找到一个非常平衡的任务, 通常通过( 约) 将服务器负载的 $@ infty$- 或 $\ kal_ 2$- norm 来最小化 。 综合这两个目标, 全部负载平衡问题要求进行一项任务, 大约将每个客户的 $\ ell_ p$- norm 的目标最小化为 1 ge 1 g$, 包括 $ p=\ inty$。 我们的主要结果是找到一个确定性的 $O (log{n) $- passlocal $(1)- a- proclutional max a- common rogrational- pal max a lign_ a ligrocal- pal- pal- lemental- pal_ a- pal- lemental- progniquenal_ a- a- plemental_ a- plemental- plemental_ lementalxn===