All-Norm Load Balancing in Graph Streams via the Multiplicative Weights Update Method

TL;DR

This paper presents a deterministic semi-streaming algorithm that approximates all $ ext{l}_p$-norm load balancing objectives simultaneously using the multiplicative weights update method, improving prior work for the $ ext{l}_ ext{infinity}$ case.

Contribution

It introduces a novel application of the multiplicative weights update method to solve the all-norm load balancing problem in semi-streaming settings.

Findings

Achieves $O( ext{log} n)$ passes with $O(1)$ approximation ratio.

Extends previous work from $ ext{l}_ ext{infinity}$ to all $ ext{l}_p$-norm objectives.

Provides a deterministic algorithm for a complex convex program with infinitely many constraints.

Abstract

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.