# Distributed load balancing: a new framework and improved guarantees

Innovations in Theoretical Computer Science (2021)

## Abstract

Inspired by applications on search engines and web servers, we consider a load balancing problem with a general \textit{convex} objective function. In this problem, we are given a bipartite graph on a set of sources $S$ and a set of workers $W$ and the goal is to distribute the load from each source among its neighboring workers such that the total load of workers are as balanced as possible. We present a new distributed algorithm that works with \textit{any} symmetric non-decreasing convex function for evaluating the balancedness of the workers' load. Our algorithm computes a nearly optimal allocation of loads in $O(\log n \log^2 d/\eps^3)$ rounds where $n$ is the number of nodes, $d$ is the maximum degree, and $\eps$ is the desired precision. If the objective is to minimize the maximum load, we modify the algorithm to obtain a nearly optimal solution in $O(\log n \log d/\eps^2)$ rounds. This improves a line of algorithms that require a polynomial number of rounds in $n$ and $d$ and appear to encounter a fundamental barrier that prevents them from obtaining poly-logarithmic runtime \cite{berenbrink2005dynamic, berenbrink2009new, subramanian1994analysis, rabani1998local}. In our paper, we introduce a novel primal-dual approach with multiplicative weight updates that allows us to circumvent this barrier. Our algorithm is inspired by \cite{agrawal2018proportional} and other distributed algorithms for optimizing linear objectives but introduces several new twists to deal with general convex objectives.