Improved Bounds for Distributed Load Balancing

TL;DR

This paper presents the first distributed algorithms achieving constant-factor approximations for load balancing in polylogarithmic rounds, improving upon prior results and also providing near-linear time algorithms in the sequential setting.

Contribution

It introduces the first polylogarithmic-round distributed algorithms for constant-factor load balancing approximations, applicable to both unweighted and weighted clients, in the CONGEST and LOCAL models.

Findings

Constant-factor approximation algorithms in polylogarithmic rounds

First near-linear time approximation in the sequential setting

Simultaneous approximation of all -norms including _{}-norms

Abstract

In the load balancing problem, the input is an $n$-vertex bipartite graph $G = (C \cup S, E)$ and a positive weight for each client $c \in C$. The algorithm must assign each client $c \in C$ to an adjacent server $s \in S$. The load of a server is then the weighted sum of all the clients assigned to it, and the goal is to compute an assignment that minimizes some function of the server loads, typically either the maximum server load (i.e., the $\ell_{\infty}$-norm) or the $\ell_p$-norm of the server loads. We study load balancing in the distributed setting. There are two existing results in the CONGEST model. Czygrinow et al. [DISC 2012] showed a 2-approximation for unweighted clients with round-complexity $O(\Delta^5)$, where $\Delta$ is the maximum degree of the input graph. Halld\'orsson et al. [SPAA 2015] showed an $O(\log{n}/\log\log{n})$-approximation for unweighted clients and $O(\log^2\!{n}/\log\log{n})$-approximation for weighted clients with round-complexity polylog$(n)$. In this paper, we show the first distributed algorithms to compute an $O(1)$-approximation to the load balancing problem in polylog$(n)$ rounds. In the CONGEST model, we give an $O(1)$-approximation algorithm in polylog$(n)$ rounds for unweighted clients. For weighted clients, the approximation ratio is $O(\log{n})$. In the less constrained LOCAL model, we give an $O(1)$-approximation algorithm for weighted clients in polylog$(n)$ rounds. Our approach also has implications for the standard sequential setting in which we obtain the first $O(1)$-approximation for this problem that runs in near-linear time. A 2-approximation is already known, but it requires solving a linear program and is hence much slower. Finally, we note that all of our results simultaneously approximate all $\ell_p$-norms, including the $\ell_{\infty}$-norm.