## Improved Bounds for Distributed Load Balancing

Authors:
Sepehr Assadi, Aaron Bernstein, Zach Langley

Conference:
International Symposium on Distributed Computing (DISC 2020)

Recipient of the Best Paper Award at DISC 2020.

Abstract:
In the load balancing problem, the input is an n-vertex bipartite graph G = (C ∪ S, E)—with the two sides of the bipartition are referred to as the clients and the servers—and a positive weight for
each client c ∈ C. The algorithm must assign each client c ∈ C to an adjacent server s ∈ S. The
load of a server is then the weighted sum of all the clients assigned to it. The goal is to compute
an assignment that minimizes some function of the server loads, typically either the maximum
server load (i.e., the l∞-norm) or the lp-norm of the server loads. This problem has a variety of
applications and has been widely studied under several different names, including: scheduling with
restricted assignment, semi-matching, and distributed backup placement.
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(∆^5), where ∆ is the maximum degree of the input graph. Halldórsson et al. [SPAA 2015] showed an O(log n/ log log n)-approximation for unweighted clients and O(log^2n/ 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:

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 lp -norms, including the l∞ -norm.

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 CONGEST, 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 lp -norms, including the l∞ -norm.

Full version:
[arXiv]