Locally Optimal Load Balancing

TL;DR

This paper investigates distributed algorithms for locally optimal load balancing on graphs, providing tight bounds for paths and polynomial solutions for general graphs, focusing on fractional and discrete load distribution.

Contribution

It establishes tight bounds for load balancing on paths and introduces polynomial-time algorithms for general graphs, advancing understanding of distributed load balancing complexity.

Findings

Fractional load balancing on paths takes O(L) rounds, which is tight.

Discrete load balancing on paths can also be achieved in O(L) rounds.

Fractional load balancing on general graphs is solvable in polynomial rounds.

Abstract

This work studies distributed algorithms for locally optimal load-balancing: We are given a graph of maximum degree $\Delta$, and each node has up to $L$ units of load. The task is to distribute the load more evenly so that the loads of adjacent nodes differ by at most $1$. If the graph is a path ($\Delta = 2$), it is easy to solve the fractional version of the problem in $O(L)$ communication rounds, independently of the number of nodes. We show that this is tight, and we show that it is possible to solve also the discrete version of the problem in $O(L)$ rounds in paths. For the general case ($\Delta > 2$), we show that fractional load balancing can be solved in $\operatorname{poly}(L,\Delta)$ rounds and discrete load balancing in $f(L,\Delta)$ rounds for some function $f$, independently of the number of nodes.