Distributed Load Balancing on Unrelated Machines

2026-07-10

AI Summary

This work addresses the unrelated-machines load balancing problem in the distributed CONGEST model, where jobs exhibit arbitrary processing times across machines. It presents the first efficient algorithm that computes a solution in polylogarithmic rounds. The key technical contribution is a (1+ε)-approximate black-box solver for mixed packing-covering linear programs, overcoming prior limitations that only handled pure packing or pure covering constraints. Leveraging this solver, the algorithm yields either a (1+ε)-approximate fractional solution or a (2+ε)-approximate integral solution. This result significantly improves upon previous distributed algorithms, which were restricted to identical job processing times, and constitutes the first efficient distributed solution for the general setting with arbitrary processing times.

Abstract

We study the well-known load balancing problem in the distributed CONGEST model of computation. We consider the unrelated machines setting, where each job $j$ specifies a size $s_{ij}$ for every machine $i$. We want to find an assignment $\varphi: J \to M$ minimizing the maximum machine load, where the load of a machine $i$ is the total size of the jobs assigned to it. In the CONGEST model, the state-of-the-art is an algorithm that runs in polylog rounds and returns a $(1+\varepsilon)$-approximate fractional solution from Ahmadian, Liu, Peng, and Zadimoghaddam (2021). However, this algorithm, as well as all previous CONGEST algorithms only solve a special case of load balancing, where each job has the same size on each machine. Our main contribution is an algorithm for general sizes $s_{ij}$. The algorithm computes a $(1+\varepsilon)$-approximate fractional solution or a $(2+\varepsilon)$-approximate integral solution in polylog rounds. The problem structure changes significantly once we allow arbitrary edge-sizes, so our techniques are very different from those used in previous algorithms for distributed load balancing. One ingredient of our result is a black-box tool of independent interest: a $(1+\varepsilon)$-approximation algorithm to arbitrary mixed packing-covering linear programs in the CONGEST model in polylog rounds. such algorithms were known in the more powerful parallel model, but previous polylog-round algorithms in the distributed CONGEST model only solved pure packing or pure covering problems. We improve upon a recent $O(D\,\mathrm{polylog})$-round CONGEST algorithm for mixed packing-covering, where $D$ is the diameter of the communication graph.

Problem

• distributed load balancing
• unrelated machines
• CONGEST model
• maximum machine load
• job assignment

Innovation

• distributed load balancing
• unrelated machines
• mixed packing-covering LP
• CONGEST model
• polylog-round algorithm