(Almost) Perfect Discrete Iterative Load Balancing

2025-10-17

AI Summary

This paper studies discrete iterative load balancing on arbitrary graphs, where tokens are migrated between nodes via matching-based operations to minimize load disparity. We propose a unified local-matching-based balancing strategy that subsumes three fundamental models—random matchings, periodic circuits, and asynchronous edge selection—and analyze its convergence using probabilistic methods and spectral graph theory. Our key contribution is the first constant-load-discrepancy bound of 3 on general (non-regular) graphs, with asymptotically optimal convergence time—overcoming prior limitations that restricted analysis to regular graphs and yielded implicit, unquantified constants. Crucially, our results demonstrate that, under matching constraints, discrete load balancing is no harder than its continuous counterpart, thereby providing a more general and tighter theoretical foundation for distributed load scheduling.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider discrete, iterative load balancing via matchings on arbitrary graphs. Initially each node holds a certain number of tokens, defining the load of the node, and the objective is to redistribute the tokens such that eventually each node has approximately the same number of tokens. We present results for a general class of simple local balancing schemes where the tokens are balanced via matchings. In each round the process averages the tokens of any two matched nodes. If the sum of their tokens is odd, the node to receive the one excess token is selected at random. Our class covers three popular models: in the matching model a new matching is generated randomly in each round, in the balancing circuit model a fixed sequence of matchings is applied periodically, and in the asynchronous model the load is balanced over a randomly chosen edge. We measure the quality of a load vector by its discrepancy, defined as the difference between the maximum and minimum load across all nodes. As our main result we show that with high probability our discrete balancing scheme reaches a discrepancy of $3$ in a number of rounds which asymptotically matches the spectral bound for continuous load balancing with fractional load. This result improves and tightens a long line of previous works, by not only achieving a small constant discrepancy (instead of a non-explicit, large constant) but also holding for arbitrary instead of regular graphs. The result also demonstrates that in the general model we consider, discrete load balancing is no harder than continuous load balancing.

Problem

• Redistributing tokens evenly across nodes in arbitrary graphs
• Achieving constant discrepancy via local iterative matching schemes
• Bridging performance gap between discrete and continuous load balancing

Innovation

• Discrete iterative load balancing via matchings
• Achieves constant discrepancy of three tokens
• Works on arbitrary graph topologies efficiently