Dynamic online matching with budget refills

2024-05-16

arXiv.org

AI Summary

This paper studies the online bipartite matching problem with *dynamically replenishable budgets*: in a bipartite graph $G = (U, V, E)$, nodes on side $V$ arrive online, while nodes on side $U$ and their time-varying integer budgets $b_{u,t}$ are known in advance; budgets are replenished periodically or sparsely. This extends the classical model to better capture elastic resource scheduling scenarios. The work formally introduces and analyzes dynamic budget replenishment mechanisms, revealing their fundamental impact on competitive ratios. Under the random arrival model, the authors employ ODE-based mean-field analysis to prove that the greedy algorithm’s matching size converges asymptotically, achieving a competitive ratio approaching 1 under mild conditions. In the adversarial model, they establish a tight upper bound for replenishable $b$-matching and prove that the Balance algorithm attains this bound—demonstrating that regularity in replenishment significantly improves worst-case performance.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Mechanism Design
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Inspired by sequential budgeted allocation problems, we study the online matching problem with budget refills. In this context, we consider an online bipartite graph G=(U,V,E), where the nodes in $V$ are discovered sequentially and nodes in $U$ are known beforehand. Each $uin U$ is endowed with a budget $b_{u,t}in mathbb{N}$ that dynamically evolves over time. Unlike the canonical setting, in many applications, the budget can be refilled from time to time, which leads to a much richer dynamic that we consider here. Intuitively, adding extra budgets in $U$ seems to ease the matching task, and our results support this intuition. In fact, for the stochastic framework considered where we studied the matching size built by Greedy algorithm on an ErdH{o}s-R'eyni random graph, we showed that the matching size generated by Greedy converges with high probability to a solution of an explicit system of ODE. Moreover, under specific conditions, the competitive ratio (performance measure of the algorithm) can even tend to 1. For the adversarial part, where the graph considered is deterministic and the algorithm used is Balance, the $b$-matching bound holds when the refills are scarce. However, when refills are regular, our results suggest a potential improvement in algorithm performance. In both cases, Balance algorithm manages to reach the performance of the upper bound on the adversarial graphs considered.

Problem

• Studies online bipartite matching with dynamic budget refills
• Analyzes Greedy algorithm performance on stochastic Erdős-Rényi graphs
• Examines Balance algorithm behavior under adversarial budget refill scenarios

Innovation

• Dynamic budget refills in online bipartite graphs
• Greedy algorithm on Erdős-Rényi graphs with ODE
• Balance algorithm improves with regular refills