Dynamic online matching with budget refills

TL;DR

This paper investigates online bipartite matching with dynamic budget refills, analyzing the performance of greedy and balance algorithms in stochastic and adversarial settings, revealing conditions for optimality and potential improvements.

Contribution

It introduces a novel model of online matching with dynamic budget refills and analyzes algorithm performance, including convergence to ODE solutions and conditions for near-optimality.

Findings

Greedy matching size converges to an explicit ODE solution in stochastic Erdős-Rényi graphs.

Balance algorithm achieves near-optimal performance in adversarial settings with scarce refills.

Regular refills can improve algorithm performance beyond traditional bounds.

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 $u\in U$ 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 Erd\H{o}s-R{\'e}yni 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.