Online Bipartite Matching with Reusable Resources

TL;DR

This paper introduces new algorithms for online bipartite matching with reusable resources, achieving better competitive ratios than the naive approach by leveraging reranking and randomized tie-breaking techniques.

Contribution

It presents the first algorithms surpassing the 0.5 competitive ratio for unit inventory reusable resources, extending classic methods with novel reranking and correlated selection strategies.

Findings

Achieved a 0.589 competitive ratio with a reranking-based algorithm.

Developed a 0.505 competitive ratio primal-dual randomized algorithm.

Extended results to weighted offline vertices.

Abstract

We study the classic online bipartite matching problem with a twist: offline vertices, called resources, are $\textit{reusable}$. In particular, when a resource is matched to an online vertex it is unavailable for a deterministic time duration $d$ after which it becomes available again for a re-match. Thus, a resource can be matched to many different online vertices over a period of time. While recent work on the problem have resolved the asymptotic case where we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of $\textit{unit inventory}$ and give the first algorithms that are provably better than the naive greedy approach which has a competitive ratio of (exactly) 0.5. Our first algorithm, which achieves a competitive ratio of $0.589$, generalizes the classic RANKING algorithm for online bipartite matching of non-reusable resources (Karp et al., 1990), by $\textit{reranking}$ resources independently over time. While reranking resources frequently has the same worst case performance as greedy, we show that reranking intermittently on a periodic schedule succeeds in addressing reusability of resources and performs significantly better than greedy in the worst case. Our second algorithm, which achieves a competitive ratio of $0.505$, is a primal-dual randomized algorithm that works by suggesting up to two resources as candidate matches for every online vertex, and then breaking the tie to make the final matching selection in a randomized correlated fashion over time. As a key component of our algorithm, we suitably adapt and extend the powerful technique of online correlated selection (Fahrbach et al., 2020) to reusable resources, in order to induce negative correlation in our tie breaking step and to beat the competitive ratio of $0.5$. Both of our results also extend to the case where offline vertices have weights.