Edge-Weighted Online Bipartite Matching

Abstract

Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal competitive ratio of 1 − 1 /e. Later, Aggarwal et al. (SODA 2011) generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial 1 /2-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing 1 /2 barrier and achieves a competitive ratio of at least 0.5086. In light of the hardness result of Kapralov, Post, and Vondrák (SODA 2013) that restricts beating a 1 /2 competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting.The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems. * This paper merges and refines the results in arXiv: 1704.05384v2, arXiv:1910.02569, and arXiv:1910. In particular, we fix a bug in arXiv:1910.03287 and have a smaller competitive ratio as a result. Appendix C discusses the connections between the primal-dual algorithm in this work and the original algorithm of Fahrbach and Zadimoghaddam.

Online Bipartite Matching with Reusable Resources

“…The above progress is particularly important, as it shows 0.5 is not a barrier for competitive ratio in the case of online bipartite matching with reusable resources. It also resonates well with other recent work on edge-weighted online bipartite matching with free-disposal [14] and Adwords [32], in which a similar paradigm is observed: it has been known for some time how to obtain the optimal (1 − 1/e) competitive ratio using fractional allocations in these problem, but it was not known a priori whether there exists an integral randomized algorithm that admits a constant improvement over 0.5. Our results also resemble other line of work on variants of online bipartite matching with different input semantics-e.g., the fully online bipartite matching [29] and the general vertex-arrival online bipartite matching [22]: it has been known that the greedy algorithm is again 0.5-competitive in these problems, but it was not clear a priori whether any improvement is possible; nevertheless, these works demonstrate that barrier 0.5 for competitive ratio can be overcome by proper usage of randomization.…”

“…More interestingly, for the inner layer, we introduce a new randomized selection problem that we refer to as the online correlated rental (OCR). Our OCR problem is indeed an artifact of a unified interpretation and a simpler proof of prior work [14,32], which introduces a powerful technique known as the online correlated selection (OCS) for resources that are non-reusable and can be matched multiple times-but contribute to the objective only once (due to free-disposal); This unified interpretation is the key to extend the existing machinery to the matching case with reusable resources. We then solve this new problem and use its solution at the heart of the inner layer of our final algorithm.…”

“…However, when two vertices are suggested, we refer to the suggestion as a randomized query, and the inner randomized algorithm is then executed to select which one to match. This is inspired by a similar idea in [14], where the online integral algorithm pushes its allocation towards more fractional allocations by suggesting two choices rather than one. 5 One simple candidate for the inner algorithm mentioned above is selecting one of the two vertices independently with probability ½.…”

“…Given the above mentioned type of guarantees by the OCR procedure, and the way that our outer algorithm picks its suggested pair of resources, we develop our primal-dual analysis framework. In fact, this primal-dual analysis is tightly related to the design of the primal-dual algorithm itself, and follows a similar logic as the recent primal-dual techniques in [32] and [14]. We end up with identifying constraints that should be satisfied by our competitive ratio Γ and our dual assignments in order to guarantee approximate dual feasibility.…”

Subgroup Fairness in Two-Sided Markets

“…In the model of [44], one part of the bipartite graph ("workers") is fixed and known in advance, while nodes from the other part ("jobs") arrive online and must immediately be matched or discarded [45][46][47]. This model has many applications in online advertising platforms [48,49]; a wide range of novel applications arise [50] in the sharing economy, including applications within task assignment in spatial crowd-sourcing, real-estate agencies, and food delivery [51][52][53][54]. Especially, in the application of ride-sharing platforms, it could be a powerful tool for taxi order dispatch systems [55].…”