Online Bipartite Matching Problem Research Articles

Online Bipartite Matching with Reusable Resources

We study the classic online bipartite matching problem with a twist: off-line vertices, called resources, are 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 rematch. Thus, a resource can be matched to many different online vertices over a period of time. Whereas recent work on the problem has resolved the asymptotic case in which we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of unit inventory and give the first algorithms that are provably better than the naïve 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 nonreusable resources (Karp et al. 1990) by reranking resources independently over time. Whereas 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 beat the competitive ratio of 0.5. Both of our results also extend to the case in which off-line vertices have weights. Funding: R. Niazadeh’s research is supported by the Asness Junior Faculty Fellowship at The University of Chicago Booth School of Business.

Secretary Matching with Vertex Arrivals and No Rejections

Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in a uniformly random order and the edge-weights are arbitrary positive numbers. For the capacitated online bipartite matching problem in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the online general (non-bipartite) matching problem, where all vertices arrive online, we give a 3.34-competitive algorithm. We also study the online roommate matching problem, in which each room (offline vertex) holds 2 persons (online vertices). Persons derive non-negative additive utilities from their room as well as roommate. In this model, with the goal of maximizing the sum of utilities of all persons, we give a 7.96-competitive algorithm. This is an improvement over the 24.72 approximation factor in prior work.

Dynamic Relaxations for Online Bipartite Matching

Online bipartite matching (OBM) is a fundamental model underpinning many important applications, including search engine advertisement, website banner and pop-up ads, and ride hailing. We study the independent and identically distributed (i.i.d.) OBM problem, in which one side of the bipartition is fixed and known in advance, whereas nodes from the other side appear sequentially as i.i.d. realizations of an underlying distribution and must immediately be matched or discarded. We introduce dynamic relaxations of the set of achievable matching probabilities; show how they theoretically dominate lower dimensional, static relaxations from previous work; and perform a polyhedral study to theoretically examine the new relaxations’ strength. We also discuss how to derive heuristic policies from the relaxations’ dual prices in a similar fashion to dynamic resource prices used in network revenue management. We finally present a computational study to demonstrate the empirical quality of the new relaxations and policies. Summary of Contribution: Online bipartite matching (OBM) is one of the fundamental problems in the area of online decision analysis with a wide variety of applications in operations research and computer science, for example, online advertising, ride sharing, and general resource allocation. Over the last decades, both communities have been interested in the design and analysis of new approaches. Our main contribution is to provide a polyhedral study that considers the problem’s sequential nature. Specifically, we achieve this via dynamic relaxations. We also discuss how to derive heuristic policies from the relaxations’ dual prices. We support our theoretical findings with a detailed computational study.

Online Learning Bipartite Matching with Non-stationary Distributions

Online bipartite matching has attracted wide interest since it can successfully model the popular online car-hailing problem and sharing economy. Existing works consider this problem under either adversary setting or i.i.d. setting. The former is too pessimistic to improve the performance in the general case; the latter is too optimistic to deal with the varying distribution of vertices. In this article, we initiate the study of the non-stationary online bipartite matching problem, which allows the distribution of vertices to vary with time and is more practical. We divide the non-stationary online bipartite matching problem into two subproblems, the matching problem and the selecting problem, and solve them individually. Combining Batch algorithms and deep Q-learning networks, we first construct a candidate algorithm set to solve the matching problem. For the selecting problem, we use a classical online learning algorithm, Exp3, as a selector algorithm and derive a theoretical bound. We further propose CDUCB as a selector algorithm by integrating distribution change detection into UCB. Rigorous theoretical analysis demonstrates that the performance of our proposed algorithms is no worse than that of any candidate algorithms in terms of competitive ratio. Finally, extensive experiments show that our proposed algorithms have much higher performance for the non-stationary online bipartite matching problem comparing to the state-of-the-art.

A tight bound for shortest augmenting paths on trees

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree T=(W⊎B,E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. It was conjectured by Chaudhuri et al. [6] that the total length of all shortest augmenting paths found is O(nlog⁡n). In this paper we prove a tight O(nlog⁡n) upper bound for the total length of shortest augmenting paths for trees improving over O(nlog2⁡n) bound [3].

Packing returning secretaries

AbstractWe study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to determine a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5‐competitive algorithm. For the online bipartite matching problem, we obtain an algorithm with ratio at least 0.5721 − o(1), and an algorithm with ratio at least 0.5459 for all n ≥ 1. We extend all algorithms and ratios to k ≥ 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed. We focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Θ(n log n) is always sufficient. For bipartite matching, we can show a tight bound of O(r log n), where r is the size of the optimum matching. For matroids, we can improve this further to a tight bound of O(r′ log(n/r′)), where r′ is the minimum rank of the matroid and the dual matroid.

Privacy preserving online matching on ridesharing platforms

Ridesharing platforms, as typical applications of spatial crowdsourcing, are becoming more and more popular in the era of mobile internet and sharing economy. One of the most fundamental issues on ridesharing platforms is to assign orders to drivers, which can be naturally modeled as online bipartite matching problem. However, conventional online matching algorithms usually lack data privacy protection mechanisms. This has become a serious issue since the spatiotemporal data of passengers is often sensitive. New policies such as EU’s General Data Protection Regulation (GDPR) also enforce protection of sensitive data, which further exacerbate the privacy issues. To deal with the problems, in this paper we propose a framework based on differential privacy (DP) techniques to preserve the privacy of individuals on ridesharing platforms. Specifically, we devise a novel approach to perturb locations in online minimum bipartite matching problem and theoretically show that the performance of the perturbed matching algorithm has the same magnitude with the unperturbed one. Experiments conducted on real datasets have also shown the effectiveness of proposed framework.

Online Bipartite Matching with Amortized O (log 2 n ) Replacements

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O (log 2 n ) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n ) lower bound. The previous best strategy known achieved amortized O (√ n ) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O ( n ) bound was known except in special cases. Our analysis immediately implies the same upper bound of O (log 2 n ) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L , then the SAP protocol makes amortized O (min { L log 2 n , √ n log n }) reassignments. We also show that this is close to tight, because Ω (min { L , √ n }) reassignments can be necessary.

Online Vertex-Weighted Bipartite Matching

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and we prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time increases, and each online vertex matches the neighbor with the highest offer at its arrival.

Shortest Augmenting Paths for Online Matchings on Trees

The shortest augmenting path (Sap) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp (J. ACM 19(2), 248–264 1972) have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although it has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph G = (W ⊎ B, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM’09) that the total length of all augmenting paths found by Sap is mathcal {O}(n log n). However, no better bound than mathcal {O}(n^{2}) is known even for trees. In this paper we prove an mathcal {O}(n log ^{2}n) upper bound for the total length of augmenting paths for trees.

A Matching Approach to Factor Scores Based on Online Sponsored Search Auction

How does a search engine company decide which advertisements to display for each query to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. In this paper, search engines decide the strategy to allocate resources with an advertiser credibility factor under budget constraints. Based on the optimal algorithm, which is the notion of a trade-off revealing LP, this paper remains the competitive ratio as 1-1/e with an advertiser credibility factor. During the ranking in the keywords auctions, the authors calculate factor scores. The new arrival keywords should match to the advertisement which has the maximum factor scores. The amount of spent money, budget constraints, CTR (Click Through Rate) and a credibility factor are added to the trade-off function using in the authors' algorithms. In the long term, users tend to use the search engine with high credibility, which will bring greater revenues. The search engine will attract advertisers to bid on keywords and improve their credibility actively.

Inventory Balancing with Online Learning

We study a general problem of allocating limited resources to heterogeneous customers over time, under model uncertainty. Each type of customer can be serviced using different actions, each of which stochastically consumes some combination of resources, and returns different rewards for the resources consumed. We consider a general model framework, where the resource consumption distribution associated with each (customer type, action) combination is not known, but is consistent and can be learned over time. In addition, the sequence of customer types to arrive over time is arbitrary and completely unknown. We achieve near optimality under both model uncertainty and customer heterogeneity by judiciously synergizing two algorithmic frameworks in the literature: inventory balancing, which reserves a portion of each resource for high-reward customer types which could later arrive; and online learning, which shows how to explore'' the resource consumption distributions of each customer type under different actions. We define an auxiliary problem, which allows for existing competitive ratio and regret bounds to be seamlessly integrated. Furthermore, we show that the performance guarantee generated by our framework is tight, using the special case of the online bipartite matching problem with unknown match probabilities. Finally, we demonstrate the practicality and efficacy of algorithms generated by our framework using a publicly available hotel data set.

A polyhedral approach to online bipartite matching

We study the i.i.d. online bipartite matching problem, a dynamic version of the classical model where one side of the bipartition is fixed and known in advance, while nodes from the other side appear one at a time as i.i.d. realizations of a uniform distribution, and must immediately be matched or discarded. We consider various relaxations of the polyhedral set of achievable matching probabilities, introduce valid inequalities, and discuss when they are facet-defining. We also show how several of these relaxations correspond to ranking policies and their time-dependent generalizations. We finally present a computational study of these relaxations and policies to determine their empirical performance.

On the advice complexity of online bipartite matching and online stable marriage

In this paper, we study the advice complexity of the online bipartite matching problem and the online stable marriage problem. We show that for both problems, ⌈log2⁡(n!)⌉ bits of advice are necessary and sufficient for a deterministic online algorithm to be optimal, where n denotes the number of vertices in one bipartition in the former problem, and the number of men in the latter.

On-line bipartite matching made simple

We examine the classic on-line bipartite matching problem studied by Karp, Vazirani, and Vazirani [8] and provide a simple proof of their result that the Ranking algorithm for this problem achieves a competitive ratio of 1 -- 1/ e .

AdWords and generalized online matching

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a trade-off revealing LP and use it to derive an optimal algorithm achieving a competitive ratio of 1−1/ e for this problem.

Online request server matching

In the following paper an alternative online variant of the matching problem in bipartite graphs is presented. It is triggered by a scheduling problem. There, a task is unknown up to its disclosure. However, when a task is revealed, it is not necessary to take a decision on the service of that particular task. On the contrary, an online scheduler has to decide on how to use the current resources. Therefore, the problem is called online request server matching ( ORSM). It differs substantially from the online bipartite matching problem of Karp et al. (Proc. 22nd Annual ACM Symp. on Theory of Computing, Baltimore, Maryland, May 14–16, 1990, ACM Press, New York, 1990). Hence, the analysis of an optimal, deterministic online algorithm for the ORSM problem results in a smaller competitive ratio of 1.5. An extension to a weighted bipartite matching problem is also introduced, and results of its investigation are presented. Additional concepts for the ORSM model (e.g. lookahead, parallel resources, deadlines) are studied. All of these modifications are realized by restrictions on the input structure. Decreased competitive ratios are presented for some of these modified models.