Online Stochastic Matching: Online Actions Based on Offline Statistics

Manshadi, Vahideh; Gharan, Shayan Oveis; Saberi, Amin

doi:10.1287/moor.1120.0551

Cited by 148 publications

(70 citation statements)

References 13 publications

(32 reference statements)

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“…due to [19]. We take a step towards closing that gap by showing that an algorithm can achieve 0.7299 > 1 − 2e −2 for both the unweighted and vertex-weighted variants with integral arrival rates.…”

Section: Overview Of Vertex-weighted Algorithm and Contributionsmentioning

confidence: 99%

“…Constraint 3 is valid because each vertex in V has an arrival rate of 1. Constraint 4 is used in [19] and [12]. It captures the fact that the expected number of matches for any edge is at most 1 − 1/e.…”

Section: Lp Benchmark For Deterministic Rewardsmentioning

confidence: 99%

“…The vertex-weighted and unweighted settings have many results starting with Feldman, Mehta, Mirrokni and Muthukrishnan [10] who were the first to beat 1 − 1/e with a competitive ratio of 0.67 for the unweighted problem. This was improved by Manshadi, Gharan, and Saberi [19] to 0.705 with an adaptive algorithm. In addition, they showed that even in the unweighted variant with integral arrival rates, no algorithm can achieve a ratio better than 1 − e −2 ≈ 0.86.…”

Section: Related Workmentioning

confidence: 99%

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Online Stochastic Matching: New Algorithms and Bounds

et al. 2020

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Online matching has received significant attention over the last 15 years due to its close connection to Internet advertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal (1−1/e) competitive ratio in the standard adversarial online model, much effort has gone into developing useful online models that incorporate some stochasticity in the arrival process. One such popular model is the "known I.I.D. model" where different customer-types arrive online from a known distribution. We develop algorithms with improved competitive ratios for some basic variants of this model with integral arrival rates, including: (a) the case of general weighted edges, where we improve the best-known ratio of 0.667 due to Haeupler, Mirrokni and Zadimoghaddam [12] to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of Jaillet and Lu [13] to 0.7299.We also consider an extension of stochastic rewards, a variant where each edge has an independent probability of being present. For the setting of stochastic rewards with non-integral arrival rates, we present a simple optimal non-adaptive algorithm with a ratio of 1 − 1/e. For the special case where each edge is unweighted and has a uniform constant probability of being present, we improve upon 1 − 1/e by proposing a strengthened LP benchmark.One of the key ingredients of our improvement is the following (offline) approach to bipartitematching polytopes with additional constraints. We first add several valid constraints in order to get a good fractional solution f; however, these give us less control over the structure of f. We next remove all these additional constraints and randomly move from f to a feasible point on the matching polytope with all coordinates being from the set {0, 1/k, 2/k, . . . , 1} for a chosen integer k. The structure of this solution is inspired by Jaillet and Lu (Mathematics of Operations Research, 2013) and is a tractable structure for algorithm design and analysis. The appropriate random move preserves many of the removed constraints (approximately with high probability and exactly in expectation). This underlies some of our improvements and could be of independent interest. * A preliminary version of this appeared in the European Symposium on Algorithms (ESA), 2016 † Claim 4. For a large edge e, EW 1 [h] (3) with parameter h achieves a competitive ratio of R[EW 1 , 2/3] = 0.67529 + (1 − h) * 0.00446. Claim 5. For a small edge e of type Γ 1 , EW 1 [h] (3) achieves a competitive ratio of R[EW 1 , 1/3] = 0.751066, regardless of the value h. Claim 6. For a small edge e of type Γ 2 , EW 1 [h] (3) achieves a competitive ratio of R[EW 1 , 1/3] = 0.72933 + h * 0.040415. By setting h = 0.537815, the two types of small edges have the same ratio and we get that EW 1 [h] achieves (R[EW 1 , 2/3], R[EW 1 , 1/3]) = (0.679417, 0.751066). Thus, this proves Lemma 2. Proof of Claim 4Proof. Consider a large edge e = (u, v 1 ) in the graph G F . Let e ′ = (u, v 2 ) be the other small edge incident to u. Edges e and e ′ can appear in [M ...

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Section: Overview Of Vertex-weighted Algorithm and Contributionsmentioning

confidence: 99%

Section: Lp Benchmark For Deterministic Rewardsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Online Stochastic Matching: New Algorithms and Bounds

et al. 2020

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“…Mahdian and Yan [14], in 2011, achieved a competitive ratio of 0.696. Manshadi et al [16] showed that you cannot do better than 0.823. If the problem also has weights, then the bestpossible competitive ratio is 0.368 by a reduction from the secretary problem as shown by Kesselheim et al [11].…”

Section: Related Workmentioning

confidence: 99%

DISPATCH: An Optimally-Competitive Algorithm for Maximum Online Perfect Bipartite Matching with i.i.d. Arrivals

Chang

Spaen

Velednitsky

2018

Approximation and Online Algorithms

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This work presents an optimally-competitive algorithm for the problem of maximum weighted online perfect bipartite matching with i.i.d. arrivals. In this problem, we are given a known set of workers, a distribution over job types, and non-negative utility weights for each pair of worker and job types. At each time step, a job is drawn i.i.d. from the distribution over job types. Upon arrival, the job must be irrevocably assigned to a worker and cannot be dropped. The goal is to maximize the expected sum of utilities after all jobs are assigned. We introduce Dispatch, a 0.5-competitive, randomized algorithm. We also prove that 0.5-competitive is the best possible. Dispatch first selects a "preferred worker" and assigns the job to this worker if it is available. The preferred worker is determined based on an optimal solution to a fractional transportation problem. If the preferred worker is not available, Dispatch randomly selects a worker from the available workers. We show that Dispatch maintains a uniform distribution over the workers even when the distribution over the job types is non-uniform.

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“…The theory of matching started with Peterson and König and was under a lot of interests in graph theory with problems like maximum matchings. It was extended to online matching setting [8,12,14] where one population is static and the other arrives according to a stochastic process. In the recent years, fully dynamic matching models have been considered where both populations are random.…”

Section: Introductionmentioning

confidence: 99%

Optimal Control of Dynamic Bipartite Matching Models

Cadas

Bušíć

Doncel

2019

Proceedings of the 12th EAI International Conference on Performance Evaluation Methodologies and Tools

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A dynamic bipartite matching model is given by a bipartite matching graph which determines the possible matchings between the various types of supply and demand items. Both supply and demand items arrive to the system according to a stochastic process. Matched pairs leave the system and the others wait in the queues, which induces a holding cost. We model this problem as a Markov Decision Process and study the discounted cost and the average cost case. We first consider a model with two types of supply and two types of demand items with an N matching graph. For linear cost function, we prove that an optimal matching policy gives priority to the end edges of the matching graph and is of threshold type for the diagonal edge. In addition, for the average cost problem, we compute the optimal threshold value. According to our preliminary numerical experiments, threshold-type policies performs also very well for more general bipartite graphs.

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Online Stochastic Matching: Online Actions Based on Offline Statistics

Cited by 148 publications

References 13 publications

Online Stochastic Matching: New Algorithms and Bounds

Online Stochastic Matching: New Algorithms and Bounds

DISPATCH: An Optimally-Competitive Algorithm for Maximum Online Perfect Bipartite Matching with i.i.d. Arrivals

Optimal Control of Dynamic Bipartite Matching Models

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