# Online Stochastic Weighted Matching: Improved Approximation Algorithms

@inproceedings{Haeupler2011OnlineSW,
title={Online Stochastic Weighted Matching: Improved Approximation Algorithms},
booktitle={WINE},
year={2011}
}
• Published in WINE 11 December 2011
• Computer Science, Mathematics
Motivated by the display ad allocation problem on the Internet, we study the online stochastic weighted matching problem. In this problem, given an edge-weighted bipartite graph, nodes of one side arrive online i.i.d. according to a known probability distribution. Recently, a sequence of results by Feldman et. al [14] and Manshadi et. al [20] result in a 0.702-approximation algorithm for the unweighted version of this problem, aka online stochastic matching, breaking the 1−1 / e barrier. Those…
114 Citations

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STOC
• 2022
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### Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

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ICALP
• 2018
This work builds 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, and proves a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order.

### Edge-Weighted Online Bipartite Matching

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2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
• 2020
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• 2022
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### Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

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### Near optimal algorithms for online weighted bipartite matching in adversary model

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• 2017
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### Stochastic Matching : New Algorithms and Bounds ∗

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• 2017
This work develops algorithms with improved competitive ratios for some basic variants of this known I.I.D. model with integral arrival rates, and presents a simple optimal non-adaptive algorithm with a ratio of 1− 1/e for the setting of stochastic rewards with non-integral arrival rates.

## References

SHOWING 1-10 OF 24 REFERENCES

### Improved Bounds for Online Stochastic Matching

• Computer Science
ESA
• 2010
It is shown that the best competitive ratio that can be obtained with the static analysis used in the d-SM algorithm is upper bounded by 0.76, thus suggesting that a dynamic analysis may be needed to improve the competitive ratio significantly.

### Online vertex-weighted bipartite matching and single-bid budgeted allocations

• Economics
SODA '11
• 2011
The main result is an optimal (1−1/e)-competitive randomized algorithm for general vertex weights that effectively solves the problem of online budgeted allocations in the case when an agent makes the same bid for any desired item, even if the bid is comparable to his budget.

### Online Stochastic Matching: Beating 1-1/e

• Computer Science
2009 50th Annual IEEE Symposium on Foundations of Computer Science
• 2009
A novel application of the idea of the power of two choices from load balancing, which compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order to characterize an upper bound for the optimum in any scenario.

### Online bipartite matching with unknown distributions

• Mathematics, Computer Science
STOC '11
• 2011
This is the first analysis to show an algorithm which breaks the natural 1 - 1/e -barrier' in the unknown distribution model (the authors' analysis in fact works in the stricter, random order model) and answers an open question in [GM08].

### Online budgeted matching in random input models with applications to Adwords

• Computer Science, Economics
SODA '08
• 2008
An online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints is studied, with a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 - 1/e for maximizing the value of the assignment.

### Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs

• Computer Science
STOC '11
• 2011
This paper studies the ranking algorithm in the random arrivals model, and shows that it has a competitive ratio of at least 0.696, beating the 1-1/e ≈ 0.632 barrier in the adversarial model.

### Online Stochastic Packing Applied to Display Ad Allocation

• Computer Science
ESA
• 2010
It is proved that a simple dual training-based algorithm achieves a (1-o(1)- approximation guarantee in the random order stochastic model, which is a significant improvement over logarithmic or constant-factor approximations for the adversarial variants of the same problems.

### Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue

• Computer Science
ESA
• 2007
A (1 - 1/e)-competitive (optimal) algorithm is designed for the online ad-auctions problem, which is based on a clean primal-dual approach, matching the competitive factor obtained in Mehta et al.

### Optimal online assignment with forecasts

• Computer Science, Economics
EC '10
• 2010
The online assignment with forecast problem is formulated, a version of the online allocation problem where the algorithm has access to random samples from the future set of arriving vertices, and it is proved that representing the primal solution using such a compact allocation plan yields a robust online algorithm which makes near-optimal online decisions.

### Near optimal online algorithms and fast approximation algorithms for resource allocation problems

• Computer Science
EC '11
• 2011
A new distributional model called the adversarial stochastic input model, which is a generalization of the i.i.d model with unknown distributions, where the distributions can change over time is introduced, and a 1-O(ε) approximation algorithm is given for the resource allocation problem.