# Online Stochastic Weighted Matching: Improved Approximation Algorithms

@inproceedings{Haeupler2011OnlineSW, title={Online Stochastic Weighted Matching: Improved Approximation Algorithms}, author={Bernhard Haeupler and Vahab S. Mirrokni and Morteza Zadimoghaddam}, booktitle={WINE}, year={2011} }

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

### Online Vertex-Weighted Bipartite Matching

- MathematicsACM Trans. Algorithms
- 2019

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.

### Online stochastic matching, poisson arrivals, and the natural linear program

- Mathematics, Computer ScienceSTOC
- 2021

This work proposes a natural linear program for the Poisson arrival model, and demonstrates how to exploit its structure by introducing a converse of Jensen's inequality, and designs an algorithmic amortization to replace the analytic one in previous work.

### (Fractional) online stochastic matching via fine-grained offline statistics

- Computer ScienceSTOC
- 2022

Fractional algorithms that are 0.718-competitive and 0.704-competitive integral algorithms for i.i.d. arrivals are the first algorithms beating the 1−1/e ≈ 0.632 barrier of the classical adversarial setting, and it is proved that no fractional algorithm can achieve a competitive ratio better than 0.75 for non i.

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

- MathematicsICALP
- 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

- Computer Science2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

This paper presents the first online algorithm that breaks the long-standing 1/2 barrier and achieves a competitive ratio of at least 0.5086, and can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting.

### The power of multiple choices in online stochastic matching

- Computer ScienceSTOC
- 2022

Two approaches for designing and analyzing algorithms that use multiple choices in online stochastic matching are introduced, and it is proved that no algorithms can be 0.703 competitive, separating this setting from the aforementioned three.

### Prophet Inequality for Bipartite Matching: Merits of Being Simple and Non Adaptive

- MathematicsEC
- 2019

This work shows existence of a vertex-additive policy with the expected payoff of at least one third of the prophet's payoff and presents gradient decent type algorithm that quickly converges to the desired vector of vertex prices.

### Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

- Computer ScienceEC
- 2021

A polynomial-time algorithm is presented which approximates optimal online within a factor of 0.51---beating the best-possible prophet inequality and showing that it is PSPACE-hard to approximate this problem within some constant α < 1.

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

- Computer Science, MathematicsJ. Comb. Optim.
- 2017

This work studies the online weighted bipartite matching problem in adversary model, and designs algorithms for both variants with competitive ratio: the bounded weight problem in which all weights are in the range $$[alpha, \beta ]$$[α,β], and the normalized summation problem inwhich each vertex in one side has the same total weights.

### Stochastic Matching : New Algorithms and Bounds ∗

- Computer Science
- 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.

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