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

@article{Huang2018OnlineVB, title={Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals}, author={Zhiyi Huang and Z. Tang and Xiaowei Wu and Yuhao Zhang}, journal={ArXiv}, year={2018}, volume={abs/1804.07458} }

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and 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… Expand

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#### References

SHOWING 1-10 OF 35 REFERENCES

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

- Computer Science, Mathematics
- 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. Expand

How to match when all vertices arrive online

- Computer Science, Mathematics
- STOC
- 2018

A fully online model of maximum cardinality matching in which all vertices arrive online, and a novel charging mechanic is brought into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. Expand

Online Stochastic Weighted Matching: Improved Approximation Algorithms

- Mathematics, Computer Science
- WINE
- 2011

The first approximation (0.667-competitive) algorithm for the online stochastic weighted matching problem beating the 1−1 / e barrier is presented and the approximation factor is improved by computing a careful third pseudo-matching along with the two offline solutions, and using it in the online algorithm. Expand

Randomized Primal-Dual analysis of RANKING for Online BiPartite Matching

- Computer Science, Mathematics
- SODA
- 2013

This is the first instance of a non-trivial randomized primal-dual algorithm in which the dual constraints only hold in expectation. Expand

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

- Mathematics, 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. Expand

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]. Expand

Online Stochastic Matching: Beating 1-1/e

- Computer Science, Mathematics
- 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. Expand

Improved Bounds for Online Stochastic Matching

- Mathematics, 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. Expand

Ranking on Arbitrary Graphs: Rematch via Continuous LP with Monotone and Boundary Condition Constraints

- Mathematics, Computer Science
- SODA
- 2014

The Ranking algorithm is revisited using the LP framework, and new duality and complementary slackness characterizations are introduced that can handle the monotone and the boundary constraints in continuous LP. Expand

Randomized Online Matching in Regular Graphs

- Computer Science
- SODA
- 2018

This work presents a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order, and shows the convergence rate of the algorithm's competitive ratio to one is nearly tight, as no algorithm achieves competitive ratio better than [EQUATION]. Expand