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On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching

Authors Jingxun Liang , Zhihao Gavin Tang, Yixuan Even Xu , Yuhao Zhang, Renfei Zhou

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Author Details

Jingxun Liang
  • IIIS, Tsinghua University, Beijing, China
Zhihao Gavin Tang
  • ITCS, Shanghai University of Finance and Economics, China
Yixuan Even Xu
  • IIIS, Tsinghua University, Beijing, China
Yuhao Zhang
  • Shanghai Jiao Tong University, China
Renfei Zhou
  • IIIS, Tsinghua University, Beijing, China

Funding

This work is supported by Science and Technology Innovation 2030 - "New Generation of Artificial Intelligence" Major Project No.(2018AAA0100903), Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities.
  • Tang, Zhihao Gavin: Zhihao Gavin Tang is partially supported by Huawei Theory Lab.
  • Zhang, Yuhao: Yuhao Zhang is supported by National Natural Science Foundation of China, Grant No. 62102251.

Acknowledgements

We thank all anonymous reviewers for their insightful comments.

Cite As Get BibTex

Jingxun Liang, Zhihao Gavin Tang, Yixuan Even Xu, Yuhao Zhang, and Renfei Zhou. On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 80:1-80:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.80

Abstract

Ranking and Balance are arguably the two most important algorithms in the online matching literature. They achieve the same optimal competitive ratio of 1-1/e for the integral version and fractional version of online bipartite matching by Karp, Vazirani, and Vazirani (STOC 1990) respectively. The two algorithms have been generalized to weighted online bipartite matching problems, including vertex-weighted online bipartite matching and AdWords, by utilizing a perturbation function. The canonical choice of the perturbation function is f(x) = 1-e^{x-1} as it leads to the optimal competitive ratio of 1-1/e in both settings.
We advance the understanding of the weighted generalizations of Ranking and Balance in this paper, with a focus on studying the effect of different perturbation functions. First, we prove that the canonical perturbation function is the unique optimal perturbation function for vertex-weighted online bipartite matching. In stark contrast, all perturbation functions achieve the optimal competitive ratio of 1-1/e in the unweighted setting. Second, we prove that the generalization of Ranking to AdWords with unknown budgets using the canonical perturbation function is at most 0.624 competitive, refuting a conjecture of Vazirani (2021). More generally, as an application of the first result, we prove that no perturbation function leads to the prominent competitive ratio of 1-1/e by establishing an upper bound of 1-1/e-0.0003. Finally, we propose the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets, and we design an optimal 1-1/e competitive algorithm by generalizing Balance.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online Matching
  • AdWords
  • Ranking
  • Water-Filling

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References

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