Revisiting Ranking for Online Bipartite Matching with Random Arrivals: the Primal-Dual Analysis
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929744538238976 |
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| author | Peng, Bo Tang, Zhihao Gavin |
| author_facet | Peng, Bo Tang, Zhihao Gavin |
| contents | We revisit the celebrated Ranking algorithm by Karp, Vazirani, and Vazirani (STOC 1990) for online bipartite matching under the random arrival model, that is shown to be $0.696$-competitive for unweighted graphs by Mahdian and Yan (STOC 2011) and $0.662$-competitive for vertex-weighted graphs by Jin and Williamson (WINE 2021).
In this work, we explore the limitation of the primal-dual analysis of Ranking and aim to bridge the gap between unweighted and vertex-weighted graphs. We show that the competitive ratio of Ranking is between $0.686$ and $0.703$, under our current knowledge of Ranking and the framework of primal-dual analysis. This confirms a conjecture by Huang, Tang, Wu, and Zhang (TALG 2019), stating that the primal-dual analysis could lead to a competitive ratio that is very close to $0.696$. Our analysis involves proper discretizations of a variational problem and uses LP solver to pin down the numerical number. As a bonus of our discretization approach, our competitive analysis of Ranking applies to a more relaxed random arrival model. E.g., we show that even when each online vertex arrives independently at an early or late stage, the Ranking algorithm is at least $0.665$-competitive, beating the $1-1/e \approx 0.632$ competitive ratio under the adversarial arrival model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisiting Ranking for Online Bipartite Matching with Random Arrivals: the Primal-Dual Analysis Peng, Bo Tang, Zhihao Gavin Data Structures and Algorithms Computer Science and Game Theory We revisit the celebrated Ranking algorithm by Karp, Vazirani, and Vazirani (STOC 1990) for online bipartite matching under the random arrival model, that is shown to be $0.696$-competitive for unweighted graphs by Mahdian and Yan (STOC 2011) and $0.662$-competitive for vertex-weighted graphs by Jin and Williamson (WINE 2021). In this work, we explore the limitation of the primal-dual analysis of Ranking and aim to bridge the gap between unweighted and vertex-weighted graphs. We show that the competitive ratio of Ranking is between $0.686$ and $0.703$, under our current knowledge of Ranking and the framework of primal-dual analysis. This confirms a conjecture by Huang, Tang, Wu, and Zhang (TALG 2019), stating that the primal-dual analysis could lead to a competitive ratio that is very close to $0.696$. Our analysis involves proper discretizations of a variational problem and uses LP solver to pin down the numerical number. As a bonus of our discretization approach, our competitive analysis of Ranking applies to a more relaxed random arrival model. E.g., we show that even when each online vertex arrives independently at an early or late stage, the Ranking algorithm is at least $0.665$-competitive, beating the $1-1/e \approx 0.632$ competitive ratio under the adversarial arrival model. |
| title | Revisiting Ranking for Online Bipartite Matching with Random Arrivals: the Primal-Dual Analysis |
| topic | Data Structures and Algorithms Computer Science and Game Theory |
| url | https://arxiv.org/abs/2503.04196 |