Abstract
In this work, we study truthful mechanisms for the rank-maximal matching problem, in the view of approximation. Our result reduces the gap from both the upper and lower bound sides. We propose a lexicographically truthful (LT) and nearly Pareto optimal (PO) randomized mechanism with an approximation ratio \(\frac{2 \sqrt{e}-1}{2 \sqrt{e}-2} \approx 1.77\). The previous best result is 2. The crucial and novel ingredients of our algorithm are preservation lemmas, which allow us to utilize techniques from online algorithms to analyze the new ratio. We also provide several hardness results in variant settings to complement our upper bound. In particular, we improve the lower bound of the approximation ratio for our LT and PO mechanism to \(18/13\approx 1.38\). To the best of our knowledge, it is the first time to obtain a lower bound by utilizing the linear programming approach along this research line.
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Notes
- 1.
A rank-maximal matching is a matching where the maximum possible number of applicants are matched to their first choice tie, and subject to that condition, the maximum possible number are matched to their second choice tie, and so on.
- 2.
Our algorithm and analysis can be easily extended to the setting where each object j can be allocated to at most b(j) applicants, e.g., graduate college admission. We can make b(j) copies of object j, add them to each agent’s preferences and each copy can be allocated to at most one agent.
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Acknowledgments
This work was supported by the National Key Research and Development Program of China (2022YFF0902005). Jianwei Yin was supported by the National Science Fund for Distinguished Young Scholars (No. 61825205). Jinshan Zhang was supported by the Key Research and Development Jianbing Program of Zhejiang Province (2023C01002), Hangzhou Major Project and Development Program (2022AIZD0140) and Yongjiang Talent Introduction Programm (2022A-236-G). Xiaotie Deng was supported by the National Natural Science Foundation of China (No. 62172012). Zhengyang Liu was supported by the National Natural Science Foundation of China (No. 62002017) and Beijing Institute of Technology Research Fund Program for Young Scholars. We are also grateful for the valuable comments from the anonymous reviewers.
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Zhang, J., Liu, Z., Deng, X., Yin, J. (2024). Improved Truthful Rank Approximation for Rank-Maximal Matchings. In: Garg, J., Klimm, M., Kong, Y. (eds) Web and Internet Economics. WINE 2023. Lecture Notes in Computer Science, vol 14413. Springer, Cham. https://doi.org/10.1007/978-3-031-48974-7_36
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