Fast Core Identification
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911630064877568 |
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| author | Aldridge, Irene |
| author_facet | Aldridge, Irene |
| contents | This paper examines the computational complexity of the \emph{Core Identification Problem} (CIP) in one-sided matching markets governed by the Top Trading Cycles (TTC) algorithm. The central contribution is a formal complexity separation: this paper proves that identifying which agents receive a core allocation is strictly easier than computing the full TTC allocation. Specifically, we show that CIP can be solved in $\bigO{Ln}$ time, where $L$ is the maximum number of preferences reported per agent, by computing the leading eigenvector of a preference-derived Markov transition matrix via randomized SVD\@. For sparse preference profiles ($L = \bigO{1}$, as in the NYC school choice where $L = 12$), this yields an algorithm $\bigO{n}$. This result strictly improves on the $\bigO{n \log n}$ complexity of the full TTC allocation (\cite{SabanSethuraman2013}) and matches the $\Omg{n}$ information-theoretic lower bound, establishing asymptotic optimality. The method inherits all properties of TTC: Pareto efficiency, individual rationality, and strategy-proofness, and is robust to preference noise for sufficiently large~$n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25954 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast Core Identification Aldridge, Irene Computer Science and Game Theory Theoretical Economics Trading and Market Microstructure H.4 This paper examines the computational complexity of the \emph{Core Identification Problem} (CIP) in one-sided matching markets governed by the Top Trading Cycles (TTC) algorithm. The central contribution is a formal complexity separation: this paper proves that identifying which agents receive a core allocation is strictly easier than computing the full TTC allocation. Specifically, we show that CIP can be solved in $\bigO{Ln}$ time, where $L$ is the maximum number of preferences reported per agent, by computing the leading eigenvector of a preference-derived Markov transition matrix via randomized SVD\@. For sparse preference profiles ($L = \bigO{1}$, as in the NYC school choice where $L = 12$), this yields an algorithm $\bigO{n}$. This result strictly improves on the $\bigO{n \log n}$ complexity of the full TTC allocation (\cite{SabanSethuraman2013}) and matches the $\Omg{n}$ information-theoretic lower bound, establishing asymptotic optimality. The method inherits all properties of TTC: Pareto efficiency, individual rationality, and strategy-proofness, and is robust to preference noise for sufficiently large~$n$. |
| title | Fast Core Identification |
| topic | Computer Science and Game Theory Theoretical Economics Trading and Market Microstructure H.4 |
| url | https://arxiv.org/abs/2604.25954 |