Fast Core Identification
TLDR
A new algorithm identifies core allocations in matching markets in optimal $\bigO{n}$ time, outperforming full TTC computation.
Key contributions
- Proves the Core Identification Problem (CIP) is computationally easier than computing full TTC allocations.
- Introduces an $\bigO{Ln}$ algorithm for CIP using randomized SVD on a preference-derived Markov matrix.
- Achieves optimal $\bigO{n}$ complexity for sparse preference profiles, improving on $\bigO{n \log n}$ methods.
- The method retains TTC properties like Pareto efficiency and strategy-proofness, and is robust to preference noise.
Why it matters
This paper offers a significantly faster and asymptotically optimal method for identifying stable allocations in large-scale matching markets. It provides a crucial complexity separation, showing that core identification is easier than full allocation computation. This advancement has practical implications for efficient market design, particularly in systems like school choice.
Original Abstract
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~$n$.
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