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~$n$.

翻译：本文研究在由顶级交易循环（TTC）算法支配的单边匹配市场中，核心识别问题（CIP）的计算复杂度。核心贡献在于形式化的复杂度分离：本文证明，识别哪些代理人获得核心分配严格比计算完整的TTC分配更容易。具体而言，我们证明CIP可在$\bigO{Ln}$时间内求解，其中$L$是每个代理人报告的最大偏好数量，通过计算偏好导出的马尔可夫转移矩阵的主特征向量（采用随机SVD方法）。对于稀疏偏好配置（$L = \bigO{1}$，如纽约市择校场景中$L = 12$），这产生一个$\bigO{n}$的算法。该结果严格优于完整TTC分配的$\bigO{n \log n}$复杂度（参见\cite{SabanSethuraman2013}），并与信息论下界$\Omg{n}$相匹配，确立了渐近最优性。该方法继承了TTC的所有性质：帕累托效率、个体理性和策略证明性，并且对于充分大的$n$，对偏好噪声具有鲁棒性。