Fast Core Identification

TL;DR

This paper introduces an efficient algorithm for identifying agents in core allocations under the Top Trading Cycles mechanism, improving computational complexity and matching the theoretical lower bound.

Contribution

It proves that core identification can be done faster than computing the full TTC allocation, achieving asymptotic optimality with a new eigenvector-based method.

Findings

Core identification solvable in O(Ln) time using randomized SVD.

Algorithm is efficient for sparse preferences, e.g., O(n) for NYC school choice.

Method retains TTC properties and is robust to preference noise.

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~$n$.