Finding Pareto frontier for one-sided matching

TL;DR

This paper introduces ITEA, an efficient algorithm to compute all Pareto-optimal allocations in one-sided matching problems, enhancing understanding of the Pareto frontier beyond traditional TTC solutions.

Contribution

The paper presents ITEA, a novel algorithm that efficiently enumerates the entire Pareto-efficient frontier in one-sided matching problems, with proven correctness and improved computational performance.

Findings

ITEA reduces redundant computations compared to brute-force enumeration.

Empirical results show efficient characterization of the Pareto frontier.

The algorithm handles cases with fewer Pareto-optimal allocations more efficiently.

Abstract

One-sided matching problems with ordinal preferences, such as hostel room allocation, are commonly solved using the Top Trading Cycles (TTC) mechanism, which guarantees Pareto-optimal (PO) outcomes. However, TTC does not yield a unique solution: multiple PO allocations may exist, and many distinct initial endowments can converge to the same outcome. Focusing on a single TTC result obscures the structure of the Pareto-efficient frontier and limits principled secondary optimization over fairness or welfare objectives. Therefore, the goal is to find the entire set of PO allocations for a given preference profile. We propose the Inverse Top Trading Cycles Enumeration Algorithm (ITEA), a novel method that efficiently computes the complete set of Pareto-optimal allocations in one-sided matching problems. We prove the soundness and completeness of the proposed algorithm and analyze its computational complexity. Although in the worst case, there can be $n!$ PO allocations; however, compared to the brute-force approach, our algorithm reduces time complexity when there are fewer PO allocations. Empirical results demonstrate substantial reductions in redundant TTC computations compared to brute-force enumeration, enabling efficient characterization of the Pareto frontier.