Finding Pareto frontier for one-sided matching

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.

• The paper introduces ITEA, an inverse TTC algorithm that efficiently enumerates Pareto-optimal allocations in one-sided matching markets by recovering TTC preimages.
• It leverages a tagging system and recursive breadth-first search to prune explored configurations and avoid redundant TTC computations.
• Empirical results demonstrate orders of magnitude speedups over brute-force methods, highlighting significant computational savings in practical settings.

Pareto Frontier Enumeration in One-Sided Matching via Inverse Top Trading Cycles

Problem Setting and Motivation

The paper "Finding Pareto frontier for one-sided matching" (2604.21306) addresses a foundational problem in mechanism design and market allocation: the explicit enumeration of all Pareto-optimal (PO) allocations in one-sided matching markets with strict ordinal preferences. Such settings, typified by assignments like hostel room allocation, require each agent to be matched to a unique item (e.g., room) based only on preference rankings, with monetary transfers disallowed.

Top Trading Cycles (TTC), a well-studied and algorithmically tractable mechanism, is known to efficiently yield a PO allocation through a directed graph process. However, PO in this context lacks uniqueness; TTC's outcome depends on the initial endowment, leading to potentially many distinct PO allocations. This multitude critically affects downstream decision-making if fairness or welfare criteria are to be subsequently optimized, necessitating full knowledge of the Pareto frontier rather than selecting a single arbitrarily obtained solution.

The Inverse-TTC Enumeration Algorithm (ITEA)

Formulation

Denote the agent set as $\mathcal{N}$ ($|\mathcal{N}|=n$), item set as $\mathcal{R}$, preferences as $\mathcal{O}$, and the space of allocations as $\mathcal{A}$ ($|\mathcal{A}|=n!$). The TTC mapping $f:\mathcal{A}\to\mathcal{P}$ maps any initial allocation to its final PO allocation, with $\mathcal{P}$ the (unknown) set of all PO allocations.

The principal task is to efficiently enumerate $\mathcal{P}$, ideally by partitioning $\mathcal{A}$ into TTC-equivalence classes, avoiding the redundant exponential work of naively executing TTC from all $|\mathcal{N}|=n$0 initial allocations.

Algorithmic Structure

The proposed ITEA avoids brute-force redundancy by only executing forward TTC once per unique PO allocation, and reconstructing (by algorithmic inversion) all initial allocations mapping to that PO outcome.

A key conceptual innovation is the enumeration of preimages under $|\mathcal{N}|=n$1, enabling the inverse traversal of the TTC mechanism. This is accomplished via a two-component tagging system on the rooms (unmarked, circled, squared) and a recursive exploration (devour) of preimage configurations using breadth-first search, augmented by a procedure for updating eligible rooms (dressup).

Figure 1: Algorithmic flow outlining ITEA's forward TTC and inverse enumeration steps.

Figure 2: Schematic depiction of the InvTTC process for reconstructing all initial allocations leading to a target PO allocation.

By updating tags and permuting eligible rooms, the algorithm recursively recovers each equivalence class $|\mathcal{N}|=n$2 for all $|\mathcal{N}|=n$3, pruning explored configurations and guaranteeing complete coverage without redundancy.

Example

Consider $|\mathcal{N}|=n$4 agents and a specified preference profile. The diagram enumerates twelve initial allocations all converging under TTC to a particular PO allocation, illustrating the multi-level preimage structure and the recursive subdivisions.

Figure 3: The twelve allocations in invTTC([4,3,2,1,5])—all distinct initial allocations that TTC maps to the same PO assignment.

Theoretical Analysis

Soundness and Completeness

The paper rigorously establishes that:

• Soundness: Every allocation produced by invTTC for a PO target $|\mathcal{N}|=n$5 satisfies $|\mathcal{N}|=n$6.
• Completeness: Every initial allocation $|\mathcal{N}|=n$7 with $|\mathcal{N}|=n$8 is generated by invTTC.

This ensures a bijective correspondence between equivalence classes of initial allocations and the set of PO outcomes, with each class precisely recovered.

Computational Complexity

Let $|\mathcal{N}|=n$9 be the number of PO allocations (typically much less than $\mathcal{R}$0):

• Brute-force: Requires $\mathcal{R}$1 TTC runs, each $\mathcal{R}$2 or $\mathcal{R}$3.
• ITEA: Requires $\mathcal{R}$4 TTC runs (one per PO allocation) and $\mathcal{R}$5 additional complexity for the inverse traversals. Thus,

$\mathcal{R}$6

(or better with pointer-based TTC).

• Empirically, for $\mathcal{R}$7 up to 9, $\mathcal{R}$8 drops sharply with $\mathcal{R}$9, making ITEA orders of magnitude faster in TTC invocations for larger $\mathcal{O}$0.

Empirical Evaluation

The authors evaluated ITEA and brute-force TTC enumeration over 700 randomly sampled profiles for $\mathcal{O}$1 to $\mathcal{O}$2, with 100 instances per $\mathcal{O}$3. For each, they measured wall-clock runtime for full Pareto frontier recovery.

ITEA outperforms brute force for all $\mathcal{O}$4, with speedup factors increasing with $\mathcal{O}$5 as $\mathcal{O}$6 shrinks. For $\mathcal{O}$7, the average Pareto frontier size is $\mathcal{O}$8 (versus $\mathcal{O}$9 allocations in total), implying a reduction of over $\mathcal{A}$0 in TTC calls.

Figure 4: Empirical runtime comparison between ITEA and brute-force as $\mathcal{A}$1 grows, averaged over 100 random instances per $\mathcal{A}$2.

Implications and Future Research

Practical and Theoretical Ramifications

ITEA provides a concrete, polynomial-space mechanism for explicit Pareto frontier enumeration in one-sided matching. This enables:

• Secondary Optimization: Once the set $\mathcal{A}$3 is enumerated, arbitrary polynomial-time computable selection criteria (e.g., maximizing utilitarian or egalitarian welfare, minimizing envy) can be optimized via brute-force search over $\mathcal{A}$4.
• Structural Analysis: The approach reveals the full structure and diversity of PO allocations, relevant for both fairness diagnostics and property testing of different selection mechanisms.
• Algorithmic Efficiency: Empirical evidence underscores that for typical preference profiles, the frontier is far smaller than the full combinatorial allocation space, validating the practical impact of the proposed methodology.

Open Directions

• Efficient PO Selection: While ITEA enumerates all PO allocations, developing algorithms for selecting (without full enumeration) PO allocations optimizing secondary criteria (e.g., welfare, fairness) remains an open question for future work.
• Scalability: Though improved over brute-force, factorial complexity remains the bottleneck in worst-case instances. Further advances could seek relaxations, parameterized complexity refinements, or succinct PO representation.
• Extensions: Exploring adaptations to many-to-one matchings, ties in preferences, or other preference/assignment domains would enhance general applicability.

Conclusion

This work introduces a principled and computationally efficient inverse enumeration approach to fully characterize the Pareto frontier in one-sided matching with strict ordinal preferences. The ITEA algorithm provably yields all and only PO allocations, offering substantial computational savings over brute-force methods in non-pathological instances. This explicit characterization of the PO frontier provides a crucial tool for mechanism designers, enabling informed secondary optimization and a deeper understanding of the trade-offs inherent in one-sided matching markets.