A matching process on a two-sided market can determine who gets which job [40, 31], school place[44],orspouse.

The rise of Large Language Models (LLMs) has driven progress in reasoning tasks -- from program synthesis to scientific hypothesis generation -- yet their ability to handle ranked preferences and structured algorithms in combinatorial domains remains underexplored. We study matching markets, a core framework behind applications like resource allocation and ride-sharing, which require reconciling individual ranked preferences to ensure stable outcomes. We evaluate several state-of-the-art models on a hierarchy of preference-based reasoning tasks -- ranging from stable-matching generation to instability detection, instability resolution, and fine-grained preference queries -- to systematically expose their logical and algorithmic limitations in handling ranked inputs. Surprisingly, even top-performing models with advanced reasoning struggle to resolve instability in large markets, often failing to identify blocking pairs or execute algorithms iteratively. We further show that parameter-efficient fine-tuning (LoRA) significantly improves performance in small markets, but fails to bring about a similar improvement on large instances, suggesting the need for more sophisticated strategies to improve LLMs' reasoning with larger-context inputs.

We study a many-to-one matching problem, such as the college admission problem, where each college can admit multiple students. Unlike classical models, colleges evaluate sets of students through non-linear utility functions that capture diversity between them. In this setting, we show that classical stable matchings may fail to exist. To address this, we propose alternative solution concepts based on Rawlsian fairness, aiming to maximize the minimum utility across colleges. We design both deterministic and stochastic algorithms that iteratively improve the outcome of the worst-off college, offering a practical approach to fair allocation when stability cannot be guaranteed.

Neural Information Processing Systems http://nips.cc/

We study parameterized approximability of three optimization problems related to stable matching: (1) Min-BP-SMI: Given a stable marriage instance and a number k, find a size-at-least-k matching that minimizes the number $β$ of blocking pairs; (2) Min-BP-SRI: Given a stable roommates instance, find a matching that minimizes the number $β$ of blocking pairs; (3) Max-SMTI: Given a stable marriage instance with preferences containing ties, find a maximum-size stable matching. The first two problems are known to be NP-hard to approximate to any constant factor and W[1]-hard with respect to $β$, making the existence of an EPTAS or FPT-algorithms unlikely. We show that they are W[1]-hard with respect to $β$ to approximate to any function of $β$. This means that unless FPT=W[1], there is no FPT-approximation scheme for the parameter $β$. The last problem (Max-SMTI) is known to be NP-hard to approximate to factor-29/33 and W[1]-hard with respect to the number of ties. We complement this and present an FPT-approximation scheme for the parameter "number of agents with ties".

The Stable Roommates problems are characterized by the preferences of agents over other agents as roommates. A solution is a partition of the agents into pairs that are acceptable to each other (i.e., they are in the preference lists of each other), and the matching is stable (i.e., there do not exist any two agents who prefer each other to their roommates, and thus block the matching). Motivated by real-world applications, and considering that stable roommates problems do not always have solutions, we continue our studies to compute "good-enough" matchings. In addition to the agents' habits and habitual preferences, we consider their networks of preferred friends, and introduce a method to generate personalized solutions to stable roommates problems. We illustrate the usefulness of our method with examples and empirical evaluations.

The Deferred Acceptance (DA) algorithm is an elegant procedure for finding a stable matching in two-sided matching markets. It ensures that no pair of agents prefers each other to their matched partners. In this work, we initiate the study of two-sided manipulations in matching markets as non-cooperative games. We introduce the accomplice manipulation game, where a man misreports to help a specific woman obtain a better partner, whenever possible. We provide a polynomial time algorithm for finding a pure strategy Nash equilibrium (NE) and show that our algorithm always yields a stable matching - although not every Nash equilibrium corresponds to a stable matching. Additionally, we show how our analytical techniques for the accomplice manipulation game can be applied to other manipulation games in matching markets, such as one-for-many and the standard self-manipulation games. We complement our theoretical findings with empirical evaluations of different properties of the resulting NE, such as the welfare of the agents.

The stable marriage problem with incomplete lists and ties (SMTI) and the hospitals/residents problem with ties (HRT) are important in matching theory with broad practical applications. In this paper, we introduce a tie-breaking based local search algorithm (TBLS) designed to achieve a weakly stable matching of maximum size for both the SMTI and HRT problems. TBLS begins by arbitrarily resolving all ties and iteratively refines the tie-breaking strategy by adjusting the relative order within ties based on preference ranks and the current stable matching. Additionally, we introduce TBLS-E, an equity-focused variant of TBLS, specifically designed for the SMTI problem. This variant maintains the objective of maximizing matching size, while enhancing equity through two simple modifications. In comparison with ten other approximation and local search algorithms, TBLS achieves the highest matching size, while TBLS-E exhibits the lowest sex equality cost. Significantly, TBLS-E preserves a matching size comparable to that of TBLS. Both our algorithms demonstrate faster computational speed than other local search algorithms in solving large-sized instances.

This paper considers the scenario in which there are multiple institutions, each with a limited capacity for candidates, and candidates, each with preferences over the institutions. A central entity evaluates the utility of each candidate to the institutions, and the goal is to select candidates for each institution in a way that maximizes utility while also considering the candidates' preferences. The paper focuses on the setting in which candidates are divided into multiple groups and the observed utilities of candidates in some groups are biased--systematically lower than their true utilities. The first result is that, in these biased settings, prior algorithms can lead to selections with sub-optimal true utility and significant discrepancies in the fraction of candidates from each group that get their preferred choices. Subsequently, an algorithm is presented along with proof that it produces selections that achieve near-optimal group fairness with respect to preferences while also nearly maximizing the true utility under distributional assumptions. Further, extensive empirical validation of these results in real-world and synthetic settings, in which the distributional assumptions may not hold, are presented.

The canonical setup is the primary approach adopted in cell-free multiple-input multiple-output (MIMO) networks, in which all access points (APs) jointly serve every user equipment (UE). This approach is not scalable in terms of computational complexity and fronthaul signaling becoming impractical in large networks. This work adopts a user-centric approach, a scalable alternative in which only a set of preferred APs jointly serve a UE. Forming the optimal cluster of APs for each UE is a challenging task, especially, when it needs to be dynamically adjusted to meet the quality of service (QoS) requirements of the UE. This complexity is even exacerbated when considering the constrained fronthaul capacity of the UE and the AP. We solve this problem with a novel many-to-many matching game. More specifically, we devise an early acceptance matching algorithm, which immediately admits or rejects UEs based on their requests and available radio resources. The proposed solution significantly reduces the fronthaul signaling while satisfying the maximum of UEs in terms of requested QoS compared to state-of-the-art approaches.