arXiv subjects

David Manlove

Publications and source records attributed to David Manlove.

3 recordsLinked to original sources

arXiv2026

Structural and Algorithmic Results for Stable Cycles and Partitions in the Roommates Problem

In the Stable Roommates problem, we seek a stable matching of the agents into pairs, in which no two agents have an incentive to deviate from their assignment. It is well known that a stable matching is unlikely to exist, but a stable partition always does and provides a succinct certificate for the unsolvability of an instance. Furthermore, apart from being a useful structural tool to study the problem, every stable partition corresponds to a stable half-matching, which has applications, for example, in sports scheduling and time-sharing. We establish new structural results for stable partitions and show how to enumerate all stable partitions and the cycles included in such structures efficiently. We also adapt optimality criteria from stable matchings to stable partitions and give complexity and approximability results for the problems of computing such "fair" and "optimal" stable partitions. Through this research, we contribute to a deeper understanding of stable partitions from a combinatorial point of view, as well as the computational complexity of computing "fair" or "optimal" stable half-matchings in practice, closing the gap between integral and fractional stable matchings and paving the way for further applications of stable partitions to unsolvable instances and computationally hard stable matching problems.

cs.DS↗

arXiv2026

Unsolvability and Beyond in Many-To-Many Non-Bipartite Stable Matching

We study the Stable Fixtures problem, a many-to-many generalisation of the classical non-bipartite Stable Roommates matching problem. Building on the foundational work of Tan on stable partitions, we extend his results to this significantly more general setting and develop a rich framework for understanding stable structures. Our main contribution, the notion of a generalised stable partition (GSP), not only characterises the solution space but also serves as a versatile tool for ordinal preference systems with capacity constraints. We show that a GSP can be computed efficiently and can provide an elegant representation of key aspects of a preference system. Leveraging a connection to stable half-matchings, we also establish an analogous Rural Hospitals Theorem for stable half-matchings and GSPs, and connect our results to recent work on near-feasible matchings, providing a simpler algorithm and tighter analysis. Our work also addresses the computational challenges of finding optimal stable half-matchings and GSPs, presenting a flexible integer linear programming model for various objectives. Beyond theoretical insights, we conduct the first empirical analysis of random Stable Fixtures instances. Our work unifies and extends classical and recent perspectives on stability in non-bipartite stable matching and establishes new tools and techniques for stable matchings and their applications.

cs.DS↗

arXiv2026

A Minimax Perspective on Almost-Stable Matchings

Stability is crucial in matching markets, yet in many real-world settings - from hospital residency allocations to roommate assignments - full stability is either unachievable or comes at the cost of leaving agents unmatched. In these cases, algorithmicists and market designers face a critical question: how should instability be measured and distributed among participants? Existing approaches to "almost-stable" matchings focus on aggregate measures, minimising the number of blocking pairs or the count of agents involved in blocking pairs. However, these objectives can result in concentrated instability on a few agents, raising concerns about fairness and incentives to deviate. We introduce a fairness-oriented approach to approximate stability based on the minimax principle: we seek matchings that minimise the maximum number of blocking pairs any agent is in. Equivalently, we minimise the maximum number of agents that anyone has justified envy towards. This distributional objective protects the worst-off agents from bearing a disproportionate amount of instability. We characterise the computational complexity of this notion across fundamental matching settings. Surprisingly, even very modest guarantees with respect to the distribution of instability prove computationally intractable: we show that it is NP-complete to decide whether a matching exists in which no agent is in more than one blocking pair, even when preference lists are bounded. This result applies to both Stable Roommates and maximum cardinality Stable Marriage. On the positive side, we provide polynomial-time algorithms when agents rank at most two others, and present approximation algorithms and integer programs for general settings. Our results map the algorithmic landscape and reveal fundamental trade-offs between distributional guarantees on justified envy and computational feasibility in matching market design.

cs.GT↗