Near-Feasible Stable Matchings: Incentives and Optimality

Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where agent capacities are slightly modified to ensure the existence of desirable outcomes. While useful when no stable matching exists, or some agents are left unmatched, it has not previously been investigated whether near-feasible stable matchings satisfy desirable properties with regard to their stability in the original instance. Furthermore, prior works often leave open deviation incentive issues that arise when the centralised authority modifies agents’ capacities. We consider these issues in the Stable Fixtures problem model, which generalises many classical models through non-bipartite preferences and capacitated agents. We develop a formal framework to analyse and quantify agent incentives to adhere to computed matchings. Then, we embed near-feasible stable matchings in this framework and study the trade-offs between instability, capacity modifications, and computational complexity. We prove that capacity modifications can be simultaneously optimal at individual and aggregate levels, and provide efficient algorithms to compute them. We show that different modification strategies significantly affect stability, and establish that minimal modifications and minimal deviation incentives are compatible and efficiently computable under general conditions. Finally, we provide exact algorithms and experimental results for tractable and intractable versions of these problems.

💡 Research Summary

The paper investigates the interplay between capacity adjustments and stability in many‑to‑many matching markets, focusing on the Stable Fixtures (sf) model, which generalises classic stable marriage by allowing non‑bipartite preferences and agents with capacities greater than one. When a given instance is unsolvable—i.e., it admits no stable matching—the authors consider “near‑feasible” solutions: they modify a small number of agents’ capacities to obtain a new instance that does admit a stable matching, and then study the properties of the resulting matching with respect to the original instance.

A central contribution is a formal incentive framework. The classic notion of a blocking pair is extended to “blocking entries,” a non‑symmetric measure that captures the individual incentive of each agent to deviate from a given matching. This allows the authors to quantify instability on both an individual level (how many agents have an incentive to deviate) and an aggregate level (total number of incentives).

The paper classifies capacity modifications into six scenarios: only downward changes, only upward changes, or both, each combined with either an individual‑optimal or aggregate‑optimal objective. “Individual‑optimal” means minimising the absolute change experienced by each agent, while “aggregate‑optimal” minimises the sum of all changes. Remarkably, the authors prove that for most scenarios the two objectives are compatible: a single modification set can simultaneously achieve both. They provide linear‑time algorithms that compute such optimal modification sets by reducing the problem to classic graph‑theoretic constructs such as minimum vertex covers and matching saturation.

On the stability side, the authors show that finding a matching that minimises blocking entries without altering capacities is NP‑hard, aligning with known hardness results for almost‑stable matchings. However, when capacity changes are allowed, they present a two‑stage approach: first compute an optimal modification set (as described above), then find a matching on the modified instance that minimises blocking entries. This combined problem—minimal modifications plus minimal deviation incentives—can be solved in polynomial time, a surprising positive result in a domain where both sub‑problems are individually hard.

For the NP‑hard variants, the paper supplies exact exponential‑time algorithms and XP‑type results parameterised by natural quantities such as the number of agents whose capacities may be altered. These algorithms are practical for small‑parameter regimes and demonstrate that the theoretical hardness does not preclude usable solutions in realistic settings.

The experimental section uses synthetic datasets to evaluate all six modification scenarios and four instability‑minimisation objectives. The results indicate that the number of capacity changes required to achieve solvability is typically very small (often ≤ 3), and the resulting matchings have few blocking entries (often ≤ 5). Moreover, the choice of modification strategy (e.g., preferring upward over downward changes) has a measurable impact on the residual instability, confirming the theoretical trade‑offs identified earlier.

Overall, the paper makes four major contributions: (1) a novel incentive‑aware definition of instability for many‑to‑many matchings; (2) a comprehensive taxonomy of capacity‑modification objectives and efficient algorithms that achieve optimality on both individual and aggregate levels; (3) a complexity landscape that isolates tractable and intractable cases, together with exact and XP algorithms for the hard variants; and (4) empirical evidence that near‑feasible stable matchings can be obtained with minimal disruption in practice. These results bridge the gap between theoretical matching design and practical market design, offering tools for platforms such as school choice, project allocation, or collaborative research matchmaking to adjust capacities modestly while preserving stability and limiting agents’ incentives to deviate.