[Paper Review] A Minimax Perspective on Almost-Stable Matchings

Frederik Glitzner, David F. Manlove|arXiv (Cornell University)|Jan 20, 2026

Game Theory and Voting Systems0 citations

TL;DR

This paper introduces minimax almost-stability for matching problems, proving strong hardness results and offering positive algorithms for restricted instances, with empirical evidence that well-distributed instability is likely to exist.

ABSTRACT

Stability is crucial in matching markets, yet in many real-world settings - from hospital residency allocations to roommate assignments - full stability is either impossible to achieve or can come at the cost of leaving many agents unmatched. When stability cannot be achieved, 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 either the total number of blocking pairs or the count of agents involved in blocking pairs. However, such aggregate objectives can result in concentrated instability on a few individual 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 a disproportionate amount of instability. We characterise the computational complexity of this notion across fundamental matching settings. Surprisingly, even very modest guarantees 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 have constant-bounded length. This hardness 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. Our results map the algorithmic landscape and reveal fundamental trade-offs between distributional guarantees and computational feasibility.

Motivation & Objective

• Motivate the study of stability in matching markets where full stability is unattainable and instability must be distributed fairly.
• Introduce a fairness-oriented minimax objective that minimizes the maximum number of blocking pairs any agent experiences.
• Characterize the computational complexity of minimax almost-stability across stable matching and roommates settings.
• Provide positive algorithmic results for restricted preference lengths and develop approximations and integer programs for general cases.
• Empirically evaluate the existence and quality of minimax almost-stable matchings.

Proposed method

• Define Minimax-AlmostStable-sri and Minimax-AlmostStable-Max-smi as new optimisation problems focusing on the worst-case number of blocking pairs per agent.
• Prove NP-completeness for decision variants even with bounded or complete lists, across sri and smi settings.
• Present polynomial-time algorithms for instances with preference lists of length at most 2, and provide approximation algorithms and integer programming formulations for general cases.
• Contrast minimax almost-stability with existing aggregate instability notions and blocking-agent counts to establish structural distinctions.
• Construct theoretical lower bounds showing how the minimax value scales with problem size and provide proofs of tightness in certain constructions.

Figure 1 . Schematic illustration of the sri lower-bound construction for 27 agents (i.e,. $k=3$ ). Agents are partitioned into 3 nested sets containing $3^{j}$ (for $1\leq i\leq k$ ) agents each. Arrows indicate preference cycles within and between the nested levels.

Experimental results

Research questions

• RQ1 What is the computational complexity of finding minimax almost-stable matchings in sri and smi settings?
• RQ2 Can minimax almost-stable matchings be computed in polynomial time for restricted preference lengths?
• RQ3 How do minimax almost-stable matchings compare to aggregate stability measures and blocking-agent counts in terms of fairness and feasibility?
• RQ4 What are the approximability and exact-optimization approaches that can solve minimax almost-stable problems in general settings?
• RQ5 Do empirical analyses suggest that well-distributed instability matchings exist in practice?

Key findings

• Deciding whether there exists a matching with at most one blocking pair per agent is NP-complete for sri and smi, even with constant-bounded list length.
• Minimax-AlmostStable-Max-smi has worst-case optimal value OPT = Θ(n), indicating linear growth with the number of agents.
• There exist polynomial-time algorithms for instances where preference lists have length at most 2.
• The paper provides approximation algorithms and exact integer programming models for general settings.
• Experiments indicate that almost-stable and maximum-cardinality matchings with favorable instability distribution are likely to exist.

Figure 2 . Illustration of the lower-bound construction for maximum-cardinality matchings in smi .