Almost-Stable Matching Problems

• Almost-stable matching problems are models that relax strict stability by permitting a limited number of blocking pairs, enabling practical solutions under complex constraints.
• They encompass various frameworks such as bounded instability, minimax criteria, and fractional approaches, each with distinct tractability and approximation characteristics.
• Recent research highlights NP-hardness in exact minimization while demonstrating viable parameterized and distributed algorithms for achieving near-stable matchings.

Almost-stable matching problems address the computation and analysis of matchings that relax strict stability conditions, typically by allowing a bounded number or type of local instabilities. This concept is central in matching theory, especially in practical settings where exact stability is infeasible due to computational constraints, market structure, preference distributions, or the presence of global or combinatorial constraints. The recent literature provides a comprehensive taxonomy, complexity landscape, and algorithmic toolkit for various almost-stable matching notions, including aggregate, distributional, approximate, and parameterized frameworks.

1. Definitions and Taxonomy of Almost-Stable Matching

At the core, a stable matching instance (bipartite or nonbipartite) is specified by a set of agents with preferences (strict, weak, or cardinal) over a subset of potential partners. A matching is stable if there are no blocking pairs—pairs where both agents strictly prefer each other to their assigned partners.

Almost-stable matching problems generalize this as follows (Chen et al., 13 Aug 2025, Glitzner et al., 20 Jan 2026, &&&2&&&, Manlove et al., 2016):

• Bounded instability (blocking pairs): Seek a matching admitting at most $k$ blocking pairs ($\min \beta(M)$ for a matching $M$).
• Size-stability trade-off: Maximize the cardinality of a matching subject to a bound on blocking pairs, or given $k$, find the largest $M$ with $\beta(M) \leq k$ (Gupta et al., 2020).
• Distributional (minimax) instability: Minimize the maximum number of blocking pairs involving any agent, i.e., $\min_M \max_i b(i)$, where $b(i)$ is the count for agent $i$ (Glitzner et al., 20 Jan 2026).
• Weak stability (subsample of agents): Require that only a specified subset of deviators be protected from blocks, with others conforming regardless (Glitzner et al., 26 Jan 2026).
• $\varepsilon$-stable or approximately stable: For cardinal and/or feasibility-constrained settings, allow deviations only if a group can achieve an improvement by more than a factor $\alpha$ or $\varepsilon$ (Kawase et al., 2019, Caragiannis et al., 2019).
• Incremental almost-stable matching: Given a stable matching $M_1$ for input $\mathcal{P}_1$, compute $M_2$ stable or nearly stable for modified input $\mathcal{P}_2$, while minimizing $\lvert M_1 \bigtriangleup M_2 \rvert$ and/or the number of blocking pairs (Boehmer et al., 2021).

These forms reflect both aggregate and agent-centric notions, and their tractability and approximation properties vary substantially across models.

2. Complexity and Inapproximability Results

The computational complexity of almost-stable problems depends acutely on:

• Metric: Whether minimizing the total number of blocking pairs, the number of agents with at least one block (blocking agents), the maximally blocked agent, or limiting instability to specified agents.
• Instance structure: Bipartite vs nonbipartite, strict vs weak/partial orders, presence of ties, cardinal vs ordinal preferences, and constraints (e.g., matroids, knapsacks).
• Parameterizations: Size of allowed instability, matching cardinality, number of ties, or structural parameters.

The principal negative results include:

• Blocking pair minimization: Both the stable marriage (SMI) and the stable roommates (SRI) versions of $\min \beta(M)$ are NP-hard and W[1]-hard parameterized by $\beta$, even for small $\beta$, and are inapproximable to any $f(\beta)$-factor for any computable function $f$ unless W[1]=FPT (Chen et al., 13 Aug 2025).
• Minimax instability: Deciding if there exists a matching where every agent is in at most $k$ blocking pairs is NP-complete for $k=1$ in both SR and maximum-cardinality SM, already for bounded (constant) preference list length. No $2-\epsilon$ approximation is possible unless P=NP (Glitzner et al., 20 Jan 2026).
• Deviator-stable matching: If only a subset of agents are potential deviators, it remains NP-complete to decide whether there is a matching with no blocking deviators, both for nonbipartite and bipartite cases, even with list length constraints (at most 3) (Glitzner et al., 26 Jan 2026).
• Parameterization by size or difference: Parameterized complexity results show W[1]-hardness, e.g., for the problem of finding a matching larger than the maximum stable matching by $t$ and with at most $k$ blocking pairs, parameterized by $k+t+d$ (Gupta et al., 2020). For incremental models, NP-hardness and W[1]-hardness persist when parameterizing by the allowed change and number of blocking pairs (Boehmer et al., 2021).
• Max-SMTI with ties: Maximizing the size of a stable matching in SMTI is NP-hard to approximate better than $29/33$ and W[1]-hard parameterized by number of agents with ties (Chen et al., 13 Aug 2025, Csáji et al., 2022).
• Three-dimensional matching: In both 3GSM (Three-Gender Stable Marriage) and 3PSA (Three Person Stable Assignment), deciding existence of a stable matching is NP-complete, and both maximally stable matching and maximum stable submatching are APX-hard (Ostrovsky et al., 2014).

3. Algorithmic Positive Results and Approximation Schemes

Despite strong intractability, certain cases and relaxations admit nontrivial polynomial-time or parameterized solutions:

• Distributed constant-time approximations: For bounded-degree graphs, a truncated Gale–Shapley process yields $\varepsilon$-stable matchings (number of blocking pairs within $\varepsilon$ times the matching size) in $O(\Delta^2/\epsilon)$ time (0812.4893). For unbounded degree, subpolynomial-round deterministic and randomized distributed algorithms achieve similar bounds (Ostrovsky et al., 2014).
• Parameter-tractable and local-search FPT: The local-search variant of ASM (almost-stable marriage), where the matching is at most $q$ away from a stable matching and at most $k$ blocking pairs are allowed, admits $2^{O(q \log d)} n^{O(1)}$ FPT solutions (Gupta et al., 2020).
• Incremental and minimally invasive stability: If the allowed matching change is $k$ and the number of blocking pairs $b$ is fixed, problems like IASM/IHR-T admit XP or FPT-algorithms in special cases, especially when parameterized by agent-type count or the number of residents (Boehmer et al., 2021).
• Fractional and approximate stability: With cardinal utilities and fractional matchings, computing an $\varepsilon$-stable matching with welfare $\varepsilon$-fraction of optimum is polynomial-time tractable (Caragiannis et al., 2019). Polytime 2-stable fractional matchings always exist with optimal welfare.
• General constraints and packing: For submodular/hard constraints, the α-stable matching paradigm leverages online packing algorithms: e.g., for $k$-matroid intersection, the greedy yields $k$-stable matchings (Kawase et al., 2019).
• Tractable cases for restricted lists: Many models become polynomial if all lists have length at most 2 (e.g., minimax (Glitzner et al., 20 Jan 2026), MIN BP HRC (Manlove et al., 2016), deviator-stability (Glitzner et al., 26 Jan 2026)).
• FPT for Max-SMTI by ties: Although Max-SMTI is hard in general, with only $\tau$ agents having ties, one can FPT-approximate the maximum stable size to any $1-\varepsilon$ fraction in $f(\varepsilon,\tau) n^{O(1)}$ time (Chen et al., 13 Aug 2025).

4. Fairness, Distribution, and Agent-Specific Instability

Recent frameworks extend almost-stability to protect or balance instability in a more agent-aware manner.

• Minimax approach: Minimize $\beta(M) = \max_i b(i)$, i.e., the worst-off agent's exposure to instability (Glitzner et al., 20 Jan 2026). This "fairness-oriented" approach leads to severe intractability: even deciding $\beta(M)\leq 1$ is NP-complete for both SR and SM with bounded lists.
• Deviator-conformist models: Given a set of deviators $D$ (agents who may initiate blocks) and conformists $C$, aim for $M$ such that $bp_D(M)=\varnothing$ (no blocking pair involves a deviator), or with limited exceptions. This model interpolates between classical full-agent and completely weak stability and is NP-complete for $k=0$, but FPT-tractable parameterized by $|D|$ and block count (Glitzner et al., 26 Jan 2026).
• Incremental minimal change vs. blocking-pair trade-off: Allowing a few blocking pairs drastically decreases the number of matching changes needed after preference perturbations (Boehmer et al., 2021).
• Maximum number of agents in blocks: In hospital-residents/couples, optimization may target minimizing the number of involved agents, not blocking pairs (Manlove et al., 2016).

5. Methodologies and Mathematical Programming

Various mathematical and algorithmic frameworks support almost-stable matching computation:

• Integer/Constraint Programming: Expressing almost-stable objectives through binary variables encoding assignments and blocking pairs enables exact IP/CP models for both bipartite and nonbipartite settings (Manlove et al., 2016, Glitzner et al., 20 Jan 2026), with symmetry-breaking and efficient presolve yielding tractability for moderate-size real-world instances.
• Distributed/local algorithms: Quantile-based proposals, randomized maximal matching, and truncate-and-prune strategies yield efficient distributed protocols (Ostrovsky et al., 2014, 0812.4893).
• Online and packing reductions: For complex constraints, mapping stability to competitive online (cardinal) packing or matroid kernel optimization enables the design and analysis of scalable approximation algorithms (Kawase et al., 2019, Csáji et al., 2022).
• Reduction to weighted stable marriage: Incremental and change-limited variants are reduced to weighted SM and solved via specific polynomial-time algorithms (Boehmer et al., 2021).
• Fractional relaxations: LP relaxations and their integrality gaps are tight for particular preference classes; in interval orders the gap is 1.5, while with arbitrary partials it's 2 (Csáji et al., 2022).

6. Applications, Empirical Observations, and Open Questions

Almost-stable matching concepts are motivated by and applicable to a range of real-world allocation and market-design settings:

• Practical matching markets: Empirical studies show that even highly constrained hospital/residents/couples markets (HRC) with random or real-world-like preferences are almost entirely stable, with solutions typically admitting at most 1-2 blocking pairs, and exact/incremental matching change is minimized by tolerating even a small fraction of instability (Manlove et al., 2016, Boehmer et al., 2021).
• Three-dimensional extension: The absence of guaranteed stability in 3GSM and 3PSA exposes the need to optimize for maximally stable matchings, typically via greedy approximation (4/9-stable in $O(n^7)$ time) (Ostrovsky et al., 2014).
• Dynamic and evolving preferences: In dynamic models where preference lists evolve, continuous Gale–Shapley–like processes maintain an $O((\log n)^2)$-almost-stable matching despite adversarial random perturbations (Kanade et al., 2015).
• Distributed systems: Local information and bounded communication suffice for almost-stable matching in large-scale networks (0812.4893).
• Open questions: Includes the precise tractability boundary for minimax and deviator-stability as list lengths vary; the existence of better approximation or FPT algorithms for these and related objectives; and scaling CP/IP and matroidal methods to massive real-world market instances (Glitzner et al., 20 Jan 2026, Glitzner et al., 26 Jan 2026, Boehmer et al., 2021, Csáji et al., 2022).

7. Comparative Table of Core Models and Complexity

Model/Class	Objective/Instability	Tractability	References
Min BP-SMI / Min BP-SRI	Total blocking pairs ($\beta$)	NP-hard, W[1]-hard in $\beta$; no FPT approx in $\beta$	(Chen et al., 13 Aug 2025, Gupta et al., 2020)
Minimax (max agent blocks)	$\max_i b(i)$	NP-complete for $k=1$; linear-time for list length $\leq 2$; $\lceil d/2\rceil$-approx	(Glitzner et al., 20 Jan 2026)
Deviator-Stability	Blockers only in $D$	NP-complete for $k=0$; FPT in $|D|$ and $k$; linear-time for list length $\leq 2$	(Glitzner et al., 26 Jan 2026)
Max-SMTI (with ties)	Maximum size, stable	NP-hard to $29/33$, W[1]-hard by ties, FPT-AS for $\tau$ ties	(Chen et al., 13 Aug 2025, Csáji et al., 2022)
Approximate/Fractional/ε-stable	$\varepsilon$-stability	Fractional: polytime for $\varepsilon$-stable; inapproximable for $\varepsilon$ below 0.03	(Caragiannis et al., 2019, Kawase et al., 2019)
HRC MIN BP	Blocking pairs w/ couples	NP-hard, inapproximable; polynomial for (2,1,2) lists	(Manlove et al., 2016)
Incremental almost-stable	Min adjustments + blocks	W[1]-hard in $(k+b+\Delta)$, but empirical: small $b$ suffices to minimize adjustments	(Boehmer et al., 2021)
3GSM, 3PSA	Max stable triples	APX-hard, $\frac{4}{9}$-approx greedy in $O(n^7)$	(Ostrovsky et al., 2014)

These results chart a nuanced but sharply drawn frontier between tractable and intractable forms of almost-stable matching, and underline the importance of careful modeling choices in both theory and applied market design.