Robust Stable Matching Polytope

• The paper introduces a polytope formulation where robust stable matchings are defined as the convex hull of matchings that remain stable across perturbed preference instances.
• It demonstrates that integrality holds under single-agent perturbations, enabling efficient LP solutions, while multiple deviations lead to fractional vertices and increased complexity.
• The study connects polyhedral combinatorics with lattice theory, revealing robust matching structures that support algorithmic optimization in controlled uncertainty settings.

A Robust Stable Matching Polytope is the convex hull of matchings that retain stability properties not just in a single instance of the matching market but across a collection of perturbed or uncertain preference profiles. Robustness in matching has multiple formalizations; the strongest postulates that a matching is stable under every preference profile within a given family of perturbations. The polyhedral study of such robust matchings reveals how far the classical combinatorial, lattice, and polytope theory of stable matchings extends to robustness against preference uncertainty.

1. Robust Stable Matchings: Definitions and Polytope Formulation

Given two-sided matching instances (e.g., workers $W$ and firms $F$), a matching $M$ is robust-stable across instances $A$ and $B$ if $M$ is stable in both, i.e., $M \in \mathcal M_A \cap \mathcal M_B$, where $\mathcal M_X$ is the set of stable matchings for instance $X$. The concept extends naturally to a collection of instances $\mathcal S$: $M$ is fully robust if $M$ is stable in all $X \in \mathcal S$.

The robust stable matching polytope corresponding to $A$ and $B$ is given by the convex hull of $\mathcal M_A \cap \mathcal M_B$ viewed in the matching indicator variables $x_{wf}$: $$\mathrm{conv}(\mathcal M_A \cap \mathcal M_B) = \mathrm{conv}\left\{x_{wf} \in \{0,1\}^{n \times n}: M \text{ stable in } A \text{ and } B \right\}.$$ The linear constraints for the robust polytope are:

• matching constraints: row- and column-sums $=1$;
• blocking-pair constraints: for all $(w,f)$,

$$\sum_{f': f' >_w^A f} x_{w f'} - \sum_{w': w' >_f^A w} x_{w' f} \leq 0,$$

plus the analogous constraints for $B$.

This polytope generalizes Rothblum's classic stable matching polytope (convex hull of the stable matchings for a fixed instance).

2. Polyhedral Structure and Integrality

A central question is whether the robust stable matching polytope is integral—i.e., whether every vertex is integral and thus corresponds to an actual robust matching. The answer depends on the perturbation regime:

• Single-agent perturbations: When at most one agent on each side may change their preference list arbitrarily (type $(1,n)$ or $(n,1)$), every vertex of the robust polytope is integral. Thus, the linear description above fully characterizes the convex hull in these regimes, and robust matchings can be found by linear programming in $O(n^3)$ time (Gangam et al., 12 Jan 2026).
• Multiple agents changing: If two or more agents per side may change, integrality fails. For example, in a type $(2,2)$ instance with $4\times4$ agents, the robust polytope has fractional extreme points (e.g., $x = \frac12 M_1 + \frac12 M_2$), as no integer weighting satisfies the blocking constraints for both $A$ and $B$ (Gangam et al., 12 Jan 2026).
• General (fully robust): The polytope approach remains valid for any finite set of instances, but integrality may be lost as soon as the set includes distinct "critical" changes on both sides.

Table: Polyhedral Integrality by Perturbation Type

Perturbation Regime	Integrality Holds	Polytope is tractable?
Single upward shift	Yes	Yes ($O(n^2)$ LP)
Single arbitrary permutation (one agent)	Yes	Yes ($O(n^2)$ LP)
Up to 1 agent per side	Yes	Yes ($O(n^3)$ LP)
$\geq$2 agents per side	No	No (fractional vertices)

3. Polytopal Algorithms and Lattice Connections

When the robust polytope is integral, one can optimize linear objectives (e.g., minimum egalitarian cost over all robust matchings) by simply solving an LP. In these regimes, the structural properties of classical stable matching—distributive lattices and a succinct Birkhoff partial order—persist:

• The set $\mathcal M_A \cap \mathcal M_B$ forms a distributive sublattice of the stable matching lattice of $A$ (and $B$) (Gangam et al., 2018, Gangam et al., 12 Jan 2026).
• There exists an explicit, succinct partial order (of meta-rotations), and all robust matchings are accessible as closed sets in this Birkhoff representation (Gangam et al., 12 Jan 2026, Gangam et al., 2018, Menon et al., 2018).
• Deferred Acceptance variants and bouquet compression algorithms efficiently enumerate and optimize robust stable matchings in polynomial time in the favorable regime.

However, when the robust polytope is non-integral, the sublattice structure and efficient enumeration break down, and algorithms revert to XP or NP-complete status (Gangam et al., 12 Jan 2026).

4. Structural Boundaries and Counterexamples

The dichotomy between integrality and fractional vertices (as well as sublattice and absence thereof) is precisely delineated:

• For at most one agent per side changing (i.e., $\min(p,q) \leq 1$ in type $(p,q)$ perturbations), the robust stable matching polytope is integral and the robust stable matchings form a sublattice (Gangam et al., 12 Jan 2026).
• For $\min(p,q) \geq 2$, both properties can fail: the intersection of stables may not be closed under meet or join, and explicit examples exist where the LP has only fractional optimal points [(Gangam et al., 12 Jan 2026), Example 3.2].

These properties are robust to various definitions of the error set: upward shifts, single-agent arbitrary permutation, and mixtures thereof all yield the same threshold for integrality.

5. Computational Complexity and Algorithmic Regimes

The integrality of the polytope yields efficient algorithms in the favorable regime:

• $O(n^3)$-time LP formulations suffice for all objectives expressible as linear functions of the matching variables (Gangam et al., 12 Jan 2026).
• Enumeration and optimization over all robust matchings can be implemented with $O(1)$ delay after preprocessing, via the Birkhoff partial order (Gangam et al., 12 Jan 2026, Gangam et al., 2018).
• For generality, if $p+q$ agents can change, robust matching decision/optimization can be performed in $\operatorname{XP}$ time: $O(n^{p+q+2})$, tractable for constant parameters (Gangam et al., 12 Jan 2026).
• For $p+q = n$, i.e. all agents may change, the problem is NP-complete, and the robust stable matching polytope neither encodes all solutions nor supports efficient search (Gangam et al., 12 Jan 2026).

6. Connections, Implications, and Broader Models

The study of the robust stable matching polytope connects polyhedral combinatorics, lattice theory, and computational social choice. It characterizes the precise impact of local uncertainty in preferences on the structure and tractability of matching markets. Robustness imposes stronger constraints than classical stability, yet surprisingly the polyhedral and lattice theory persists for broad, practically motivated perturbation classes (e.g., survey revision by a single participant). However, robustness across many changes quickly induces combinatorial intractability and destroys the favorable properties of the polytope.

A plausible implication is that, in real-world markets where only modest numbers of agents revise or misstate preferences, polyhedral and lattice-based algorithms for robust stable matching remain efficient and expressive. By contrast, markets with broad uncertainty or coordinated preference changes—even by just two agents per side—experience a qualitative shift in complexity and structural behavior (Gangam et al., 12 Jan 2026).

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References:

• "Robust Stable Matchings: Dealing with Changes in Preferences" (Gangam et al., 12 Jan 2026)
• "Matchings under Preferences: Strength of Stability and Trade-offs" (Chen et al., 2019)
• "A Structural and Algorithmic Study of Stable Matching Lattices of 'Nearby' Instances, with Applications" (Gangam et al., 2018)
• "Robust and Approximately Stable Marriages under Partial Information" (Menon et al., 2018)