[Paper Review] Stable Matchings, Robust Solutions, and Distributive Lattices

Tung Mai, Vijay V. Vazirani|arXiv (Cornell University)|Apr 16, 2018

Game Theory and Voting Systems11 references4 citations

TL;DR

This paper presents an O(|S| p(n)) algorithm to compute a matching stable across a base stable matching instance A and |S| perturbed versions of A, each with a single error in a preference list. The method leverages novel structural properties of distributive lattices of stable matchings and a generalized Birkhoff's theorem for sublattices, enabling polynomial-time robust solutions when |S| is polynomially bounded.

ABSTRACT

A stable matching instance $A$ is transmitted over a channel which introduces {\em one error} from a super-exponentially large set, $T$; the error consists of arbitrarily permuting the preference list of any one boy or any one girl. An arbitrary set $S \subseteq T$ is specified. We give an $O(|S| p(n))$ time algorithm for finding a matching that is stable for $A$ and for each of the $|S|$ instances that result by introducing one error from $S$, where $p$ is a polynomial function. In particular, if $S$ is polynomial sized, then our algorithm runs in polynomial time. Our algorithm is based on new, non-trivial structural properties of the lattice of stable matchings. A second ingredient of our algorithm is a generalization of Birkhoff's Theorem for finite distributive lattices which states that corresponding to such a lattice, $\mathcal{L}$, there is a partial order, say $\Pi$, such that the lattice of closed sets of $\Pi$, $L(\Pi)$, is isomorphic to $\mathcal{L}$. Our generalization shows that corresponding to each sublattice $\mathcal{L}'$ of $\mathcal{L}$, there is a partial order $\Pi'$ that can be obtained from $\Pi$ (via a specified operation) such that $L(\Pi') \cong \mathcal{L}'$. This generalization is of independent interest.

Motivation & Objective

• To develop an efficient algorithm that computes a single matching stable across a base stable matching instance and multiple perturbed variants, each with one error in a preference list.
• To identify structural properties of the lattice of stable matchings that enable robustness against single-list perturbations.
• To generalize Birkhoff's theorem for finite distributive lattices to sublattices, establishing a correspondence between sublattices and derived partial orders.
• To ensure the algorithm runs in polynomial time when the number of perturbed instances |S| is polynomially bounded.

Proposed method

• Leverage the distributive lattice structure of stable matchings to model and analyze stability under perturbations.
• Define a partial order Π from the original lattice of stable matchings, such that the lattice of closed sets of Π is isomorphic to the original lattice.
• Generalize Birkhoff's theorem to show that for any sublattice L′ of the original lattice L, there exists a derived partial order Π′ such that the lattice of closed sets of Π′ is isomorphic to L′.
• Construct a polynomial-time algorithm that checks stability across the base instance and each of the |S| error-affected instances by exploiting lattice closure and sublattice properties.
• Use the generalized lattice isomorphism to efficiently compute a common stable matching across all instances in the set S.
• Apply the algorithm to instances where S is polynomial-sized, ensuring overall runtime remains O(|S| p(n)) for some polynomial p(n).

Experimental results

Research questions

• RQ1 Can a single matching be computed that remains stable across a base stable matching instance and multiple variants with one error in any preference list?
• RQ2 What structural properties of the lattice of stable matchings enable robustness against single-list perturbations?
• RQ3 How can Birkhoff's theorem for finite distributive lattices be generalized to sublattices, and what is the resulting partial order construction?
• RQ4 Under what conditions does the algorithm achieve polynomial-time complexity for robust matching computation?

Key findings

• The algorithm computes a matching stable for the original instance A and for each of |S| perturbed instances with a single error, in O(|S| p(n)) time.
• The method relies on a novel generalization of Birkhoff's theorem, establishing a lattice isomorphism between the closed sets of a derived partial order Π′ and any sublattice of the original lattice.
• When |S| is polynomially bounded, the algorithm runs in polynomial time, making robust stable matching computationally feasible for practical error sets.
• The structural properties of the distributive lattice of stable matchings are shown to be resilient to single-list perturbations, enabling efficient search for common stable matchings.
• The derived partial order Π′ is constructed from the original Π via a specified operation, ensuring the lattice of closed sets of Π′ is isomorphic to the target sublattice.
• The framework provides a foundation for designing robust algorithms in matching markets under uncertainty or communication errors.