Affinely representable lattices, stable matchings, and choice functions

TL;DR

This paper introduces affine representability of distributive lattices, enabling efficient linear optimization and convex hull descriptions, and applies it to stable matchings with choice functions, generalizing previous models.

Contribution

It defines affine representability for distributive lattices and uses it to develop efficient algorithms for stable matchings with choice functions, expanding beyond classical methods.

Findings

Efficient algorithms for stable matchings with choice functions.

Polyhedral descriptions of the convex hull of lattice elements.

Generalization of existing models in stable matching literature.

Abstract

Birkhoff's representation theorem (Birkhoff, 1937) defines a bijection between elements of a distributive lattice and the family of upper sets of an associated poset. Although not used explicitly, this result is at the backbone of the combinatorial algorithm by Irving et al. (1987) for maximizing a linear function over the set of stable matchings in Gale and Shapley's stable marriage model (Gale and Shapley, 1962). In this paper, we introduce a property of distributive lattices, which we term as affine representability, and show its role in efficiently solving linear optimization problems over the elements of a distributive lattice, as well as describing the convex hull of the characteristic vectors of the lattice elements. We apply this concept to the stable matching model with path-independent quota-filling choice functions, thus giving efficient algorithms and a compact polyhedral description for this model. To the best of our knowledge, this model generalizes all models from the literature for which similar results were known, and our paper is the first that proposes efficient algorithms for stable matchings with choice functions, beyond classical extensions of the Deferred Acceptance algorithm.