Affinely Representable Lattices, Stable Matchings, and Choice Functions

Abstract

Birkhoff’s representation theorem [11] defines a bijection between elements of a distributive lattice \(\mathcal{L}\) and the family of upper sets of an associated poset \(\mathcal{B}\). When elements of \(\mathcal{L}\) are the stable matchings in an instance of Gale and Shapley’s marriage model, Irving et al. [22] showed how to use \(\mathcal{B}\) to devise a combinatorial algorithm for maximizing a linear function over the set of stable matchings. In this paper, we introduce a general 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 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 extension of the Deferred Acceptance algorithm [31].