Search for: All records

Gale and Shapley’s stable assignment problem has been extensively studied, applied, and extended. In the context of school choice, mechanisms often aim at finding an assignment that is more favorable to students. We investigate two extensions introduced in this framework—legal assignments and the efficiency adjusted deferred acceptance mechanism (EADAM) algorithm—through the lens of the classic theory of stable matchings. In any instance, the set [Formula: see text] of legal assignments is known to contain all stable assignments. We prove that [Formula: see text] is exactly the set of stable assignments in another instance. Moreover, we show that essentially all optimization problems over [Formula: see text] can be solved within the same time bound needed for solving it over the set of stable assignments. A key tool for this latter result is an algorithm that finds the student-optimal legal assignment. We then generalize our algorithm to obtain the assignment output of EADAM with any given set of consenting students without sacrificing the running time, hence largely improving in both theory and practice over known algorithms. Finally, we show that the set [Formula: see text] can be much larger than the set of stable matchings, connecting legal matchings with certain concepts and open problems in the literature. 

Birkhoff’s representation theorem defines a bijection between elements of a distributive lattice L and the family of upper sets of an associated poset B. When elements of L are the stable matchings in an instance of Gale and Shapley’s marriage model, Irving et al. showed how to use 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 quotafilling 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.