Stable Matchings, Robust Solutions, and Distributive Lattices

We build on our recent paper [MV18a], which initiated the problem of finding stable matchings that are robust to errors in the input and gave a polynomial time algorithm for a special case in which the domain of errors was polynomially large. We had observed that this set of errors is too restrictive and had asked the question of extending the domain. In [MV18a] we had also initiated work on a new structural question regarding stable matchings, namely finding relationships between the lattices of solutions of two nearby instances (in this case, the given and the perturbed instances). In this paper, we make substantial progress on both fronts. First, we extend the domain of errors to a super-exponential one, though we need to somewhat weaken the notion of robust. Underlying our polynomial time algorithms are new structural properties, of the type described above. A key to deriving these structural properties is a purely combinatorial proof of a generalization of Birkhoffu0027s Theorem for finite distributive lattices, derived here in the context of stable matching lattices. This proof yields crucial new notions, such as that of a {em meta-rotation}, and is therefore of independent interest.