Refined Inequalities for Stable Marriage

We consider two approaches to the stable marriage problem: proposal algorithms and describing the stable matching polytope using linear inequalities. We illuminate the relationship between the two approaches. Beginning with a set of linear inequalities that describe the stable matching polytope, we describe a process of refining the set of linear inequalities by eliminating redundant constraints and pruning the preference lists to eliminate unattainable assignments. We show that it is trivial to use the pruned preference lists to find the firm-optimal and worker-optimal stable matchings. We then describe a new procedure that combines a proposal algorithm and our refining process to find a stable matching that does not favor one group over the other. Finally, we apply our refining process to problems in which couples submit preferences over pairs of positions.