The Stable Matching Linear Program and an Approximate Rural Hospital Theorem with Couples

The deferred acceptance algorithm has been the most commonly studied tool for computing stable matchings. An alternate less-studied approach is to use integer programming formulations and linear programming relaxations to compute optimal stable matchings. Papers in this area tend to focus on the simple ordinal preferences of the stable marriage problem. This paper advocates the use of linear programming for computing stable matchings with more general preferences: complements, substitutes and responsiveness, by presenting a series of qualitative and computational results. First, we show how linear programming relaxations can provide strong qualitative insights by deriving a new approximate rural hospital theorem. The standard rural hospital theorem, which states that every stable outcome matches the same doctors and hospitals, is known to fail in the presence of couples. We show that the total number of doctors and hospitals that change from matched to unmatched, and vice versa, between stable matchings is, at most, twice the number of couples. Next, we move from qualitative to computational insights, by outlining sufficient conditions for when our linear program returns a stable matchings. We show solving the stable matching linear program will yield a stable matching (i) for the doctor-optimal objective (or hospital-optimal), when agent preferences obey substitutes and the law of aggregate demand, and (ii) for any objective, when agent preferences over sets of contracts are responsive. Finally, we demonstrate the computational power of our linear program via synthetic experiments for finding stable matchings in markets with couples. Our linear program more frequently finds stable matchings than a deferred acceptance algorithm that accommodates couples.