Stable Matchings with Flexible Quotas

We consider the problem of assigning agents to programs in the presence of two-sided preferences, commonly known as the Hospital Residents problem. In the standard setting each program has a rigid upper quota which cannot be violated. Motivated by applications where quotas may be flexible, we propose and study the problem of computing stable matchings under flexible quotas -- denoted as the SMFQ setting. In the SMFQ setting we have a cost associated with every program and these costs control the quotas. Our goal is to compute a stable matching that matches all agents and is optimal with respect to the cost criteria. We study two optimization problems with respect to the costs and show that there is a sharp contrast in the complexity status of the two optimization criteria. We also establish the connection of our model with the stable extension problem in an apparently different two-round setting of the stable matching problem. We show that our results in the SMFQ setting generalize the stable extension problem.