Stable Matchings with Flexible Quotas

We consider the problem of assigning agents to programs in the presence of two-sided preferences. Each agent has to be assigned to at most one program and each program can accommodate multiple agents. However unlike the standard setting of school choice, we do not have fixed upper quotas with programs as an input -- instead we abstract it as a cost associated with every program. This setting enables the programs to control the number of agents assigned to them and is also applicable in the two-round setting (Gajulapalli et al., FSTTCS 2020) where largest stable extension is computed. In this setting, our goal is to compute a min-cost stable matching that matches all the agents. We show that the problem is NP-hard even under very severe restrictions on the instance. We complement our negative results by presenting approximation algorithms for general case and a fast exponential algorithm for a special case.