Approximately Stable Matchings with General Constraints

This paper focuses on two-sided matching where one side (a hospital or firm) is matched to the other side (a doctor or worker) so as to maximize a cardinal objective under general feasibility constraints. In a standard model, even though multiple doctors can be matched to a single hospital, a hospital has a responsive preference and a maximum quota. However, in practical applications, a hospital has some complicated cardinal preference and constraints. With such preferences (e.g., submodular) and constraints (e.g., knapsack or matroid intersection), stable matchings may fail to exist. This paper first determines the complexity of checking and computing stable matchings based on preference class and constraint class. Second, we establish a framework to analyze this problem on packing problems and the framework enables us to access the wealth of online packing algorithms so that we construct approximately stable algorithms as a variant of generalized deferred acceptance algorithm. We further provide some inapproximability results.