Hospitals/Residents Problems with Quota Lower Bounds

Years and Authors of Summarized Original Work

• 2010; Biró, Fleiner, Irving, Manlove;

• 2010; Huang;

• 2011; Hamada, Iwama, Miyazaki;

• 2012; Fleiner, Kamiyama

Problem Definition

The Hospitals/Residents (HR) problem is the many-to-one version of the stable marriage problem introduced by Gale and Shapley. In this problem, a bipartite graph $G = (\mathcal{R}\cup \mathcal{H},E)$ is given. Each vertex in $\mathcal{H}$ represents a hospital and each vertex in $\mathcal{R}$ a resident. Each vertex has a preference over its neighboring vertices. Each hospital h has an upper quota u(h) specifying the maximum number of residents it can take in a matching. The goal is to find a stable matching while respecting the upper quotas of the hospitals.

The original HR has been well studied in the past decades. A recent trend is to assume that each hospital h also comes with a lower quota l(h). In this context, it is required (if possible) that a matching satisfies both the upper and the lower quotas of...