Hospitals/Residents Problem

Synonyms

College admissions problem; Stable admissions problem; Stable assignment problem; Stable b-matching problem; University admissions problem

Years and Authors of Summarized Original Work

1962; Gale, Shapley

Problem Definition

An instance I of the Hospitals/Residents problem (HR) [6, 7, 18] involves a set $R =\{ r_{1},\ldots ,r_{n}\}$ of residents and a set $H =\{ h_{1},\ldots ,h_{m}\}$ of hospitals. Each hospital h _j ∈ H has a positive integral capacity, denoted by c _j . Also, each resident r _i ∈ R has a preference list in which he ranks in strict order a subset of H. A pair (r _i , h _j ) ∈ R × H is said to be acceptable if h _j appears in r _i ’s preference list; in this case r _i is said to find h_jacceptable. Similarly each hospital h _j ∈ H has a preference list in which it ranks in strict order those residents who find h _j acceptable. Given any three agents x, y, z ∈ R ∪ H, x is said to prefer y to z if x finds each of y and z acceptable, and y precedes z on x’s preference...