The Hospitals / Residents Problem (1962; Gale, Shapley)

An instance I of the Hospitals / Residents problem (HR) [6, 7, 15] involves a set R = {r1, . . . , rn} of residents and a set H = {h1, . . . , hm} of hospitals. Each hospital hj ∈ H has a positive integral capacity, denoted by cj . Also, each resident ri ∈ R has a preference list in which he ranks in strict order a subset of H. A pair (ri, hj) ∈ R ×H is said to be acceptable if hj appears in ri’s preference list; in this case ri is said to find hj acceptable. Similarly each hospital hj ∈ H has a preference list in which it ranks in strict order those residents who find hj 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 list. Let C = ∑ hj∈H cj . Let A denote the set of acceptable pairs in I, and let L = |A|. An assignment M is a subset of A. If (ri, hj) ∈ M , ri is said to be assigned to hj , and hj is assigned ri. For each q ∈ R ∪ H, the set of assignees of q in M is denoted by M(q). If ri ∈ R and M(ri) = ∅, ri is said to be unassigned, otherwise ri is assigned. Similarly, any hospital hj ∈ H is under-subscribed, full or over-subscribed according as |M(hj)| is less than, equal to, or greater than cj , respectively. A matching M is an assignment such that |M(ri)| ≤ 1 for each ri ∈ R and |M(hj)| ≤ cj for each hj ∈ H (i.e., no resident is assigned to an unacceptable hospital, each resident is assigned to at most one hospital, and no hospital is over-subscribed). For notational convenience, given a matching M and a resident ri ∈ R such that M(ri) 6= ∅, where there is no ambiguity the notation M(ri) is also used to refer to the single member of M(ri). A pair (ri, hj) ∈ A\M blocks a matching M , or is a blocking pair for M , if the following conditions are satisfied relative to M :