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...