House Allocation Problems with Existing Tenants and Priorities for Teacher Recruitment

We study a variant of house allocation problems with application to a real-world job assignment problem where some applicants cannot result unmatched. Such applicants hold posts initially, which enter the market if they can get strictly better posts. All applicants are strictly ordered by priority in a single master list. Their preference lists may be incomplete and may contain ties. We seek a matching that assigns the best possible post to each applicant, taking into account their preferences, priorities and initial posts. We give algorithms for solving the problem in polynomial time for three different cases.