A computational approach to the multi-period many-to-one matching with ties

The many-to-one matching problem is commonly referred as the hospitals/residents problem which assigns each resident a hospital in an efficient and fair way. This paper considers a multi-period hospitals/residents problem that consists of assigning positions to overlapping generations of residents. From one period to another, residents can either retain their current positions or can choose a more preferred one. In this situation, a fairness criterion is introduced with the condition of the individual rationality for the matching. Moreover, it has been proven that the matching satisfying such criterion always exists and can be obtained by iteratively eliminating Pareto improvement cycles and unjustified claim cycles from any acceptable matching. This paper presents a novel algorithm to compute a matching with minimal unjustified claims under the premise of satisfying the individual rationality and the Pareto efficiency. The complexity of the proposed algorithm is bounded by $$O(Q^{3}n^{3})$$ O ( Q 3 n 3 ) in each period, where n and Q are the number of the residents and the total number of positions of the hospitals in the corresponding period, respectively.