Provision-after-wait with preferences ordered by difference: Tighter complexity and better approximation

• Published in: European journal of operational research Vol. 289; no. 3; pp. 1008 - 1012
• Main Authors: Kovalyov, Mikhail Y., Pesch, Erwin, Quilliot, Alain
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 16.03.2021; Elsevier
• Subjects: Computer Science; FPTAS; Healthcare; Knapsack problem; Operations Research; Resource allocation; Scheduling; Healthcare; Scheduling; FPTAS; Resource allocation; Knapsack problem
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2019.07.047

•Find an envy-free assignment of patients and their waiting times to hospitals.•Social welfare is maximized.•The problem is reduced to Ordered Knapsack.•Our NP-hardness proof is simpler and our approximation scheme is faster. Braverman et al. [Math. Oper. Res. 41(1), (2016), pp. 352–376], introduce the problem Provision-after-Wait which is to find a stable (envy free) assignment of n patients to m hospitals, and their waiting times before admission, such that the social welfare is maximized, subject to a limited budget. Chan et al. [ACM Trans. Econ. Comput. 5(2), (2017), Article 12, pp. 12:1–12:36] focus on a natural case of d-ordered preferences, in which patients are ordered according to the differences of their values between consecutive hospitals. For this case, they provide a sophisticated proof of ordinary NP-hardness, reduce it to the problem called Ordered Knapsack, and develop a fully polynomial time approximation scheme for Ordered Knapsack. We present a simple proof that Ordered Knapsack is NP-hard, which implies NP-hardness of a more restrictive case of the original problem, and present an alternative fully polynomial time approximation scheme with a reduced run time by a quadratic factor of n, for a fixed m. A similar algorithm is developed to find a solution for which the social welfare is as high as for the optimal solution of Ordered Knapsack, and the budget limit can be exceeded by at most 1+ε times. We also present polynomial algorithms for the cases of Ordered Knapsack, in which the number of distinct input parameters is fixed.