Provision-after-wait with preferences ordered by difference: Tighter complexity and better approximation - Kovalyov, Mikhail Y.; Pesch, Erwin; Quilliot, Alain (European journal of operational research, 2021)
•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.