Mikhail Y. Kovalyov, Erwin Pesch, Alain Quilliot

• Braverman et al. [Math. Oper. Res. 41(1), (2016), pp. 352–376], introduce the problem Provision-after-Wait which is to find a stable (ey 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.