Resource allocation in rooted trees subject to sum constraints and nonlinear cost functions - Halman, Nir; Wimer, Shmuel (Information processing letters, 2021)
•We study resource allocation problems in rooted trees in which demand values are given in the leaves.•Single-type resources (weights) are to be assigned in the tree nodes such that the total weight in the rooted path from each leaf to the root equals its demand.•The goal is to minimize the total costs of the allocated resources.•It is known that when the cost of a resource is linearly proportional to its weight, the problem is solvable in linear time.•We show that when costs are arbitrary monotone nondecreasing functions, the problem becomes intractable, and design for it a fully polynomial time approximation scheme. We study resource allocation problems in rooted trees in which demand values are given in the leaves. Single-type resources (weights) are to be assigned in the tree nodes such that the total weight in the rooted path from each leaf to the root equals its demand. The goal is to minimize the total costs of the allocated resources. It is known that the linear cost case, i.e., when the cost of a resource is proportional to its weight, is solvable in linear time. In this paper we show that when costs are monotone nondecreasing functions, which reflect, e.g., (dis)economies of scale, the problem becomes intractable, and design for it a fully polynomial time approximation scheme by formulating it as a dynamic program and using the technique of K-approximation sets and functions.