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.

我们研究了在叶子中给出需求值的生根树中的资源分配问题。将在树节点中分配单一类型的资源（权重），以使从每个叶子到根的根路径中的总权重等于其需求。目标是最大程度地减少分配的资源的总成本。已知线性成本情况，即当资源成本与其权重成正比时，可以在线性时间内解决。在本文中，我们表明，当成本是单调非递减函数时，它反映了（例如）规模的（非）经济问题，则该问题变得棘手，并通过将其公式化为动态程序并使用其来设计完全多项式时间逼近方案。K逼近集和函数的技巧。