In this paper we consider how to increase the capacities of the elements in a set E efficiently so that the capacity of a given family ℱ of subsets of E can be increased to the maximum extent while the total cost for the increment of capacity is within a given budget bound. We transform this problem into finding the minimum weight element of ℱ when the weight of each element of E is a linear function of a single parameter and propose an algorithm for solving this problem. We further discuss the problem for some special cases. Especially, when E is the edge set of a network and the family ℱ consists of all spanning trees, we give a strongly polynomial algorithm.
