A class of bottleneck expansion problems

Abstract 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. Scope and Purpose There are many capacity expansion problems in real life. For example, a transportation network may need to increase the flow capacity, and a production system may need to enhance its productive ability. It is quite often that the capacity of a scheme is decided by its bottleneck capacity, i.e. the capacity of the weakest part, and people need to choose from all possible schemes the one with the largest bottleneck capacity. Mathematically, this is a max–min problem. Capacity expansion is unavoidably related to the available budget. So, we need to consider how to use the limited budget to expand the bottleneck capacity to the largest extent. The purpose of the paper is to give a general method for a class of bottleneck expansion problems which is easy to use and obtains solutions in polynomial time.