Bottleneck Capacity Expansion Problems with General Budget Constraints

¹ Institut für Mathematik B, TU Graz, Steyrergasse 30, 8010 Graz, Austria.
² Department of Mathematics, City University of Hong Kong, Hong Kong.

Received: May 2000
Accepted: February 2001

Abstract

This paper presents a unified approach for bottleneck capacity expansion problems. In the bottleneck capacity expansion problem, BCEP, we are given a finite ground set E, a family F of feasible subsets of E and a nonnegative real capacity ĉ_e for all e ∈ E. Moreover, we are given monotone increasing cost functions f_e for increasing the capacity of the elements e ∈ E as well as a budget B. The task is to determine new capacities c_e ≥ ĉ_e such that the objective function given by max_F∈Fmin_e∈Fc_e is maximized under the side constraint that the overall expansion cost does not exceed the budget B. We introduce an algebraic model for defining the overall expansion cost and for formulating the budget constraint. This models allows to capture various types of budget constraints in one general model. Moreover, we discuss solution approaches for the general bottleneck capacity expansion problem. For an important subclass of bottleneck capacity expansion problems we propose algorithms which perform a strongly polynomial number of steps. In this manner we generalize and improve a recent result of Zhang et al. [15].