Min-max-min Geometric Facility Location Problems

We propose algorithms for a special type of geometric facility location problem in which customers may choose not to use the facility. We minimize the maximum cost incurred by a customer, where the cost itself is a minimum between two costs, according to whether the facility is used or not. We therefore call this type of location problem a min-max-min geometric facility location problem. As a first example, we describe the Closer Post Office problem, a generalization of the minimum spanning circle problem. We show that this problem can be solved in O(n) randomized expected time. We also show that the proposed algorithm solves two other min-max-min geometric facility location problems. One, which we call the Moving Walkway problem, seems to be the first instance of a facility location problem using time metrics.