Approximate hierarchical facility location and applications to the bounded depth Steiner tree and range assignment problems

The paper concerns a new variant of the hierarchical facility location problem   on metric powers (HFLβ[h]HFLβ[h]), which is a multi-level uncapacitated facility location problem defined as follows. The input consists of a set F   of locations that may open a facility, subsets D1,D2,…,Dh−1D1,D2,…,Dh−1 of locations that may open an intermediate transmission station and a set DhDh of locations of clients. Each client in DhDh must be serviced by an open transmission station in Dh−1Dh−1 and every open transmission station in DlDl must be serviced by an open transmission station on the next lower level, Dl−1Dl−1. An open transmission station on the first level, D1D1 must be serviced by an open facility. The cost of assigning a station j   on level l⩾1l⩾1 to a station i   on level l−1l−1 is cijcij. For i∈Fi∈F, the cost of opening a facility at location i   is fi⩾0fi⩾0. It is required to find a feasible assignment that minimizes the total cost. A constant ratio approximation algorithm is established for this problem. This algorithm is then used to develop constant ratio approximation algorithms for the bounded depth Steiner tree problem and the bounded hop strong-connectivity range assignment problem.