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 (HFL_β [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, ..., D_{h - 1} of locations that may open an intermediate transmission station and a set D_h of locations of clients. Each client in D_h must be serviced by an open transmission station in D_{h - 1} and every open transmission station in D_l must be serviced by an open transmission station on the next lower level, D_{l - 1}. An open transmission station on the first level, D₁ must be serviced by an open facility. The cost of assigning a station j on level l ≥ 1 to a station i on level l - 1 is c_{i j}. For i ∈ F, the cost of opening a facility at location i is f_i ≥ 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.