An Approximation Algorithm for the k-Level Uncapacitated Facility Location Problem with Penalties

The k-level Uncapacitated Facility Location (UFL) problem is a generalization of the UFL and the k-median problems. A significant shortcoming of the classical UFL problem is that often a few very distant customers, known as outliers, can leave an undesirable effect on the final solution. This deficiency is considered in a new variant called UFL with outliers, in which, in contrast to the other problems that need all of the customers to be serviced, there is no need to service the entire set of customers. UFL with Penalties (UFLWP) is a variant of the UFL with outliers problem in which we will decide off whether to provide service for each customer and pay the connection cost, of, to reject it and pay the penalty. In this paper we will propose a new 4-approximation algorithm for the UFLWP which is the first algorithm for this kind of problem.