Multi-level facility location as the maximization of a submodular set function

In this paper we model the multi-level uncapacitated facility location problem as two different combinatorial optimization problems. The first one uses a set of vertices as combinatorial objects to represent solutions whereas the second one uses a set of paths. An interesting observation is that the real-valued set function associated with the first combinatorial problem does not satisfy the submodular property, whereas the set function associated with the second problem does satisfy this property. This illustrates the fact that submodularity is not a property intrinsic to an optimization problem but rather to its mathematical representation.