The set cover with pairs problem

We consider a generalization of the set cover problem, in which elements are covered by pairs of objects, and we are required to find a minimum cost subset of objects that induces a collection of pairs covering all elements. Formally, let U be a ground set of elements and let S be a set of objects, where each object i has a non-negative cost w_i. For every {i, j} ⊆ S, let C(i, j) be the collection of elements in U covered by the pair {i, j}. The set cover with pairs problem asks to find a subset A ⊆ S such that U_{{i, j}⊆A} C(i, j) = U and such that ∑ _i∈A w_i minimized. In addition to studying this general problem, we are also concerned with developing polynomial time approximation algorithms for interesting special cases. The problems we consider in this framework arise in the context of domination in metric spaces and separation of point sets.