The vertex cover problem is the following. We have a graph with vertices and every vertex has a weight. Our goal is to find a sub-set of the vertices of the graph in a way that all the edges are covered, that means that for every edge of the graph at least one of the two end points of the edge is in the set cover. A feasible solution is a collection of vertices such that all the edges are covered. In our integer linear programming formulation, we assign a variable x(v) at every vertex v. If x(v) is equal to 1 then the vertex is taken in the vertex cover. So the objective function is the sum over the vertices of variables x(v) times w(v), the weight of vertex v.

---