Approximation theory in combinatorial optimization. Application to the generalized minimum spanning tree problem

We present an overview of the approximation theory in combinatorial optimization. As an application we consider the Generalized Minimum Spanning Tree (GMST) problem which is defined on an undirected complete graph with the nodes partitioned into clusters and non-negative costs are associated to the edges. This problem is NP-hard and it is known that a polynomial approximation algorithm cannot exist. We present an in-approximability result for the GMST problem and under special assumptions: cost function satisfying the triangle inequality and with cluster sizes bounded by $\rho$, we give an approximation algorithm with ratio $2 \rho$

The connected facility location polytope

We analyze the polytope associated with a combinatorial problem that combines the Steiner tree problem and the uncapacitated facility location problem. The problem, called connected facility location problem, is motivated by a real-world application in the design of a telecommunication network, and concerns with deciding the facilities to open, the assignment of customers to open facilities, and the connection of the open facilities through a Steiner tree. Several solution approaches are proposed in the literature, and the contribution of our work is a polyhedral analysis for the problem. We compute the dimension of the polytope, present valid inequalities, and analyze conditions for these inequalities to be facet defining. Some inequalities are taken from the Steiner tree polytope and the uncapacitated facility location polytope. Other inequalities are new