Models and cutting-plane strategies for the tree-star problem

Let G = (V, E) be a connected graph of set of nodes V and set of edges E. Let T = (V_T, E_T), with V_T = V and E_T ⊆ E, be a spanning tree of G. With each edge e ∈ E there is associated a routing cost $c_e^R$ if e connects two internal nodes of T; or an access cost $c_e^A$, otherwise. The problem is to determine a spanning tree (tree-star) considering access and routing edge costs of minimum cost. We present two new formulations and a cutting-plane algorithm. One is based on a classical spanning tree model. The novelty relies on the way we capture access and routing edges depending on the internal nodes of the tree. The second model is completely new and is based on the concept of dicycle to represent routing edges as quadratic variables that are linearized accordingly to obtain a tree-star equivalent structure. Computational experiments performed on benchmark instances for models P_Flow and P_HR from the literature and for the new ones (P_ST and P_DC) indicate that this problem is very difficult to deal with. Only a very small number of instances was solved to optimality in a given time limit. Models P_DC and P_HR, improved with cutting-plane strategies, although they do not solve optimally almost instances, performed better for this problem, with the dicycle-based model presenting the smallest gaps for instances for which some feasible solution was found.