Reformulation by Intersection Method on the MST Problem with Lower Bound on the Number of Leaves

Abstract

We consider a variant of the the minimum spanning tree with a constraint imposing a minimum number of leaves. This paper is motivated by the computational results taken from a small set of instances with an enhanced directed model where it is shown that the corresponding linear programming bound values strongly depend on the choice of the root node. Thus, we present a new formulation for this problem that is based on “intersecting” all the rooted tree models at the same time. We will show that the linear programming bound of the new model is, in general, substantially better than the linear programming bound obtained by the best directed model. The computational results indicate that the model is too large to solve efficiently medium sized instances. In order to overcome this disadvantage, we present an iterative procedure that starts with a single rooted model and sequentially adds other rooted models. The idea of this method is to obtain an intermediate intersection model (that is, a model where only some of the rooted models are considered in the intersection) and such that the corresponding linear programming bound will be close to the bound obtained by the model which results from intersecting all the rooted models. The computational results show that the iterative procedure is worth using and should be further investigated when using the reformulation by intersection for other problems.