Exact solution for branch vertices constrained spanning problems

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is strictly greater than 2. The Minimum Branch Vertices Spanning Tree problem (MBVST) consists in finding a spanning tree of G with the minimum number of branch vertices. This problem has been well studied in the literature and has several applications specially for routing in optical networks. However, this kind of applications do not explicitly impose a sub-graph as solution. A more flexible structure called hierarchy is proposed. Hierarchy, which can be seen as a generalization of trees, is defined as a homomorphism of a tree in a graph. Since minimizing the number of branch vertices in a hierarchy does not make sense, we propose to search the minimum cost spanning hierarchy such that the number of branch vertices is less than or equal to an integer R. We introduce the Branch Vertices Constrained Minimum Spanning Hierarchy (BVCMSH) problem which is NP-hard. The Integer Linear Program (ILP) formulation of this new problem is given. To evaluate the difference of cost between trees and hierarchies, we confront the BVCMSH problem to the Branch Vertices Constrained Minimum Spanning Tree (BVCMST) problem by comparing its exact solutions. It appears from this comparison that when R ⩽ 2 , the hierarchies improve the average cost from more than 8% when | V G | = 20 and from more than 12% when | V G | = 30 .