A Generalization of the Minimum Branch Vertices Spanning Tree Problem

Abstract

Given a connected graph $\mathcal {G}=(\mathcal {V}, \mathcal {E})$ and its spanning tree $\mathcal {T}$, a vertex $v \in \mathcal {V}$ is said to be a branch vertex if its degree is strictly greater than 2 in $\mathcal {T}$. The Minimum Branch Vertices Spanning Tree (MBVST) problem is to find a spanning tree of $\mathcal {G}$ with the minimum number of branch vertices. This problem has been extensively studied in the literature and has well-developed applications notably related to routing in optical networks. In this paper, we propose a generalization of this problem, where we begin by introducing the notion of a k-branch vertex, which is a vertex with degree strictly greater than $k+2$. Our goal is to determine a spanning tree of $\mathcal {G}$ with the minimum number of k-branch vertices (k-MBVST problem). In the context of optical networks, the parameter k can be seen as the limiting capacity of optical splitters to duplicate the input light signal and forward to k destinations. Proofs of NP-hardness and non-inclusion in the APX class of the k-MBVST problem are established for a generic value of k, and then an ILP formulation of the k-MBVST problem based on single commodity flow balance constraints is derived. Computational results based on randomly generated graphs show that the number of k-branch vertices included in the spanning tree increases with the size of the vertex set $\mathcal {V}$, but decreases with k as well as graph density. We also show that when $k\ge 4$, the number of k-branch vertices in the optimal solution is close to zero, regardless of the size and the density of the underlying graph.