# A Generalization of the Minimum Branch Vertices Spanning Tree Problem

* Corresponding author
2 MAORE - Méthodes Algorithmes pour l'Ordonnancement et les Réseaux
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
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.
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Conference papers
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https://hal.archives-ouvertes.fr/hal-01983084
Contributor : Miklos Molnar <>
Submitted on : Wednesday, January 16, 2019 - 11:07:49 AM
Last modification on : Thursday, June 6, 2019 - 10:50:42 AM

### Citation

Massinissa Merabet, Jitamitra Desai, Miklós Molnár. A Generalization of the Minimum Branch Vertices Spanning Tree Problem. ISCO: International Symposium on Combinatorial Optimization, Apr 2018, Marrakech, Morocco. pp.338-351, ⟨10.1007/978-3-319-96151-4_29⟩. ⟨hal-01983084⟩

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