# Min-Degree Constrained Minimum Spanning Tree Problem: Complexity, properties and formulations

### Authors

### Abstract

This paper addresses a new combinatorial problem, the Min-Degree Constrained Minimum Spanning Tree (md-MST), that can be stated as: given a weighted undirected graph G = (V,E) with positive costs on the edges and a node-degrees function d : V ? N, the md-MST is to find a minimum cost spanning tree T of G, where each node i of T either has at least a degree of d(i) or is a leaf node. This problem is closely related to the well-known Degree Constrained Minimum Spanning Tree (d-MST) problem, where the degree constraint is an upper bound instead. The general NP-hardness for the md-MST is established and some properties related to the feasibility of the solutions for this problem are presented, in particular we prove some bounds on the number of internal and leaf nodes. Flow based formulations are proposed and computational experiments involving the associated LP relaxations are presented.### Keywords

degree constrained spanning tree problems;computational complexity;single-commodity flow formulations;multicommodity flow formulations### Subject

Operations Research and Complexity Theory### Journal

ITOR International transactions in Operational Research, Vol. 19, #3, pp. 323-352, Wiley, May 2012### Cited by

#### Year 2016 : 2 citations

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#### Year 2015 : 1 citations

Ali Mashreghi , Alireza Zarei . "When Diameter Matters: Parameterized Approximation Algorithms for Bounded Diameter Minimum Steiner Tree Problem". Theory of Computing Systems (Fev. 2016), Vol. 58, Issue 2, pp 287-303. (available online: 24 March 2015)

#### Year 2014 : 3 citations

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Leonardo Conegundes Martinez , Alexandre Salles da Cunha, "The min-degree constrained minimum spanning tree problem: Formulations and Branch-and-cut algorithm", Discrete Applied Mathematics, 164(1):210–224, 2014.

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#### Year 2013 : 5 citations

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Akbari Torkestani, Javad, "Mobility-Based Backbone Formation in Wireless Mobile Ad-hoc Networks", Wireless Personal Communications, 71(4):563-2586, Springer US, doi=10.1007/s11277-012-0955-1

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http://dx.doi.org/10.1007/s10589-013-9556-5

#### Year 2012 : 1 citations

Martinez, L. and da Cunha, A., ”A Parallel Lagrangian Relaxation Algorithm for the Min- Degree Constrained Minimum Spanning Tree Problem”, Combinatorial Optimization, 7422:237- 248, Springer US, 2012.

#### Year 2011 : 1 citations

Martinez, L. and da Cunha, A."The min-degree constrained minimum spanning tree problem: Formulations and Branch-and-cut algorithm", Discrete Applied Mathematics, Volume 164, Part 1, 2014, Pages 210–224

#### Year 2010 : 1 citations

Martinez, L., da Cunha, A.,"Finding min-degree constrained spanning trees faster with a Branch-and-cut algorithm", Electronic Notes in Discrete Mathematics, Volume 36, 1 August 2010, Pages 311–318