A parallel lagrangian relaxation algorithm for the min-degree constrained minimum spanning tree problem

Authors:
Leonardo Conegundes Martinez;Alexandre Salles da Cunha
Affiliations:
Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil;Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Venue:
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
Year:
2012

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Abstract

Given an edge weighted undirected graph G and a positive integer d, the Min-Degree Constrained Minimum Spanning Tree Problem (MDMST) asks for a minimum cost spanning tree of G, such that each vertex is either a leaf or has degree at least d in the tree. The strongest known MDMST lower bounds, provided by a reformulation by intersection, are very expensive to be evaluated directly, by Linear Programming solvers. Therefore, we propose a Lagrangian Relaxation algorithm for approximating them. The reformulation makes use of a large number of variables and the relaxation involves the dualization of a large number of constraints. Attempting to speed up the computation of the Lagrangian Dual bounds, we implemented a parallel Subgradient Method. We also introduced a Lagrangian heuristic based on a Local Branching algorithm. With the proposed methods, respectively 26 and 14 new best upper and lower bounds are presented.