A 2-path approach for odd-diameter-constrained minimum spanning and Steiner trees

  • Authors:

  • Luis Gouveia;Thomas L. Magnanti;Cristina Requejo

  • Affiliations:

  • Departamento de Estatística e Investigação Operacional, Centro de Investigação Operacional Faculdade de Ciências da Universidade de Lisboa, Portugal;Department of Electrical Engineering and Computer Science and Sloan School of Management, MIT, Cambridge, Massachusetts 02139;Departamento de Matemática, Centro de Estudos em Optimização e Controlo, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal

  • Venue:
  • Networks
  • Year:
  • 2004

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Abstract

In a previous article, using underlying graph theoretical properties, Gouveia and Magnanti (2003) described several network flow-based formulations for diameter-constrained tree problems. Their computational results showed that, even with several enhancements, models for situations when the tree diameter D is odd proved to be more difficult to solve than those when D is even. In this article we provide an alternative modeling approach for the situation when D is odd. The approach views the diameter-constrained minimum spanning tree as being composed of a variant of a directed spanning tree (from an artificial root node) together with two constrained paths, a shortest and a longest path, from the root node to any node in the tree. We also show how to view the feasible set of the linear programming relaxation of the new formulation as the intersection of two integer polyhedra, a so-called triangle-tree polyhedron and a constrained path polyhedron. This characterization improves upon a model of Gouveia and Magnanti (2003) whose linear programming relaxation feasible set is the intersection of three rather than two integer polyhedra. The linear programming gaps for the tightened model are very small, typically less than 0.5%, and are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover, using the new model, we have been able to solve large Euclidean problem instances that are not solvable by the previous approaches. © 2004 Wiley Periodicals, Inc.