Multistars and directed flow formulations

  1. Luis Gouveia1,*,
  2. Leslie Hall2

Article first published online: 11 OCT 2002

DOI: 10.1002/net.10050

Issue

Networks

Networks

Volume 40, Issue 4, pages 188–201, December 2002

Additional Information

How to Cite

Gouveia, L. and Hall, L. (2002), Multistars and directed flow formulations. Networks, 40: 188–201. doi: 10.1002/net.10050

Author Information

  1. 1

    DEIO–CIO, Faculdade de Ciências da Universidade de Lisboa, Bloco C/2 - Campo Grande, CIDADE UNIVERSITARIA, 1749-016 Lisbon, Portugal

  2. 2

    Department of Mathematical Sciences, Johns Hopkins University, Baltimore, Maryland 21218

*DEIO–CIO, Faculdade de Ciências da Universidade de Lisboa, Bloco C/2 - Campo Grande, CIDADE UNIVERSITARIA, 1749-016 Lisbon, Portugal

Publication History

  1. Issue published online: 11 OCT 2002
  2. Article first published online: 11 OCT 2002
  3. Manuscript Accepted:
  4. Manuscript Received:

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Keywords:

  • capacitated trees;
  • integer programming;
  • polyhedral projection;
  • separation algorithms

Abstract

The Capacitated Minimum Spanning Tree Problem seeks a least-cost spanning tree subject to a bound imposed on the number of nodes in each subtree pending from a given root node. Araque et al. (Technical Report SOR-90-12, Princeton University, 1990) introduced several classes of facet-defining inequalities for the undirected version of the problem, most of which have straightforward analogs to the directed version and are also facet-defining in that case (see Zhang, Master's thesis, 1993). The multistar constraints are one such class. Gouveia [Telecommun Syst 1 (1993), 51–56] showed that a directed flow formulation gives a polynomial representation of the class of directed multistar constraints. This equivalence shows how to obtain a polynomial-time separation algorithm for this class of inequalities. In this paper, we show that the previous equivalence result implies that we can also separate in polynomial time the exponential-sized class of undirected multistar constraints. We also show that “using a directed model” plays a key role in obtaining a polynomial-time separation algorithm for this class of inequalities, that is, using a directed flow model seems to be crucial for obtaining a polynomial-time separation algorithm for the class of undirected multistar constraints. © 2002 Wiley Periodicals, Inc.

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