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Some insight into characterizations of minimally nonideal matrices

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  • Gabriela Argiroffo
  • Silvia Bianchi
  • Graciela Nasini

Abstract

Lehman (Polyhedral combinatorics 1 of DIMACS series in discrete math. and theoretical computer science, pp 101–105, 1990) described some conditions regular minimally nonideal (mni) matrices must satisfy. Although, there are few results on sufficient conditions for mni matrices. In most of these results, the covering polyhedron must have a unique fractional extreme point. This condition corresponds to ask the matrix to be the blocker of a near-ideal matrix, defined by the authors in a previous work (2006). In this paper we prove that, having the blocker of a near-ideal matrix, only a few very easy conditions have to be checked in order to decide if the matrix is regular mni. In doing so, we define the class of quasi mni matrices, containing regular mni matrices, and we find a generalization on the number of integer extreme points adjacent to the fractional extreme point in the covering polyhedron. We also give a relationship between the covering and stability number of regular mni matrices which allows to prove when a regular mni matrix can be a proper minor of a quasi mni. Copyright Springer-Verlag 2008

Suggested Citation

  • Gabriela Argiroffo & Silvia Bianchi & Graciela Nasini, 2008. "Some insight into characterizations of minimally nonideal matrices," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(2), pages 245-256, April.
    • Handle: RePEc:spr:mathme:v:67:y:2008:i:2:p:245-256
    DOI: 10.1007/s00186-007-0176-7
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    References listed on IDEAS

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    1. Christine Lütolf & François Margot, 1998. "A catalog of minimally nonideal matrices," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 47(2), pages 221-241, June.
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