Advertisement

Mathematical Programming

, Volume 55, Issue 1–3, pp 129–168 | Cite as

Structural properties and decomposition of linear balanced matrices

  • Michele Conforti
  • M. R. Rao
Article

Abstract

Claude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all ones.A(0, 1) matrixA is balanced ifA does not contain a square submatrix of odd order with two ones per row and column.

The contraction of a rowi of a matrix consists of the removal of rowi and all the columns that have a 1 in the entry corresponding to rowi.

In this paper we show that if a linear balanced matrixA does not belong to a subclass of totally unimodular matrices, thenA orAT contains a rowi such that the submatrix obtained by contracting rowi has a block-diagonal structure.

Key words

Polyhedral combinatorics integrality of polytopes decomposition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R.F. Anstee and M. Farber, “Characterizations of totally balanced matrices,”Journal of Algorithms 5 (1984) 215–230.Google Scholar
  2. C. Berge, “Balanced matrices,”Mathematical Programming 2 (1972) 19–31.Google Scholar
  3. C. Berge, “Balanced matrices and the property G,”Mathematical Programming Study 12 (1980) 163–175.Google Scholar
  4. C. Berge, “Minimax theorems for normal and balanced hypergraphs: A survey,”Annals of Discrete Mathematics 21 (1984) 3–21.Google Scholar
  5. C. Berge and M. Las Vergnas, “Sur un theoreme du type Konig pour les hypergraphes,”Annals of New York Academy of Sciences 175 (1970) 32–40.Google Scholar
  6. M. Conforti and M.R. Rao, “Structural properties and decomposition of restricted and strongly unimodular matrices,”Mathematical Programming 38 (1987a) 17–27.Google Scholar
  7. M. Conforti and M.R. Rao, “Odd cycles and 0, 1 matrices,” to appear in:Mathematical Programming (series B) (1987b).Google Scholar
  8. M. Conforti and M.R. Rao, “Testing balancedness and perfection of linear graphs,” preprint, New York University (New York, 1988).Google Scholar
  9. J. Fonlupt and A. Zemirline, “A polynomial recognition algorithm for perfect (K 4e)-free perfect graphs,” Research Report, University of Grenoble (Grenoble, 1987).Google Scholar
  10. A. Frank, “On a class of balanced hypergraphs,” mimeo, Research Institute for Telecommunications (Budapest, 1979).Google Scholar
  11. D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, “On balanced matrices,”Mathematical Programming Study 1 (1974), 120–132.Google Scholar
  12. A. Hoffman, A. Kolen and M. Sakarovitch, “Totally balanced and greedy matrices,”SIAM Journal of Algebraic and Discrete Methods 6 (1985) 721–730.Google Scholar
  13. M. Padberg, “Characterizations of totally unimodular, balanced and perfect matrices,”Combinatorial Programming, Methods and Applications (1975) 279–289.Google Scholar
  14. P.D. Seymour, “Decomposition of regular matroids,”Journal of Combinatorial Theory B-28 (1980) 305–359.Google Scholar
  15. M. Yannakakis, “On a class of totally unimodular matrices,”Mathematics of Operations Research 10 (2) (1985) 280–304.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • Michele Conforti
    • 1
  • M. R. Rao
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PadoraPadovaItaly
  2. 2.Graduate School of Business AdministrationNew York UniversityUSA

Personalised recommendations