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Polyhedra and optimization in connection with a weak majorization ordering

  • Geir Dahl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 920)

Abstract

We introduce the concept of weak k-majorization extending the classical notion of weak sub-majorization. For integers k and n with kn a vector x∈ℝn is weakly k-majorized by a vector q∈ℝk if the sum of the r largest components of x does not exceed the sum of the r largest components of q, for r=1,⋯, k. For a given q the set of vectors weakly k-majorized by q defines a polyhedron P(q; k), and we determine all its vertices. We also determine the vertices and a complete and nonredundant linear description of the integer hull of P(q; k). The results are used to give simple and efficient (polynomial time) algorithms for associated linear and integer linear programming problems.

Keywords

Majorization polyhedral combinatorics 

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References

  1. 1.
    Dahl, G., Pseudo experiments and majorization. Thesis, University of Oslo, 1984.Google Scholar
  2. 2.
    Dahl, G., Polyhedra and Optimization in Connection with a Weak Majorization Ordering, Preprint 10, Dec. 1994, University of Oslo, Department of Informatics.Google Scholar
  3. 3.
    Edmonds, J., Matroids and the greedy algorithm, Mathematical Programming 1 (1971) 127–136.Google Scholar
  4. 4.
    Folkman, J.H. and Fulkerson, D.R., “Edge colorings in bipartite graphs,” In Combinatorial mathematics and its applications (R.C. Bose and T.A. Dowling, eds.), Chapter 31, pp. 561–577, Univ. of North Carolina, Chapel Hill, 1969.Google Scholar
  5. 5.
    Fujishige, S. Submodular functions and optimization, Annals of discrete mathematics, Vol. 47. Amsterdam, North Holland, 1991.Google Scholar
  6. 6.
    Grötschel, M., Lovász, L. and Schrijver, A., Geometric algorithms and combinatorial optimization, Springer, 1988.Google Scholar
  7. 7.
    Hardy, G.H, Littlewood, J.E. and Polya, G., Inequalities (2. ed.), Cambridge Mathematical Library, Cambridge University Press, 1988.Google Scholar
  8. 8.
    Marshall, A.W. and Olkin, I., Inequalities: Theory of Majorization and Its Applications, New York, Academic Press, 1979.Google Scholar
  9. 9.
    Pulleyblank, W.R., “Polyhedral combinatorics,” in Handbooks in Operations Research, Vol. 1, Optimization, ed. Nemhauser et al., North-Holland, 1989.Google Scholar
  10. 10.
    Schrijver, A., Theory of linear and integer programming, Wiley, 1986.Google Scholar
  11. 11.
    Torgersen, E., Comparison of Statistical Experiments, Cambridge University Press, Cambridge, 1992.Google Scholar
  12. 12.
    Ziegler, G., Lectures on polytopes, Springer, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Geir Dahl
    • 1
  1. 1.Institute of InformaticsUniversity of OsloBlindernNorway

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