Advertisement

Perfect Zero-One Matrices — II

Conference paper
Part of the Proceedings in Operations Research book series (ORP, volume 1973)

Abstract

We consider combinatorial programming problems of the form (LP): max {cx|Ax ≤e, xj=0 or 1, vj}, where A is a mxn matrix of zeroes and ones, e is a column vector of m ones and c is an arbitrary (non-negative) vector of reals. Applications of this general problem include crew scheduling, political districting and others. In this paper we first summarize (without proofs) the results of a companion paper that completely characterize matrices A for Which.(IP) can be solved as an ordinary linear programming problem, i.e. where the relaxed linear program (LP): max{cx|Ax ≤e, xj≥0,vj} produces an integral solution no matter what linear form cx is maximized. (Zero-one matrices with this property are termed “perfect”). Some additional concepts and results are stated. It is shown that every totally unimodular zero-one matrix as well as every “balanced” zero-one matrix is perfect. Finally, in the concluding remarks, a reformulation of the strong perfect graph due to C. BERGE is given and some recent trends in zezo-one programming are delineated.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. (1).
    Arabeyre, J.P.,J. Feamley, F. Steiger and W. Teather: “The Airline Crew Scheduling Problem: A Survey”, Transportation Science, 3(1969), pp. 140–163.CrossRefGoogle Scholar
  2. (2).
    Balas, E.: “Facets of the Kiapsack Polytope”, Carnegie-Mellon University, Management Sciences Research Report No. 323, September 1973.Google Scholar
  3. (3).
    Balas, E., and M.W. Padberg: “On the Set Covering Problem”, Operations Research, 20(1972), pp. 1152–1161.CrossRefGoogle Scholar
  4. (4).
    Balas, E., and M.W. Padberg: “On the Set Covering Prcfolem II: An Algorithm”, Management Sciences Research Report No. 278, GSIA, Carnegie-Mellon University, Pittsburgh, Pa. (presented at the Joint National Meeting of ORSA, TIMS, AIEE, at Atlantic City, November 8–10, 1972).Google Scholar
  5. (5).
    Berge, C.: Graphes et Hypergraphes, Diaiod, Paris 1970.Google Scholar
  6. (6).
    Berge, C.: Introduction à la Theorie des Hypergraphes, Lectures notes, Université de Montreal, Summer 1971.Google Scholar
  7. (7).
    Berge, C.: “Balanced Matrices”, Mathematical Programming, 2(1972), pp. 19–31.CrossRefGoogle Scholar
  8. (8).
    Chvátal, V.: “Edmonds Polyhedra and a Hierarchy of Combinatorial Problems” Discrete Mathematics, 4(1973), pp. 305–337.CrossRefGoogle Scholar
  9. (9).
    Edmonds, J.: “Maximum Matching and a Polyhedron with 0,1 Vertices”, Journal of Research of the National Bureau of Standards, 69B(1965), pp. 125–130.Google Scholar
  10. (10).
    Fulkerson, D.R.: “Blocking and Anti-blocking Pairs of Polyhedra”, Mathematical Programming, 1(1971), pp. 168–194.CrossRefGoogle Scholar
  11. (11).
    Garfinkel, R. and G. Nemhauser: Integer Programing, John Wiley & Sons, 1972.Google Scholar
  12. (12).
    Glover, F.: “Uhi t-Cœf f icient Inequalities for Zero-One Prograitming”, University of Colorado, Management Sciences Report Series, No. 73–7, July 1973.Google Scholar
  13. (13).
    Hammer, P.L., E.L. Johnson and U.N. Peled: “Regular 0–1 Programs”, University of Waterloo, Ccmbinatorics and Optimization Research Report, CORR No. 73–18, Septenfoer 1973.Google Scholar
  14. (14).
    Hammer, P.L., E.L. Johnson and U.N. Peled: “Facets of Regular 0–1 Polytopes”, University of Waterloo, Combinatorics and Optimization Research Report, CORR No. 73–19, October 1973.Google Scholar
  15. (15).
    Höffmann, A. J.: “On Goirbinatorical Problems and Linear Inequalities”, IBM Watson Research Center, Yorktown Heights, N.Y., paper presented at the 8th International Syirposium on Mathematical Programming, Stanford, August 1973.Google Scholar
  16. (16).
    Lovász, L.: “Normal Hypergraphs and the Perfect Graph Conjecture”, Discrete Mathematics, 2(1972), pp. 253–268.CrossRefGoogle Scholar
  17. (17).
    Padberg, M.W.: Essays in Integer Programming, Ph.D. Thesis, Carnegie-Mellon University, Pittsburg, Pa. (April 1971), unpublished.Google Scholar
  18. (18).
    Padberg, M.W.: “On the Facial Structure of Set Packing Polyhedra”, IIM-Report No. I/72–13, International Institute of Management, Berlin, forthcoming in Mathematical Programming.Google Scholar
  19. (19).
    Padberg, M.W.: “Perfect Zero-One Matrices”, IM-Report No. I/73–8, International Institute of Management, Berlin, forthcoming in Mathematical Programming.Google Scholar
  20. (20).
    Padberg, M.W. “A Note on Zero-One Prograitming”, University of Waterloo, Combinatorics and Optimization Research Report, CORR No. 73–5, March 1973.Google Scholar
  21. (21).
    Trotter, L.: Solution Characteristics and Algorithms for the Vertex Packing Problem, Technical Report No. 168, Ph.D. Thesis, Cornell University, January 1973.Google Scholar
  22. (22).
    Wolsey, L.A.: “Faces for Linear Inequalities in 0–1 Variables”, Université Catholique de Louvain, CORE Discussion paper No. 7338, November 1973.Google Scholar

Copyright information

© Physica-Verlag, Rudolf Liebing KG, Würzburg 1974

Authors and Affiliations

  1. 1.BerlinDeutschland

Personalised recommendations