Skip to main content
Springer Nature Link
Log in

The Packing Property

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1412))

  • 1175 Accesses

Abstract

A clutter (V,E) packs if the smallest number of vertices needed to intersect all the edges (i.e. a transversal) is equal to the maxi- mum number of pairwise disjoint edges (i.e. a matching). This terminol- ogy is due to Seymour 1977. A clutter is minimally nonpacking if it does not pack but all its minors pack. A 0,1 matrix is minimally nonpacking if it is the edge-vertex incidence matrix of a minimally nonpacking clutter. Minimally nonpacking matrices can be viewed as the counterpart for the set covering problem of minimally imperfect matrices for the set packing problem. This paper proves several properties of minimally nonpacking clutters and matrices.

This work was supported in part by NSF grants DMI-9424348, DMS-9509581, ONR grant N00014-9710196, a William Larimer Mellon Fellowship, and the Swiss National Research Fund (FNRS).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. C. Berge. Färbung von Graphen deren sämtliche bzw. deren ungerade Kreize starr sind (Zusammenfassung), Wisenschaftliche Zeitschritch, Martin Luther Universität Halle-Wittenberg, Mathematisch-Naturwissenschaftliche Reihe, 114–115, 1960.

    Google Scholar 

  2. W. G. Bridges and H. J. Ryser. Combinatorial designs and related systems. J. Algebra, 13:432–446, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Conforti and G. Cornuéjols. Clutters that pack and the max-flow min-cut property: A conjecture. In W. R. Pulleyblank and F. B. Shepherd, editors, The Fourth Bellairs Workshop on Combinatorial Optimization, 1993.

    Google Scholar 

  4. M. Conforti, G. Cornuéjols, A. Kapoor, and K. Vušković. Balanced matrices. In J. R. Birge and K. G Murty, editors, Math. Programming, State of the Art 1994, pages 1–33, 1994.

    Google Scholar 

  5. G. Cornuéjols and B. Novick. Ideal 0, 1 matrices. J. Comb. Theory Ser. B, 60:145–157, 1994.

    Article  MATH  Google Scholar 

  6. G. Ding. Clutters with τ 2 = 2τ. Discrete Math., 115:141–152, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Math., 1:185–204, 1977.

    Article  MathSciNet  Google Scholar 

  8. B. Guenin. Packing and covering problems. Thesis proposal, GSIA, Carnegie Mellon University, 1997.

    Google Scholar 

  9. A. Lehman. On the width-length inequality. Mathematical Programming, 17:403–417, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Lehman. On the width-length inequality and degenerate projective planes. In W. Cook and P.D. Seymour, editors, Polyhedral Combinatorics, DIMACS Series in Discrete Math. and Theoretical Computer Science, Vol. 1, pages 101–105, 1990.

    Google Scholar 

  11. L. Lovász. Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2:253–267, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. Luetolf and F. Margot. A catalog of minimally nonideal matrices. Mathematical Methods of Operations Research, 1998. To appear.

    Google Scholar 

  13. M. W. Padberg. Lehman’s forbidden minor characterization of ideal 0–1 matrices. Discrete Math., 111:409–420, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Schrijver. A counterexample to a conjecture of Edmonds and Giles. Discrete Math., 32:213–214, 1980.

    MATH  MathSciNet  Google Scholar 

  15. A. Schrijver. Theory of Linear and Integer Programming, Wiley, 1986.

    Google Scholar 

  16. P. D. Seymour. The matroids with the max-flow min-cut property. J. Comb. Theory Ser. B, 23:189–222, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. D. Seymour. On Lehman’s width-length characterization. In W. Cook and P. D. Seymour, editors, Polyhedral Combinatorics, DIMACS Series in Discrete Math. and Theoretical Computer Science, Vol. 1, pages 107–117, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols & Bertrand Guenin

  2. Department of Mathematical Science, Michigan Technical University, Houghton, MI, 49931, USA

    François Margot

Authors
  1. Gérard Cornuéjols
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bertrand Guenin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. François Margot
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, 6020 Annapolis, Houston, TX, 77005, USA

    Robert E. Bixby

  2. PROS Strategic Solutions, 3223 Smith Street, Houston, TX, 77006, USA

    E. Andrew Boyd

  3. Department of Industrial, Engineering, Texas A&M University, 1000 Country Place Dr. Apt. 69, Houston, TX, 77079, USA

    Roger Z. Ríos-Mercado

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cornuéjols, G., Guenin, B., Margot, F. (1998). The Packing Property. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds) Integer Programming and Combinatorial Optimization. IPCO 1998. Lecture Notes in Computer Science, vol 1412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69346-7_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-69346-7_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64590-0

  • Online ISBN: 978-3-540-69346-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics