Skip to main content
SpringerLink
Log in

Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems

  • Chapter
  • First Online:
Approaches to Integer Programming

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 2))

Abstract

Two minimum cardinality set covering problems of similar structure are presented as difficult test problems for evaluating the computational efficiency of integer programming and set covering algorithms. The smaller problem has 117 constraints and 27 variables, and the larger one, constructed by H.J. Ryser, has 330 constraints and 45 variables. The constraint matrices of the two set covering problems are incidence matrices of Steiner triple systems. An optimal solution to the problem that we were able to solve (the smaller one) gives some new information on the 1-widths of members of this class of (0,1)-matrices.

This research was supported, in part, by National Science Foundation Grants GK-32282X and GP-32316X and Office of Naval Research Contract No. N00014-67-A-0077-0028 to Cornell University. A substantial amount of the computer time was provided by the Mathematics Research Center of the University of Wisconsin under U.S. Army Contract No. DA-31-124-ARO-D-462.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Bellmore and H.D. Ratliff, “Set covering and involutory bases”, Management Science 18 (1971) 194–206.

    Article  MATH  MathSciNet  Google Scholar 

  2. G.H. Bradley and P.N. Wahi, “An algorithm for integer linear programming: A combined algebraic and enumeration approach”, Operations Research 21 (1973) 45–60.

    Article  MATH  MathSciNet  Google Scholar 

  3. D.R. Fulkerson and H.J. Ryser, “Widths and heights of (0,1)-matrices”, Canadian Journal of Mathematics 13 (1961) 239–255.

    MATH  MathSciNet  Google Scholar 

  4. D.R. Fulkerson and H.J. Ryser, “Multiplicities and minimal widths for (0,1)-matrices”, Canadian Journal of Mathematics 14 (1962) 498–508.

    MATH  MathSciNet  Google Scholar 

  5. D.R. Fulkerson and H.J. Ryser, “Width sequences for special classes of (0,1)-matrices”, Canadian Journal of Mathematics 15 (1963) 371–396.

    MATH  MathSciNet  Google Scholar 

  6. A.M. Geoffrion, “An improved implicit enumeration approach for integer programming”, Operations Research 17 (1969) 437–454.

    Article  MATH  Google Scholar 

  7. R.E. Gomory, “An algorithm for integer solutions to linear programs”, in: R.L. Graves and P. Wolfe, eds., Recent advances in mathematical programming (McGraw-Hill, New York, 1963) pp. 269–302.

    Google Scholar 

  8. R.E. Gomory, “On the relation between integer and non-integer solutions to linear programs”, Proceedings of the National Academy of Sciences 53 (1965) 260–265.

    Article  MATH  MathSciNet  Google Scholar 

  9. G.A. Gorry and J.F. Shapiro, “An adaptive group theoretic algorithm for integer programming problems”, Management Science 17 (1971) 285–306.

    Article  MATH  MathSciNet  Google Scholar 

  10. G.A. Gorry, W.D. Northrup and J.F. Shapiro, “Computational experience with a group theoretic integer programming algorithm”, Mathematical Programming 4 (1973) 171–192.

    Article  MATH  MathSciNet  Google Scholar 

  11. R.W. House, L.D. Nelson and J. Rado, “Computer studies of a certain class of linear integer problems”, in: A. Lavi and T. Vogl, eds., Recent advances in optimization techniques (Wiley, New York, 1966) pp. 241–280.

    Google Scholar 

  12. R.G. Jeroslow, “Trivial integer programs unsolvable by branch-and-bound”, Mathematical Programming 6 (1974) 105–109.

    Article  MATH  Google Scholar 

  13. C.E. Lemke, H.M. Salkin and K. Spielberg, “Set covering by single branch enumeration with linear programming subproblems”, Operations Research 19 (1971) 998–1022.

    Article  MATH  MathSciNet  Google Scholar 

  14. H.J. Ryser, Combinatorial mathematics, Carus Mathematical Monograph No. 14 (Wiley, New York, 1963).

    MATH  Google Scholar 

  15. J.F. Shapiro, Private communication (September 1973).

    Google Scholar 

  16. H. Thiriez, “The set covering problem: A group theoretic approach”, Revue Francaise d’Informatique et de Recherche Operationelle V3 (1971) 83–104.

    Google Scholar 

  17. L.E. Trotter, Jr. and C.M. Shetty, “An algorithm for the bounded variable integer programming problem”, Journal of the Association for Computing Machinery 21 (1974) 505–513.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. L. E. Trotter

    Present address: Yale University, New Haven, Conn., USA

Authors and Affiliations

  1. Cornell University, Ithaca, N.Y., USA

    D. R. Fulkerson & G. L. Nemhauser

  2. Mathematics Research Center, University of Wisconsin, Wisconsin, USA

    L. E. Trotter

Authors
  1. D. R. Fulkerson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. L. Nemhauser
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. E. Trotter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

M. L. Balinski

Rights and permissions

Reprints and permissions

Copyright information

© 1974 The Mathematical Programming Society

About this chapter

Cite this chapter

Fulkerson, D.R., Nemhauser, G.L., Trotter, L.E. (1974). Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. In: Balinski, M.L. (eds) Approaches to Integer Programming. Mathematical Programming Studies, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120689

Download citation

  • DOI: https://doi.org/10.1007/BFb0120689

  • Received:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00739-2

  • Online ISBN: 978-3-642-00740-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation