Skip to main content
SpringerLink
Log in

Optimizing over the first Chvátal closure

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

How difficult is, in practice, to optimize exactly over the first Chvátal closure of a generic ILP? Which fraction of the integrality gap can be closed this way, e.g., for some hard problems in the MIPLIB library? Can the first-closure optimization be useful as a research (off-line) tool to guess the structure of some relevant classes of inequalities, when a specific combinatorial problem is addressed? In this paper we give answers to the above questions, based on an extensive computational analysis. Our approach is to model the rank-1 Chvátal-Gomory separation problem, which is known to be NP-hard, through a MIP model, which is then solved through a general-purpose MIP solver. As far as we know, this approach was never implemented and evaluated computationally by previous authors, though it gives a very useful separation tool for general ILP problems. We report the optimal value over the first Chvátal closure for a set of ILP problems from MIPLIB 3.0 and 2003. We also report, for the first time, the optimal solution of a very hard instance from MIPLIB 2003, namely nsrand-ipx, obtained by using our cut separation procedure to preprocess the original ILP model. Finally, we describe a new class of ATSP facets found with the help of our separation procedure.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achterberg, T., Koch, T., Martin, A.: The mixed integer programming library: MIPLIB 2003, http://www.miplib.zib.de (2003)

  2. Balas E. (1989). The asymmetric assignment problem and some new facets of the traveling salesman polytope on a directed graph. SIAM J. Discrete Math. 2: 425–451

    Article  MATH  MathSciNet  Google Scholar 

  3. Balas E. and Fischetti M. (1993). A lifting procedure for the Asymmetric Traveling Salesman Polytope and a large new class of facets. Math. Program. 58: 325–352

    Article  MathSciNet  Google Scholar 

  4. Balas, E., Saxena, A.: Optimizing over the split closure, Technical Report 2006-E5, Tepper School of Business, CMU (2005)

  5. Bonami, P., Cornuejols, G., Dash, S., Fischetti, M., Lodi, A.: Projected Chvatal-Gomory cuts for mixed integer linear programs. Technical Report 2006-E4, Tepper School of Business, CMU, to appear Math. Program. (in press)

  6. Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: MIPLIB 3.0, http://www.caam. rice.edu/~bixby/miplib/miplib.html

  7. Caprara A. and Letchford A.N. (2003). On the separation of split cuts and related inequalities. Math. Program. 94: 279–294

    Article  MATH  MathSciNet  Google Scholar 

  8. Christof, T., Löbel, A.: PORTA - POlyhedron representation transformation algorithm, http://www.zib.de/Optimization/Software/Porta/

  9. Chvátal V. (1973). Edmonds polytopes and a hierarchy of combinatorial problems.. Discrete Math. 4: 305–337

    Article  MATH  MathSciNet  Google Scholar 

  10. Dash, S., Günlük, O., Lodi, A.: On the MIR closure of polyhedra. IBM, T.J. Watson Research, Working paper, (2005)

  11. Edmonds J. (1965). Maximum matching and a polyhedron with {0,1}-vertices. J. Res. Nat. Bur. Stand. B 69: 125–130

    MATH  MathSciNet  Google Scholar 

  12. Edmonds J. and Johnson H.L. (1970). Matching: a well-solved class of integer linear programs. In: Guy, R.K. (eds) Combinatorial Structures and their Applications., pp 89–92. Gordon and Breach, New York

    Google Scholar 

  13. Eisenbrand F. (1999). On the membership problem for the elementary closure of a polyhedron. Combinatorica 19: 297–300

    Article  MATH  MathSciNet  Google Scholar 

  14. Fischetti M. and Lodi A. (2005). Optimizing over the first Chvátal closure. In: Jünger, M. and Kaibel, V. (eds) Integer programming and combinatorial optimization—IPCO 2005, LNCS 3509, pp 12–22. Springer, Berlin Heidelberg New York

    Google Scholar 

  15. Gomory R.E. (1958). Outline of an algorithm for integer solutions to linear programs. Bull. AMS 64: 275–278

    MATH  MathSciNet  Google Scholar 

  16. Gomory R.E. (1963). An algorithm for integer solutions to linear programs. In: Graves, R.L. and Wolfe, P. (eds) Recent Advances in Mathematical Programming, pp 275. McGraw-Hill, New York

    Google Scholar 

  17. ILOG Cplex 9.1: User’s manual and reference manual, ILOG, S.A. http://www.ilog.com/(2005)

  18. Letchford A.N., Reinelt G. and Theis D.O. (2004). A faster exact separation algorithm for blossom inequalities. In: Bienstock, D. and Nemhauser, G. (eds) Integer programming and combinatorial optimization—IPCO 2004, LNCS 3064, pp 196–205. Springer, Berlin Heidelberg New York

    Google Scholar 

  19. Nemhauser G.L. and Wolsey L.A. (1988). Integer and Combinatorial Optimization. Wiley, New York

    MATH  Google Scholar 

  20. Padberg M.W. and Rao M.R. (1982). Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7: 67–80

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DEI, University of Padova, via Gradenigo 6A, 35131, Padova, Italy

    Matteo Fischetti

  2. DEIS, University of Bologna, viale Risorgimento 2, 40136, Bologna, Italy

    Andrea Lodi

Authors
  1. Matteo Fischetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Lodi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matteo Fischetti.

Additional information

Work partially supported by MIUR, Italy, and by the EU projects ADONET (contract n. MRTN-CT-2003-504438) and ARRIVAL (contract no. FP6-021235-2).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fischetti, M., Lodi, A. Optimizing over the first Chvátal closure. Math. Program. 110, 3–20 (2007). https://doi.org/10.1007/s10107-006-0054-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-006-0054-8

Keywords

Search

Navigation