Mathematical Programming

, Volume 110, Issue 1, pp 3–20 | Cite as

Optimizing over the first Chvátal closure

  • Matteo FischettiEmail author
  • Andrea Lodi


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.


Integer programs Separation problems Chvátal–Gomory cuts Computational analysis 


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  1. 1.
    Achterberg, T., Koch, T., Martin, A.: The mixed integer programming library: MIPLIB 2003, (2003)Google Scholar
  2. 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 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 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 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Balas, E., Saxena, A.: Optimizing over the split closure, Technical Report 2006-E5, Tepper School of Business, CMU (2005)Google Scholar
  5. 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)Google Scholar
  6. 6.
    Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: MIPLIB 3.0, http://www.caam. Scholar
  7. 7.
    Caprara A. and Letchford A.N. (2003). On the separation of split cuts and related inequalities. Math. Program. 94: 279–294 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Christof, T., Löbel, A.: PORTA - POlyhedron representation transformation algorithm, Scholar
  9. 9.
    Chvátal V. (1973). Edmonds polytopes and a hierarchy of combinatorial problems.. Discrete Math. 4: 305–337 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dash, S., Günlük, O., Lodi, A.: On the MIR closure of polyhedra. IBM, T.J. Watson Research, Working paper, (2005)Google Scholar
  11. 11.
    Edmonds J. (1965). Maximum matching and a polyhedron with {0,1}-vertices. J. Res. Nat. Bur. Stand. B 69: 125–130 zbMATHMathSciNetGoogle Scholar
  12. 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. 13.
    Eisenbrand F. (1999). On the membership problem for the elementary closure of a polyhedron. Combinatorica 19: 297–300 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 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. 15.
    Gomory R.E. (1958). Outline of an algorithm for integer solutions to linear programs. Bull. AMS 64: 275–278 zbMATHMathSciNetGoogle Scholar
  16. 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. 17.
    ILOG Cplex 9.1: User’s manual and reference manual, ILOG, S.A. Scholar
  18. 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. 19.
    Nemhauser G.L. and Wolsey L.A. (1988). Integer and Combinatorial Optimization. Wiley, New York zbMATHGoogle Scholar
  20. 20.
    Padberg M.W. and Rao M.R. (1982). Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7: 67–80 zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.DEIUniversity of PadovaPadovaItaly
  2. 2.DEISUniversity of BolognaBolognaItaly

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