Optimizing over the first Chvátal closure FULL LENGTH PAPER First Online: 15 December 2006 Received: 28 July 2005 Accepted: 25 June 2006 DOI :
10.1007/s10107-006-0054-8

Cite this article as: Fischetti, M. & Lodi, A. Math. Program. (2007) 110: 3. doi:10.1007/s10107-006-0054-8
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.

Keywords Integer programs Separation problems Chvátal–Gomory cuts Computational analysis 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).

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

MATH CrossRef 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

CrossRef 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

MATH CrossRef 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

MATH CrossRef 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

MATH CrossRef 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

MATH MathSciNet CrossRef Google Scholar Authors and Affiliations 1. DEI University of Padova Padova Italy 2. DEIS University of Bologna Bologna Italy