Optimizing over the first Chvátal closure
 Matteo Fischetti,
 Andrea Lodi
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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 firstclosure optimization be useful as a research (offline) 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 rank1 ChvátalGomory separation problem, which is known to be NPhard, through a MIP model, which is then solved through a generalpurpose 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 nsrandipx, 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.
 Achterberg, T., Koch, T., Martin, A.: The mixed integer programming library: MIPLIB 2003, http://www.miplib.zib.de (2003)
 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 CrossRef
 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
 Balas, E., Saxena, A.: Optimizing over the split closure, Technical Report 2006E5, Tepper School of Business, CMU (2005)
 Bonami, P., Cornuejols, G., Dash, S., Fischetti, M., Lodi, A.: Projected ChvatalGomory cuts for mixed integer linear programs. Technical Report 2006E4, Tepper School of Business, CMU, to appear Math. Program. (in press)
 Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: MIPLIB 3.0, http://www.caam. rice.edu/~bixby/miplib/miplib.html
 Caprara A. and Letchford A.N. (2003). On the separation of split cuts and related inequalities. Math. Program. 94: 279–294 CrossRef
 Christof, T., Löbel, A.: PORTA  POlyhedron representation transformation algorithm, http://www.zib.de/Optimization/Software/Porta/
 Chvátal V. (1973). Edmonds polytopes and a hierarchy of combinatorial problems.. Discrete Math. 4: 305–337 CrossRef
 Dash, S., Günlük, O., Lodi, A.: On the MIR closure of polyhedra. IBM, T.J. Watson Research, Working paper, (2005)
 Edmonds J. (1965). Maximum matching and a polyhedron with {0,1}vertices. J. Res. Nat. Bur. Stand. B 69: 125–130
 Edmonds J. and Johnson H.L. (1970). Matching: a wellsolved class of integer linear programs. In: Guy, R.K. (eds) Combinatorial Structures and their Applications., pp 89–92. Gordon and Breach, New York
 Eisenbrand F. (1999). On the membership problem for the elementary closure of a polyhedron. Combinatorica 19: 297–300 CrossRef
 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
 Gomory R.E. (1958). Outline of an algorithm for integer solutions to linear programs. Bull. AMS 64: 275–278
 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. McGrawHill, New York
 ILOG Cplex 9.1: User’s manual and reference manual, ILOG, S.A. http://www.ilog.com/(2005)
 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
 Nemhauser G.L. and Wolsey L.A. (1988). Integer and Combinatorial Optimization. Wiley, New York
 Padberg M.W. and Rao M.R. (1982). Odd minimum cutsets and bmatchings. Math. Oper. Res. 7: 67–80 CrossRef
 Title
 Optimizing over the first Chvátal closure
 Journal

Mathematical Programming
Volume 110, Issue 1 , pp 320
 Cover Date
 20070601
 DOI
 10.1007/s1010700600548
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Integer programs
 Separation problems
 Chvátal–Gomory cuts
 Computational analysis
 Industry Sectors
 Authors

 Matteo Fischetti ^{(1)}
 Andrea Lodi ^{(2)}
 Author Affiliations

 1. DEI, University of Padova, via Gradenigo 6A, 35131, Padova, Italy
 2. DEIS, University of Bologna, viale Risorgimento 2, 40136, Bologna, Italy