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International Conference on Integer Programming and Combinatorial Optimization

IPCO 2002: Integer Programming and Combinatorial Optimization pp 145–160Cite as

An Exponential Lower Bound on the Length of Some Classes of Branch-and-Cut Proofs

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2337))

Abstract

Branch-and-cut methods are among the more successful techniques for solving integer programming problems. They can also be used to prove that all solutions of an integer program satisfy a given linear inequality. We examine the complexity of branch-and-cut proofs in the context of 0-1 integer programs. We prove an exponential lower bound on the length of branch-and-cut proofs in the case where branching is on the variables and the cutting planes used are lift-and-project cuts (also called simple disjunctive cuts by some authors), Gomory-Chvátal cuts, and cuts arising from the N0 matrix-cut operator of Lovász and Schrijver. A consequence of the lower-bound result in this paper is that branch-and-cut methods of the type described above have exponential running time in the worst case.

This work was supported by ONR Grant N00014-98-1-0014.

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Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08544, USA

    Sanjeeb Dash

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  1. Sanjeeb Dash
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Editors and Affiliations

  1. Program in Applied and Computational Mathematics, University of Princeton, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    William J. Cook

  2. Sloan School of Management, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Andreas S. Schulz

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Dash, S. (2002). An Exponential Lower Bound on the Length of Some Classes of Branch-and-Cut Proofs. In: Cook, W.J., Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization. IPCO 2002. Lecture Notes in Computer Science, vol 2337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47867-1_11

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  • DOI: https://doi.org/10.1007/3-540-47867-1_11

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  • Print ISBN: 978-3-540-43676-8

  • Online ISBN: 978-3-540-47867-6

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