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Deciding Emptiness of the Gomory-Chvátal Closure is NP-Complete, Even for a Rational Polyhedron Containing No Integer Point

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Integer Programming and Combinatorial Optimization (IPCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9682))

Abstract

Gomory-Chvátal cuts are prominent in integer programming. The Gomory-Chvátal closure of a polyhedron is the intersection of all half spaces defined by its Gomory-Chvátal cuts. In this paper, we show that it is \(\mathcal {NP}\)-complete to decide whether the Gomory-Chvátal closure of a rational polyhedron is empty, even when this polyhedron contains no integer point. This implies that the problem of deciding whether the Gomory-Chvátal closure of a rational polyhedron P is identical to the integer hull of P is \(\mathcal {NP}\)-hard. Similar results are also proved for the \(\{-1,0,1\}\)-cuts and \(\{0,1\}\)-cuts, two special types of Gomory-Chvátal cuts with coefficients restricted in \(\{-1, 0, 1\}\) and \(\{0, 1\}\), respectively.

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References

  1. Borosh, I., Treybig, L.B.: Bounds on positive integral solutions to linear Diophantine equations. Proc. Am. Math. Soc. 55, 299–304 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Campelo, M., Cornuéjols, G.: Stable sets, corner polyhedra and the Chvátal closure. Oper. Res. Lett. 37, 375–378 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer, Switzerland (2014)

    Book  MATH  Google Scholar 

  5. Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stan. B 69, 125–130 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  8. Gerards, A.M.H., Schrijver, A.: Matrices with the Edmonds-Johnson property. Combinatorica 6, 365–379 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lueker, G.S.: Two NP-complete Problems in Non-negative Integer Programming. Report No. 178, Department of Computer Science, Princeton University, Princeton, N.J. (1975)

    Google Scholar 

  13. Mahajan, A., Ralphs, T.: On the complexity of selecting disjunctions in integer programming. SIAM J. Optim. 20, 2181–2198 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)

    MATH  Google Scholar 

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Acknowledgement

We thank Michele Conforti for pointing out to us that the problem of deciding whether \(P_I = \emptyset \) is in \(\mathcal {NP}\) \(\cap \) co-\(\mathcal {NP}\) when we know that \(P' = P_I\).

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Authors and Affiliations

  1. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  2. Krannert School of Management, Purdue University, West Lafayette, IN, 47906, USA

    Yanjun Li

Authors
  1. Gérard Cornuéjols
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  2. Yanjun Li
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Corresponding author

Correspondence to Yanjun Li .

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Editors and Affiliations

  1. Université de Liège, Liège, Belgium

    Quentin Louveaux

  2. Technische Universität Berlin, Berlin, Germany

    Martin Skutella

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Cornuéjols, G., Li, Y. (2016). Deciding Emptiness of the Gomory-Chvátal Closure is NP-Complete, Even for a Rational Polyhedron Containing No Integer Point. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_32

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  • DOI: https://doi.org/10.1007/978-3-319-33461-5_32

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-33460-8

  • Online ISBN: 978-3-319-33461-5

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