Mathematical Programming

, Volume 145, Issue 1–2, pp 199–222 | Cite as

Design and verify: a new scheme for generating cutting-planes

  • Santanu S. Dey
  • Sebastian Pokutta
Full Length Paper Series A


A cutting-plane procedure for integer programming (IP) problems usually involves invoking a black-box procedure (such as the Gomory–Chvátal procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. This involves two steps. In the first step, we design an inequality \(cx \le d\) where \(c\) and \(d\) are integral, independent of the cutting-plane black-box. In the second step, we verify that the designed inequality is a valid inequality by verifying that the set \(P \cap \{x\in \mathbb R ^n \mid cx \ge d + 1\} \cap \mathbb Z ^n\) is empty using cutting-planes from the black-box. Here \(P\) is the feasible region of the linear-programming relaxation of the IP. We refer to the closure of all cutting-planes that can be verified to be valid using a specific cutting-plane black-box as the verification closure of the considered cutting-plane black-box. This paper undertakes a systematic study of properties of verification closures of various cutting-plane black-box procedures.


Verification scheme Cutting planes Integer programming 

Mathematics Subject Classification

90C10 90C57 



The authors are most grateful to the anonymous referees for their detailed remarks that considerably improved the presentation and simplified some of the proofs.


  1. 1.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed integer 0–1 programs. Math. Program. 58, 295–324 (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discret. Math. 4, 305–337 (1973)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chvátal, V., Cook, W., Hartmann, M.: On cutting-plane proofs in combinatorial optimization. Linear Algebra Appl. 114, 455–499 (1989)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Cook, W., Coullard, C.R., Turan, G.: On the complexity of cutting plane proof. Math. Program. 47, 11–18 (1990)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cook, W., Dash, S.: On the matrix cut rank of polyhedra. Math. Oper. Res. 26, 19–30 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Program. 58, 155–174 (1990)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cornúejols, G., Li, Y.: Elementary closures for integer programs. Oper. Res. Lett. 28, 1–8 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cornuéjols, G., Li, Y.: On the rank of mixed 0–1 polyhedra. Math. Program. 91, 391–397 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dadush, D., Dey, S.S., Vielma, J.P.: On the Chvátal-Gomory closure of a compact convex set. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011, Proceedings Lecture Notes in Computer Science, vol. 6655, pp. 130–142. Springer, Berlin (2011)Google Scholar
  10. 10.
    Dadush, D., Dey, S.S., Vielma, J.P.: The Chvátal-Gomory closure of strictly convex body. Math. Oper. Res. 36, 227–239 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dey, S.S., Pokutta, S.: Design and verify: a new scheme for generating cutting-planes. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011, Proceedings Lecture Notes in Computer Science, vol. 6655, pp. 143–155. Springer, Berlin (2011)Google Scholar
  12. 12.
    Dey, S.S., Richard, J.P.P.: Some relations between facets of low- and high-dimensional group problems. Math. Program. 123, 285–313 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Eisenbrand, F., Schulz, A.S.: Bounds on the Chvátal rank of polytopes in the 0/1-cube. Combinatorica 23, 245–262 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am Math. Soc. 64, 275–278 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lovász, L.: Geometry of numbers and integer programming. Mathematical Programming: Recent Developments and Applications (1989)Google Scholar
  16. 16.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)Google Scholar
  18. 18.
    Pokutta, S., Schulz, A.S.: Characterization of integer-free 0/1 polytopes with maximal rank (xxxx) (submitted)Google Scholar
  19. 19.
    Pokutta, S., Schulz, A.S.: On the rank of generic cutting-plane proof systems. In: Eisenbrand, F., Shepherd, B. (eds.) Integer Programming and Combinatorial Optimization, 14th International IPCO Conference, Proceedings, Lecture Notes in Computer Science, Lausanne, Switzerland, June 9–11, 2010, pp. 450–463. Springer, Berlin (2010)Google Scholar
  20. 20.
    Pokutta, S., Stauffer, G.: Lower bounds for the Chvátal-Gomory rank in the 0/1 cube. Oper. Res. Lett (2011) (to appear)Google Scholar
  21. 21.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations