Mathematical Programming Computation

, Volume 10, Issue 3, pp 423–455 | Cite as

Intersection cuts for single row corner relaxations

  • Ricardo Fukasawa
  • Laurent Poirrier
  • Álinson S. XavierEmail author
Full Length Paper


We consider the problem of generating inequalities that are valid for one-row relaxations of a simplex tableau, with the integrality constraints preserved for one or more non-basic variables. These relaxations are interesting because they can be used to generate cutting planes for general mixed-integer problems. We first consider the case of a single non-basic integer variable. This relaxation is related to a simple knapsack set with two integer variables and two continuous variables. We study its facial structure by rewriting it as a constrained two-row model, and prove that all its facets arise from a finite number of maximal \(\left( \mathbb {Z}\times \mathbb {Z}_+\right) \)-free splits and wedges. The resulting cuts generalize both MIR and 2-step MIR inequalities. Then, we describe an algorithm for enumerating all the maximal \(\left( \mathbb {Z}\times \mathbb {Z}_+\right) \)-free sets corresponding to facet-defining inequalities, and we provide an upper bound on the split rank of those inequalities. Finally, we run computational experiments to compare the strength of wedge cuts against MIR cuts. In our computations, we use the so-called trivial fill-in function to exploit the integrality of more non-basic variables. To that end, we present a practical algorithm for computing the coefficients of this lifting function.


Integer programming Lifting Cutting planes 

Mathematics Subject Classification

90C11 90C57 



We would like to thank two anonymous referees for valuable suggestions that led to significant improvements to this manuscript. Fukasawa was supported by NSERC Discovery Grant RGPIN 2014-05623. Poirrier and Xavier were supported by Early Researcher Award Grant ER-11-08-174.

Supplementary material


  1. 1.
    Agra, A., Constantino, M.F.: Description of 2-integer continuous knapsack polyhedra. Discrete Optim. 3(2), 95–110 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agra, A., Constantino, M.F.: Lifting two-integer knapsack inequalities. Math. Program. 109(1), 115–154 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.: Cutting planes from two rows of a simplex tableau (extended version). Working Paper (2006).
  4. 4.
    Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.: Inequalities from two rows of a simplex tableau. In: Fischetti, M., Williamson, D. (eds.) Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 4513, pp. 1–15. Springer, Berlin (2007). CrossRefGoogle Scholar
  5. 5.
    Atamtürk, A., Rajan, D.: Valid inequalities for mixed-integer knapsack from two-integer variable restrictions. Research Report BCOL.04.02, IEOR, University of California, Berkeley (December 2004)Google Scholar
  6. 6.
    Balas, E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 1(19), 19–39 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Balas, E., Jeroslow, R.G.: Combinational optimization strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4(4), 224–234 (1980). CrossRefzbMATHGoogle Scholar
  8. 8.
    Balas, E., Margot, F.: Generalized intersection cuts and a new cut generating paradigm. Math. Program. (2011).
  9. 9.
    Balas, E., Ceria, S., Cornuéjols, G., Natraj, N.R.: Gomory cuts revisited. Oper. Res. Lett. 19, 1–9 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Basu, A., Campelo, M., Conforti, M., Cornuéjols, G.: On lifting integer variables in minimal inequalities. In: Eisenbrand, F., Shepherd, F.B. (eds.) Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 6080, pp. 85–95. Springer, Berlin (2010). CrossRefGoogle Scholar
  11. 11.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Minimal inequalities for an infinite relaxation of integer programs. SIAM J. Discrete Math. 24(1), 158–168 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: Experiments with two-row cuts from degenerate tableaux. INFORMS J. Comput. 23, 578–590 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Basu, A., Hildebrand, R., Köppe, M.: Algorithmic and complexity results for cutting planes derived from maximal lattice-free convex sets (2011).
  14. 14.
    Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34(3), 538–546 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Conforti, M., Cornuéjols, G., Zambelli, G.: A geometric perspective on lifting. Oper. Res. 59, 569–577 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer (2014). ISBN 3319110071, 9783319110073Google Scholar
  17. 17.
    Cook, W.J., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Progr. 47, 155–174 (1990)CrossRefzbMATHGoogle Scholar
  18. 18.
    Cook, W.J., Hartmann, M., Kannan, R., McDiarmid, C.: On integer points in polyhedra. Combinatorica 12(1), 27–37 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cornuéjols, G., Margot, F.: On the facets of mixed integer programs with two integer variables and two constraints. Math. Progr. 120(2), 429–456 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dash, S., Günlük, O.: On the strength of gomory mixed-integer cuts as group cuts. Math. Progr. 115(2), 387–407 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dash, S., Goycoolea, M., Günlük, O.: Two-step MIR inequalities for mixed integer programs. INFORMS J. Comput. 22(2), 236–249 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dey, S.S., Wolsey, L.A.: Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.), 13th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2008, Bertinoro, Italy, May 26–28, 2008, Proceedings of Lecture Notes in Computer Science, vol. 5035, Springer, pp. 463–475 (2008)Google Scholar
  23. 23.
    Dey, S.S., Wolsey, L.A.: Constrained infinite group relaxations of MIPs. CORE Discussion Papers 2009033, Université Catholique de Louvain, Center for Operations Research and Econometrics (CORE), (May 2009). URL
  24. 24.
    Dey, S.S., Wolsey, L.A.: Two row mixed-integer cuts via lifting. Math. Progr. 124, 143–174 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dey, S.S., Wolsey, L.A.: Composite lifting of group inequalities and an application to two-row mixing inequalities. Discrete Optim. 7(4), 256–268 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fischetti M., Saturni, C.: Mixed-integer cuts from cyclic groups. In: Jünger, M., Kaibel, V. (eds.), 11th International IPCO Conference on Integer Programming and Combinatorial Optimization, Berlin, Germany, June 8–10, 2005. Proceedings, pp. 1–11, Springer, Berlin (2005).
  27. 27.
    Fukasawa, R., Günlük, O.: Strengthening lattice-free cuts using non-negativity. Discrete Optim. 8(2), 229–245 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fukasawa, R., Goycoolea, M.: On the exact separation of mixed integer knapsack cuts. Math. Progr. 128(1–2), 19–41 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fukasawa, R., Poirrier, L., Xavier, Á.S.: The (not so) trivial lifting in two dimensions (2016).
  30. 30.
    Fukasawa, R., Poirrier, L., Xavier, Á.S.: Intersection cuts for single row corner relaxations: source code (January 2018).
  31. 31.
    Gomory, R.E.: An algorithm for the mixed integer problem. Technical Report RM-2597, The Rand Corporation (1960)Google Scholar
  32. 32.
    Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2(4), 451–558 (1969). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, part II. Math. Progr. 3(1), 359–389 (1972). CrossRefzbMATHGoogle Scholar
  34. 34.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, part I. Math. Progr. 3, 23–85 (1972)CrossRefzbMATHGoogle Scholar
  35. 35.
    Granlund, T., GMP Development Team: GNU MP: The GNU Multiple Precision Arithmetic Library, 6.1.0 edn (2015).
  36. 36.
    Harvey, W.: Computing two-dimensional integer hulls. SIAM J. Comput. 28(6), 2285–2299 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hirschberg, D.S., Wong, C.K.: A polynomial-time algorithm for the knapsack problem with two variables. J. ACM 23(1), 147–154 (1976). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010. Math. Progr. Comput. 3(2), 103–163 (2011). CrossRefGoogle Scholar
  39. 39.
    Louveaux, Q., Poirrier, L.: An algorithm for the separation of two-row cuts. Math. Progr. 143(1–2), 111–146 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Louveaux, Q., Poirrier, L., Salvagnin, D.: The strength of multi-row models. Math. Progr. Comput. 7(2), 113–148 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lovász, L.: Geometry of numbers and integer programming. Proc. Math. Appl. Jpn. Ser. 6, 177–201 (1989)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 2001 (1998)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Nemhauser, G.L., Wolsey, L.A.: A recursive procedure to generate all cuts for 0–1 mixed integer programs. Math. Progr. 46, 379–390 (1990). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  • Ricardo Fukasawa
    • 1
  • Laurent Poirrier
    • 1
  • Álinson S. Xavier
    • 1
    Email author
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

Personalised recommendations