Mixed-Integer Linear Representability, Disjunctions, and Variable Elimination

  • Amitabh BasuEmail author
  • Kipp Martin
  • Christopher Thomas Ryan
  • Guanyi Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


Jeroslow and Lowe gave an exact geometric characterization of subsets of \(\mathbb R^n\) that are projections of mixed-integer linear sets, a.k.a MILP-representable sets. We give an alternate algebraic characterization by showing that a set is MILP-representable if and only if the set can be described as the intersection of finitely many affine Chvátal inequalities. These inequalities are a modification of a concept introduced by Blair and Jeroslow. This gives a sequential variable elimination scheme that, when applied to the MILP representation of a set, explicitly gives the affine Chvátal inequalities characterizing the set. This is related to the elimination scheme of Wiliams and Williams-Hooker, who describe projections of integer sets using disjunctions of affine Chvátal systems. Our scheme extends their work in two ways. First, we show that disjunctions are unnecessary, by showing how to find the affine Chvátal inequalities that cannot be discovered by the Williams-Hooker scheme. Second, disjunctions of Chvátal systems can give sets that are not projections of mixed-integer linear sets; so the Williams-Hooker approach does not give an exact characterization of MILP representability.


Binary Tree Linear Inequality Integer Variable Elimination Scheme Algebraic Characterization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Balas, E.: Projecting systems of linear inequalities with binary variables. Ann. Oper. Res. 188(1), 19–31 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blair, C., Jeroslow, R.: The value function of an integer program. Math. Program. 23, 237–273 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer, Heidelberg (2014)CrossRefzbMATHGoogle Scholar
  4. 4.
    Jeroslow, R., Lowe, J.: Modelling with integer variables. In: Korte, B., Ritter, K. (eds.) Mathematical Programming at Oberwolfach II, vol. 22, pp. 167–184. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  5. 5.
    Ryan, J.: Integral monoid duality models. Technical report, Cornell University Operations Research and Industrial Engineering (1986)Google Scholar
  6. 6.
    Ryan, J.: Decomposing finitely generated integral monoids by elimination. Linear Algebra Appl. 153, 209–217 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar
  8. 8.
    Vielma, J.P.: Mixed integer linear programming formulation techniques. SIAM Rev. 57(1), 3–57 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Williams, H.P.: Fourier-Motzkin elimination extension to integer programming problems. J. Comb. Theor. A 21, 118–123 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Williams, H.P.: The elimination of integer variables. J. Oper. Res. Soc. 43, 387–393 (1992)CrossRefzbMATHGoogle Scholar
  11. 11.
    Williams, H., Hooker, J.: Integer programming as projection. Discrete Optim. 22, 291–311 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wolsey, L.: The b-hull of an integer program. Discrete Appl. Math. 3(3), 193–201 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Amitabh Basu
    • 1
    Email author
  • Kipp Martin
    • 2
  • Christopher Thomas Ryan
    • 2
  • Guanyi Wang
    • 3
  1. 1.Department of Applied Mathematics and StatisticsJohns Hopkins UniversityBaltimoreUSA
  2. 2.Booth School of BusinessUniversity of ChicagoChicagoUSA
  3. 3.Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations