Matrices with lexicographically-ordered rows

  • Gustavo Angulo
Original Paper


The lexicographic order can be used to force a collection of decision vectors to be all different, i.e., to take on different values in some coordinates. We consider the set of fixed-size matrices with bounded integer entries and rows in lexicographic order. We present a dynamic program to optimize a linear function over this set, from which we obtain a compact extended formulation for its convex hull.


Lexicographic order Extended formulation Dynamic programming 



The author thanks an anonymous referee whose comments and suggestions greatly helped to improve the paper.


  1. 1.
    Angulo, G., Ahmed, S., Dey, S.S., Kaibel, V.: Forbidden vertices. Math. Oper. Res. 40(2), 350–360 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barbato, M., Grappe, R., Lacroix, M., Pira, C.: Lexicographical polytopes. Discret. Appl. Math. 240, 3–7 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Di Summa, M.: A short convex-hull proof for the all-different system with the inclusion property. Oper. Res. Lett. 43(1), 69–73 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gupte, A.: Convex hulls of superincreasing knapsacks and lexicographic orderings. Discret. Appl. Math. 201, 150–163 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hojny, C., Pfetsch, M.E.: Polytopes associated with symmetry handling. Math. Program. (2018).
  6. 6.
    Kaibel, V., Loos, A.: Branched Polyhedral Systems, Integer Programming and Combinatorial Optimization, pp. 177–190. Springer, Berlin (2010)CrossRefGoogle Scholar
  7. 7.
    Kaibel, V., Pfetsch, M.E.: Packing and partitioning orbitopes. Math. Program. 114(1), 1–36 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lee, J.: All-different polytopes. J. Comb. Optim. 6(3), 335–352 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lee, J., Margot, F.: On a binary-encoded ILP coloring formulation. INFORMS J. Comput. 19(3), 406–415 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Loos, A.: Describing Orbitopes by Linear Inequalities and Projection Based Tools, Ph.D. thesis (2011)Google Scholar
  11. 11.
    Magos, D., Mourtos, I., Appa, G.: A polyhedral approach to the all different system. Math. Program. 132(1–2), 209–260 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Martin, R.K., Rardin, R.L., Campbell, B.A.: Polyhedral characterization of discrete dynamic programming. Oper. Res. 38(1), 127–138 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1998)zbMATHGoogle Scholar
  14. 14.
    Williams, H.P., Yan, H.: Representations of the all\_different predicate of constraint satisfaction in integer programming. INFORMS J. Comput. 13(2), 96–103 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringPontificia Universidad Católica de ChileSantiagoChile

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