Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf’s Theorem

  • Iskander Aliev
  • Jesús A. De Loera
  • Quentin Louveaux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)


In this paper we study a generalization of the classical feasibility problem in integer linear programming, where an ILP needs to have a prescribed number of solutions to be considered solved.

We first provide a generalization of the famous Doignon-Bell-Scarf theorem: Given an integer k, we prove that there exists a constant c(k,n), depending only on the dimension n and k, such that if a polyhedron {x : Ax ≤ b} contains exactly k integer solutions, then there exists a subset of the rows of cardinality no more than c(k,n), defining a polyhedron that contains exactly the same k integer solutions.

The second contribution of the article presents a structure theory that characterizes precisely the set sg  ≥ k (A) of all vectors b such that the problem Ax = b, x ≥ 0, x ∈ ℤ n , has at least k-solutions. We demonstrate that this set is finitely generated, a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computation. Similar results can be derived for those right-hand-side vectors that have exactly k solutions or fewer than k solutions.

Finally we show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of sg  ≥ k (A) as a generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors that have exactly k solutions (similarly for at least k or less than k solutions). Under the same assumptions we prove that the k-Frobenius number can be computed in polynomial time.


Polynomial Time Lattice Point Knapsack Problem Integer Solution Numeric Semigroup 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    4ti2 team. 4ti2–Software package for algebraic, geometric and combinatorial problems on linear spaces,
  2. 2.
    Aardal, K., Lenstra, A.K.: Hard equality constrained integer knapsacks. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 350–366. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Aliev, I., Fukshansky, L., Henk, M.: Generalized Frobenius Numbers: Bounds and Average Behavior. Acta Arith. 155, 53–62 (2012)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Aliev, I., Henk, M., Linke, E.: Integer Points in Knapsack Polytopes and s-covering Radius. Electron. J. Combin. 20(2), Paper 42, 17 (2013)Google Scholar
  5. 5.
    Barvinok, A.I.: Polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. of Operations Research 19, 769–779 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Barvinok, A.I., Woods, K.: Short rational generating functions for lattice point problems. J. Amer. Math. Soc. 16, 957–979 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bell, D.E.: A theorem concerning the integer lattice. Studies in Applied Mathematics 56(1), 187–188 (1977)zbMATHGoogle Scholar
  8. 8.
    Beck, M., Robins, S.: A formula related to the Frobenius problem in two dimensions. In: Number Theory (New York, 2003), pp. 17–23. Springer, New York (2004)Google Scholar
  9. 9.
    Bruns, W., Gubeladze, J., Trung, N.V.: Problems and algorithms for affine semigroups. Semigroup Forum 64, 180–212 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bruns, W., Koch, R.: NORMALIZ, computing normalizations of affine semigroups,
  11. 11.
    Clarkson, K.L.: Las Vegas algorithms for linear and integer programming when the dimension is small. Journal of the ACM 42(2), 488–499 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms, Undergraduate Texts in Mathematics. Springer, New York (1992)CrossRefGoogle Scholar
  13. 13.
    De Loera, J.A., Hemmecke, R., Tauzer, J., Yoshida, R.: Effective lattice point counting in rational convex polytopes. Journal of Symbolic Computation 38(4), 1273–1302 (2004)CrossRefMathSciNetGoogle Scholar
  14. 14.
    De Loera, J.A., Haws, D.C., Hemmecke, R., Huggins, P., Sturmfels, B., Yoshida, R.: Short rational functions for toric algebra and applications. Journal of Symbolic Computation 38(2), 959–973 (2004)CrossRefMathSciNetGoogle Scholar
  15. 15.
    De Loera, J.A., Hemmecke, R., Köppe, M.: Algebraic and geometric ideas in the theory of discrete optimization. MOS-SIAM Series on Optimization, vol. 14, p. xx+322. Society for Industrial and Applied Mathematics (SIAM), Philadelphia; Mathematical Optimization Society, Philadelphia (2013)Google Scholar
  16. 16.
    Dobra, A., Karr, A.F., Sanil, P.A.: Preserving confidentiality of high-dimensional tabulated data: statistical and computational issues. Stat. Comput. 13, 363–370 (2003)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Doignon, J.-P.: Convexity in cristallographical lattices. Journal of Geometry 3(1), 71–85 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Eisenbrand, F., Hähnle, N.: Minimizing the number of lattice points in a translated polygon. In: Proceedings of SODA, pp. 1123–1130 (2013)Google Scholar
  19. 19.
    Fukshansky, L., Schürmann, A.: Bounds on generalized Frobenius numbers. European J. Combin. 3, 361–368 (2011)CrossRefGoogle Scholar
  20. 20.
    Haase, C., Nill, B., Payne, S.: Cayley decompositions of lattice polytopes and upper bounds for h *-polynomials. J. Reine Angew. Math. 637, 207–216 (2009)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Hemmecke, R., Takemura, A., Yoshida, R.: Computing holes in semi-groups and its application to transportation problems. Contributions to Discrete Mathematics 4, 81–91 (2009)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Kannan, R.: Lattice translates of a polytope and the Frobenius problem. Combinatorica 12(2), 161–177 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lagarias, J.C., Ziegler, G.M.: Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canad. J. Math. 43(5), 1022–1035 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Pikhurko, O.: Lattice points in lattice polytopes. Mathematika 48(1-2), 15–24 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Ramírez Alfonsín, J.L.: Gaps in semigroups. Discrete Mathematics 308(18), 4177–4184 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Ramírez Alfonsín, J.L.: Complexity of the Frobenius problem. Combinatorica 16(1), 143–147 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Ramírez Alfonsín, J.L.: The Diophantine Frobenius Problem. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, New York (2006)Google Scholar
  28. 28.
    Schrijver, A.: Theory of linear and integer programming. Wiley (1998)Google Scholar
  29. 29.
    Scarf, H.E.: An observation on the structure of production sets with indivisibilities. Proceedings of the National Academy of Sciences 74(9), 3637–3641 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Stanley, R.P.: Combinatorics and Commutative Algebra, 2nd edn. Progress in Mathematics, vol. 41. Birkhäuser, Basel (1996)zbMATHGoogle Scholar
  31. 31.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. AMS, Providence (1995)Google Scholar
  32. 32.
    Takemura, A., Yoshida, R.: A generalization of the integer linear infeasibility problem. Discrete Optimization 5, 36–52 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Iskander Aliev
    • 1
  • Jesús A. De Loera
    • 2
  • Quentin Louveaux
    • 3
  1. 1.Cardiff UniversityUK
  2. 2.University of CaliforniaDavisUSA
  3. 3.Université de LiègeBelgium

Personalised recommendations