A quantitative Doignon-Bell-Scarf theorem

Abstract

The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer k, we prove that there exists a constant c(n,k), depending only on the dimension n and k, such that if a polyhedron {x∈Rn: Axb} contains exactly k integer points, then there exists a subset of the rows, of cardinality no more than c(n,k), defining a polyhedron that contains exactly the same k integer points. In this case c(n,0)=2n as in the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. We work on both upper and lower bounds for the constant c(n,k) and discuss some consequences, including a Clarkson-style algorithm to find the l-th best solution of an integer program with respect to the ordering induced by the objective function.

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References

  1. [1]

    I. Aliev, J. A. De Loera and Q. Louveaux: Integer Programs with Prescribed Number of Solutions and a Weighted Version of Doignon-Bell-Scarf’s Theorem, in: Proceedings of Integer Programming and Combinatorial Optimization, 17th International IPCO Conference, Bonn Germany, June, 2014.

    Google Scholar 

  2. [2]

    N. Amenta: Helly-type theorems and generalized linear programming, Discrete and Computational Geometry 12 (1994), 241–261.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    G. Averkov and R. Weismantel: Transversal numbers over subsets of linear spaces, Adv. Geom. 12 (2012), 19–28.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    K. Andersen, Q. Louveaux and R. Weismantel: Certificates of linear mixed integer infeasibility, Operations Research Letters 36 (2008), 734–738.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    K. Andersen, Q. Louveaux and R. Weismantel: An analysis of mixed integer linear sets based on lattice point free convex sets, Math of Operations Research 35 (2010), 233–256.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    K. Andersen, Q. Louveaux, R. Weismantel and L. Wolsey: Inequalities from two rows of the simplex tableau, in: M. Fischetti & D. P., Williamson (Eds.) Integer Programming and Combinatorial Optimization, 12th International IPCO Conference, Ithaca, NY, USA, June 25-27, 2007, Proceedings, Lecture Notes in Computer Science 4513, 1-15.

    Google Scholar 

  7. [7]

    I. Bárány, M. Katchalski and J. Pach: Quantitative Helly-type theorems, Proc. Amer. Math. Soc. 86 (1982), 109–114.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    I. Bárány, M. Katchalski and J. Pach: Helly’s theorem with volumes, Amer. Math. Monthly 91 (1984), 362–365.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    A. Barvinok and J. Pommersheim: An algorithmic theory of lattice points in polyhedra, New perspectives in algebraic combinatorics, Math. Sci. res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, (1999), 91–147.

    Google Scholar 

  10. [10]

    D. E. Bell: A theorem concerning the integer lattice, Studies in Applied Mathematics 56 (1977), 187–188.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    V. Borozan and G. Cornuéjols: Minimal valid inequalities for integer constraints, Math. Oper. Res. 34 (2009), 538–546.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    K. L. Clarkson: Las Vegas algorithms for linear and integer programming when the dimension is small, Journal of the ACM 42 (1995), 488–499.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    M. Conforti, G. Cornuéjols and G. Zambelli: Corner polyhedron and intersection cuts, Surveys in Operations Research and Management Science 16 (2011), 105–120.

    Article  MATH  Google Scholar 

  14. [14]

    L. Danzer, B. Grünbaum and V. Klee: Helly’s theorem and its relatives, in: 1963 Proc. Sympos. Pure Math., Vol. VII pp. 101-180 Amer. Math. Soc., Providence, R.I.

  15. [15]

    J. A. De Loera, R. Hemmecke and M. Köppe: Algebraic and geometric ideas in the theory of discrete optimization, MOS-SIAM Series on Optimization, 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2013.

    Google Scholar 

  16. [16]

    S. Dey and L. Wolsey: Constrained infinite group relaxations of MIPs, SIAM J. Optim. 20 (2010), 2890–2912.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    J-P. Doignon: Convexity in cristallographical lattices, Journal of Geometry 3 (1973), 71–85.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    J. Eckhoff: Helly, Radon, and Carathéodory type theorems, in: Handbook of convex geometry, Vol. A, B, 389-448, North-Holland, Amsterdam, 1993.

    Google Scholar 

  19. [19]

    F. Eisenbrand: Fast integer programming in fixed dimension, Algorithms-ESA (2003), 196–207.

    Google Scholar 

  20. [20]

    B. Gärtner, J. Matoušek, L. Rüst and P Škovroň: Violator spaces: structure and algorithms, Discrete Applied Mathematics 156 (2008), 2124–2141.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    H. W. Hamacher and M. Queyranne: K best solutions to combinatorial optimization problems, Ann. Oper. Res. 4 (1985), 123–143.

    MathSciNet  Article  Google Scholar 

  22. [22]

    A.J. Hoffman: Binding constraints and Helly numbers, in: Proceedings of Second International Conference on Combinatorial Mathematics (New York, 1978), 284–288, Ann. New York Acad. Sci., 319, New York Acad. Sci., New York, 1979.

    Google Scholar 

  23. [23]

    J. C. Lagarias and G. M. Ziegler: Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Canad. J. Math. 43 (1991), 1022–1035.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    E. L. Lawler: A procedure for computing the K-best solutions to discrete optimization problems and its application to the shortest path problem, Management Sci. 18 (1971/72), 401–405.

    MathSciNet  Article  MATH  Google Scholar 

  25. [25]

    O. Pikhurko: Lattice points in lattice polytopes, Mathematika 48 (2003), 15–24.

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    H. E. Scarf: An observation on the structure of production sets with indivisibilities, Proceedings of the National Academy of Sciences 74.9 (1977), 3637–3641.

    MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    A. Schrijver: Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1986.

    Google Scholar 

  28. [28]

    M. Sharir and E. Welzl: A combinatorial bound for linear programming and related problems, in: Proceedings of 9th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science 577, Springer-Verlag, (1992), 567–579.

    Google Scholar 

  29. [29]

    R. Wenger: Helly-type theorems and geometric transversals, in: Handbook of discrete and computational geometry, Handbook of discrete and computational geometry. Second edition. Edited by Jacob E. Goodman and Joseph O’Rourke. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2004.

    Google Scholar 

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Correspondence to Quentin Louveaux.

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Aliev, I., Bassett, R., De Loera, J.A. et al. A quantitative Doignon-Bell-Scarf theorem. Combinatorica 37, 313–332 (2017). https://doi.org/10.1007/s00493-015-3266-9

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Mathematics Subject Classification (2000)

  • 52A35
  • 52C07
  • 90C10