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Combinatorica

, Volume 37, Issue 3, pp 313–332 | Cite as

A quantitative Doignon-Bell-Scarf theorem

  • Iskander Aliev
  • Robert Bassett
  • Jesús A. De Loera
  • Quentin LouveauxEmail author
Original Paper

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∈R n : 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)=2 n 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.

Mathematics Subject Classification (2000)

52A35 52C07 90C10 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Iskander Aliev
    • 1
  • Robert Bassett
    • 2
  • Jesús A. De Loera
    • 2
  • Quentin Louveaux
    • 3
    Email author
  1. 1.Cardiff UniversityCardiffUK
  2. 2.University of CaliforniaDavisUSA
  3. 3.Université de LiègeLiègeBelgium

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