A counterexample to an integer analogue of Carathéodory's theorem

Bruns, Winfried; Gubeladze, Joseph; Henk, Martin; Martín, Alexander; Weismantel, Robert

doi:10.1515/crll.1999.045

Cited by 49 publications

(54 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It remains an open question to determine a sharp upper bound in the Hilbert basis setting, in [9] an example is provided where M 0 (X) = 7 6 d . For an arbitrary finite set X ⊂ Z d , Eisenbrand and Shmonin [16] obtained the bound…”

Section: Introductionmentioning

confidence: 99%

Sparse Solutions of Linear Diophantine Equations

Aliev¹,

Loera²,

Oertel³

et al. 2017

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Sparse Solutions of Linear Diophantine Equations

Aliev¹,

Loera²,

Oertel³

et al. 2017

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

show abstract

“…It has been conjectured by Cook, Fonlupt and Schrijver [3] that this integer analogue of Carathéodory's theorem holds true for any rational pointed cone. This conjecture was proven by Sebő [14] in dimensions ≤ 3 and has recently been disproved by Bruns, Gubeladze, Henk, Martin and Weismantel [2] in dimensions n ≥ 6.…”

Section: Introductionmentioning

confidence: 87%

Diophantine Approximations and Integer Points of Cones

Henk

Weismantel

2002

Combinatorica

Self Cite

View full text Add to dashboard Cite

show abstract

“…In general it is not. It is not even true that a Hilbert basis has a unimodular partition or a unimodular covering [2] and this counterexample inspires two more remarks. First, it cannot be expected that the equivalence of (i) and (v) can be reduced to Sturmfels' generic case.…”

Section: For Instance Ifmentioning

confidence: 99%

“…In general, the converse does not hold and the most that is known in this direction is the existence of just one full dimensional subset of the columns of A which is unimodular [16]. Not even a "unimodular covering" of a Hilbert basis may be possible [2]. However, the converse does hold for normal fans of integral set packing polytopes.…”

Section: Introductionmentioning

confidence: 99%

Alternatives for testing total dual integrality

O'Shea

Sebő

2010

Math. Program.

View full text Add to dashboard Cite

Abstract. In this paper we provide characterizing properties of TDI systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system's dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and integer programming and they contain Applegate, Cook and McCormick's sufficient condition for the TDI property and Sturmfels' theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. We also study the theoretical and practical efficiency and limits of the characterizations of the TDI property presented here.In the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional, computationally interesting, geometric feature: the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones.

show abstract

A counterexample to an integer analogue of Carathéodory's theorem

Cited by 49 publications

References 3 publications

Sparse Solutions of Linear Diophantine Equations

Sparse Solutions of Linear Diophantine Equations

Diophantine Approximations and Integer Points of Cones

Alternatives for testing total dual integrality

Contact Info

Product

Resources

About