Normality and covering properties of affine semigroups

Bruns, Winfried; Gubeladze, Joseph

doi:10.1515/crll.1999.044

Cited by 39 publications

(69 citation statements)

References 9 publications

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“…Our construction is a simple enumeration of the distinct columns of A 2 . It is not difficult to combine Constructions 1 and 2 in order to solve (4) for C in (5). We state this as a lemma without proof.…”

Section: ⊓ ⊔mentioning

confidence: 99%

The Integrality Number of an Integer Program

Paat

Schlöter

Weismantel

2020

Integer Programming and Combinatorial Optimization

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We introduce the integrality number of an integer program (IP) in inequality form. Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor ∆ of the constraint matrix, our analysis allows us to make statements of the following form: there exist numbers τ (∆) and κ(∆) such that an IP with n ≥ τ (∆) many variables and n + κ(∆) · √ n many inequality constraints can be solved via a MIP relaxation with fewer than n integer constraints. A special instance of our results shows that IPs defined by only n constraints can be solved via a MIP relaxation with O( √ ∆) many integer constraints.1 Introduction.Let A ∈ Z m×n satisfy rank(A) = n and c ∈ Z n . We denote the integer linear program parameterized by right hand side b ∈ Z m by IP A,c (b) := max{c ⊺ x : Ax ≤ b and x ∈ Z n }.See [8] for more on parametric integer programs. We are interested in solving IP A,c (b) by relaxing it to have fewer integer constraints. In special cases we can solve IP A,c (b) with zero integer constraints by only solving its linear relaxation LP A,c (b) := max{c ⊺ x : Ax ≤ b}.However, these special cases require the underlying polyhedron to have optimal integral vertices, which occurs for instance when A is totally unimodular. Our target is to consider general matrices A and relaxations in the form of a mixed integer program:where k ∈ Z ≥0 and W ∈ Z k×n satisfies rank(W ) = k.One sufficient condition for solving IP A,c (b) using a mixed integer relaxation is that the vertices of W-MIP A,c (b) are integral. The vertices of W-MIP A,c (b) are the vertices of the polyhedron conv {x ∈ R n : Ax ≤ b, W x ∈ Z k } .

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Section: ⊓ ⊔mentioning

confidence: 99%

The Integrality Number of an Integer Program

Paat

Schlöter

Weismantel

2020

Integer Programming and Combinatorial Optimization

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“…One should note that if P has a unimodular triangulation, then P has the IDP. However, there are examples of polytopes which have the IDP, yet do not even admit a unimodular cover, that is, a covering of P by unimodular simplices, see [BG99,Sec. 3].…”

Section: Introductionmentioning

confidence: 99%

Unconditional Reflexive Polytopes

Kohl

Olsen

Sanyal

2020

Discrete Comput Geom

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A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets.

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“…The value σ asy (A) can be thought of as the smallest k such that almost all feasible integer programs with constraint matrix A have solutions with support of cardinality at most k. The function σ asy (·) was introduced by Bruns and Gubeladze in [6], where it was shown that σ asy (A) ≤ 2m − 3 for matrices with the Hilbert basis property. In general, an average case analysis of the support question has not been provided in the literature.…”

Section: Introductionmentioning

confidence: 99%

Sparsity of Integer Solutions in the Average Case

Oertel

Paat

Weismantel

2019

Integer Programming and Combinatorial Optimization

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We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right-hand side vectors. For a problem in standard form with m equations, there exist LP feasible solutions with at most m many nonzero entries. We show that under relatively mild assumptions, integer programs in standard form have feasible solutions with O(m) many nonzero entries, on average. Our proof uses ideas from the theory of groups, lattices, and Ehrhart polynomials. From our main theorem we obtain the best known upper bounds on the integer Carathéodory number provided that the determinants in the data are small.

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Normality and covering properties of affine semigroups

Cited by 39 publications

References 9 publications

The Integrality Number of an Integer Program

The Integrality Number of an Integer Program

Unconditional Reflexive Polytopes

Sparsity of Integer Solutions in the Average Case

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