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## Filters

##
###
Reverse Chvátal-Gomory Rank
[chapter]

2013
*
Lecture Notes in Computer Science
*

We introduce

doi:10.1007/978-3-642-36694-9_12
fatcat:ioxzxzj4gfa43dctjydb73ulie
*the*reverse Chvátal-Gomory rank r * (P )*of*an integral*polyhedron*P , defined as*the*supremum*of**the*Chvátal-Gomory ranks*of*all rational polyhedra whose*integer**hull*is P . ... We provide*a*geometric characterization*of*polyhedra with this property*in*general dimension, and investigate upper bounds on r * (P ) when this value is finite. ... Introduction*A**polyhedron*is integral if it is*the*convex*hull**of*its*integer**points*. ...##
###
Recognition of Digital Polyhedra with a Fixed Number of Faces
[chapter]

2016
*
Lecture Notes in Computer Science
*

*contains*an

*integer*

*point*), it has only

*a*finite number

*of*lattice jewels. ...

*In*this case, we provide an algorithm

*of*recognition

*of*

*a*digital

*polyhedron*with n facets which always finishes. ... -If

*the*interior

*of*

*the*convex

*hull*

*of*S

*contains*an

*integer*

*point*(i) or, if d = 2, and

*the*affine dimension

*of*S is 2 (ii), then Algorithm 1 ends

*in*

*a*finite time. Proof. ...

##
###
Two-halfspace closure
[article]

2021
*
arXiv
*
pre-print

*A*key step

*of*our analysis shows that

*the*split closure

*of*

*a*rational

*polyhedron*can be obtained by considering

*the*split closures

*of*all k-dimensional (rational) projections

*of*

*the*

*polyhedron*, for any ... We define

*a*new cutting plane closure for pure

*integer*programs called

*the*two-halfspace closure. It is

*a*natural generalization

*of*

*the*well-known Chvátal-Gomory closure. ... Their suggestions and pointers helped to improve

*the*paper from its initial versions.

*In*particular, one

*of*

*the*referees suggested

*a*shorter and more elegant proof for Theorem 2.9 which we adopted. ...

##
###
Periodic Polyhedra
[chapter]

2004
*
Lecture Notes in Computer Science
*

This paper presents

doi:10.1007/978-3-540-24723-4_10
fatcat:z5zbcbsejzekhaupwivfig2seq
*a*new method for computing*the**integer**hull**of**a*parameterized rational*polyhedron*by introducing*the*concept*of*periodic*polyhedron*. ... Besides concerning generally parametric combinatorial optimization,*the*method has many applications for*the*analysis, optimization and parallelization*of*loop nests, especially*in*compilers. ... If such*a*set*contains*several*points*,*the**integer*lexicographic maximum is*in**the*subset*of*X1 having*the*maximal*integer*value for i2, and so on. ...##
###
On lattice points in polyhedral cross-sections

1993
*
Discrete & Computational Geometry
*

*a*) We prove that

*the*convex

*hull*

*of*any k d + 1

*points*

*of*

*a*d-dimensional lattice

*contains*k + 1 collinear lattice

*points*. ... (b) For

*a*convex

*polyhedron*we consider

*the*numbers

*of*its lattice

*points*

*in*consecutive parallel lattice hyperplanes (levels). ...

*The*convex

*hull*

*of*k d + 1 distinct lattice

*points*

*in*R ~ with no k

*of*them on

*a*common line

*contains*at least kd + k distinct lattice

*points*. ...

##
###
How to Find the Convex Hull of All Integer Points in a Polyhedron?
[article]

2020
*
arXiv
*
pre-print

We propose

arXiv:2010.13147v1
fatcat:32iqejtrofgencfc4jrfyqd43e
*a*cut-based algorithm for finding all vertices and all facets*of**the*convex*hull**of*all*integer**points**of**a**polyhedron*defined by*a*system*of*linear inequalities. ... We describe*the*computer implementation*of**the*algorithm and present*the*results*of*computational experiments comparing our algorithm with*a*naive one. ... Denote by P I*the*convex*hull**of*all*integer**points**contained**in*P , i.e. P I = conv(P ∩ Z d ). We consider*the*problem*of*finding all vertices and all facets*of*P I . ...##
###
On the convergence of the affine hull of the Chvátal-Gomory closures
[article]

2012
*
arXiv
*
pre-print

closure operator have to be performed on Q to obtain

arXiv:1210.6280v2
fatcat:uyonuwniwzhoxg7mosyxirrnsu
*a**polyhedron**contained**in**the*affine*hull**of*P. ... Given an integral*polyhedron*P and*a*rational*polyhedron*Q living*in**the*same n-dimensional space and*containing**the*same*integer**points*as P, we investigate how many iterations*of**the*Chv\'atal-Gomory ... This example shows actually something stronger: it can take arbitrarily many rounds*of**the*CG closure already for*a**polyhedron*to be*contained**in**the*affine*hull**of*its*integer**points*(*the*affine*hull*...##
###
Note on the complexity of the mixed-integer hull of a polyhedron

2015
*
Operations Research Letters
*

We study

doi:10.1016/j.orl.2015.03.002
fatcat:wt7t24whyjadlcxvhsqkvvaofe
*the*complexity*of*computing*the*mixed-*integer**hull*conv(P ∩ Z n × R d )*of**a**polyhedron*P . ... Given P = conv(V ) and n fixed, we compute*a*vertex description*of**the*mixed-*integer**hull**in*polynomial time and give bounds on*the*number*of*vertices*of**the*mixed*integer**hull*. ... Hartman [8] gave*a*polynomial time algorithm*in*fixeddimension to enumerate*the*vertices*of**the**integer**hull**of**a**polyhedron*. ...##
###
Integer hulls of linear polyhedra and scl in families
[article]

2011
*
arXiv
*
pre-print

*The*

*integer*

*hull*

*of*

*a*

*polyhedron*is

*the*convex

*hull*

*of*

*the*

*integer*

*points*

*contained*

*in*it. ... We show that

*the*vertices

*of*

*the*

*integer*

*hulls*

*of*

*a*rational family

*of*polyhedra

*of*size O(n) have quasipolynomial coordinates. ...

*The*

*integer*

*hull*

*of*

*a*

*polyhedron*is

*the*convex

*hull*

*of*

*the*

*integer*lattice

*points*

*contained*

*in*it.

*The*

*integer*

*hull*is itself

*a*

*polyhedron*, and may be described by enumerating its extremal vectors. ...

##
###
Integer hulls of linear polyhedra and scl in families

2013
*
Transactions of the American Mathematical Society
*

*The*

*integer*

*hull*

*of*

*a*

*polyhedron*is

*the*convex

*hull*

*of*

*the*

*integer*

*points*

*contained*

*in*it. ... We show that

*the*vertices

*of*

*the*

*integer*

*hulls*

*of*

*a*rational family

*of*polyhedra

*of*size O(n) have eventually quasipolynomial coordinates. ...

*The*authors also thank

*the*anonymous referees for helpful suggestions and corrections. ...

##
###
Convex Hulls in a 3-Dimensional Space
[chapter]

2004
*
Lecture Notes in Computer Science
*

This paper describes

doi:10.1007/978-3-540-30503-3_14
fatcat:22uhww5renh6tk3lnsx3ef2aaq
*a*new algorithm*of*computing*the*convex*hull**of**a*3-dimensional object. ...*The*convex*hull*generated by this algorithm is an abstract*polyhedron*being described by*a*new data structure,*the*cell list, suggested by one*of**the*authors. ... Definition CH:*The*convex*hull**of**a*finite set S*of**points*is*the*smallest convex AG-*polyhedron*P G*containing*all*points**of**the*set S. ...##
###
Note on the Complexity of the Mixed-Integer Hull of a Polyhedron
[article]

2015
*
arXiv
*
pre-print

We study

arXiv:1412.2520v2
fatcat:vxqmcyloarahfikrjuo45onn2u
*the*complexity*of*computing*the*mixed-*integer**hull*conv(P∩Z^n×R^d)*of**a**polyhedron*P. ... Given P=conv(V) and n fixed, we compute*a*vertex description*of**the*mixed-*integer**hull**in*polynomial time and give bounds on*the*number*of*vertices*of**the*mixed*integer**hull*. ... Hartman [8] gave*a*polynomial time algorithm*in*fixeddimension to enumerate*the*vertices*of**the**integer**hull**of**a**polyhedron*. ...##
###
Reverse Chvátal-Gomory rank
[article]

2014
*
arXiv
*
pre-print

We introduce

arXiv:1211.0388v3
fatcat:las3jsa6irdotfv2wp7eeh5rb4
*the*reverse Chv\'atal-Gomory rank r*(P)*of*an integral*polyhedron*P, defined as*the*supremum*of**the*Chv\'atal-Gomory ranks*of*all rational polyhedra whose*integer**hull*is P. ... We provide*a*geometric characterization*of*polyhedra with this property*in*general dimension, and investigate upper bounds on r*(P) when this value is finite. ... Acknowledgments We thank Gennadiy Averkov for*pointing*out*the*connection between Theorem 1 and [21] , that lead to Lemma 18. ...##
###
Relaxations of mixed integer sets from lattice-free polyhedra

2012
*
4OR
*

*The*main focus is to provide

*a*review

*of*families

*of*lattice-free polyhedra and their use

*in*

*a*disjunctive programming approach. ... This paper gives an introduction to

*a*recently established link between

*the*geometry

*of*numbers and mixed

*integer*optimization. ... Note that every strip L

*in*S satisfies

*the*following condition: for every facet F

*of*L ,

*the*affine

*hull*

*of*F

*contains*integral

*points*. ...

##
###
Klein polyhedra and lattices with positive norm minima

2007
*
Journal de Théorie des Nombres de Bordeaux
*

*The*

*integer*length

*of*

*a*segment with endpoints

*in*Λ α is defined as

*the*number

*of*lattice

*points*

*contained*

*in*

*the*interior

*of*this segment plus one. ...

*The*convex

*hull*

*of*

*the*nonzero

*points*

*of*

*the*lattice Λ α with nonnegative coordinates (

*in*

*the*initial unit basis

*of*R 2 ) is called

*a*Klein polygon. ...

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