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### Reverse Chvátal-Gomory Rank [chapter]

Michele Conforti, Alberto Del Pia, Marco Di Summa, Yuri Faenza, Roland Grappe
2013 Lecture Notes in Computer Science
We introduce 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]

Yan Gérard
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]

Amitabh Basu, Hongyi Jiang
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]

Benoît Meister
2004 Lecture Notes in Computer Science
This paper presents 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

Nickolay M. Korneenko, Nickolay N. Metelskij
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]

S. O. Semenov, N. Yu. Zolotykh
2020 arXiv   pre-print
We propose 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]

Gennadiy Averkov, Michele Conforti, Alberto Del Pia, Marco Di Summa,, Yuri Faenza
2012 arXiv   pre-print
closure operator have to be performed on Q to obtain 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

Robert Hildebrand, Timm Oertel, Robert Weismantel
2015 Operations Research Letters
We study 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  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]

Danny Calegari, Alden Walker
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

Danny Calegari, Alden Walker
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]

Vladimir Kovalevsky, Henrik Schulz
2004 Lecture Notes in Computer Science
This paper describes 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]

Robert Hildebrand, Timm Oertel, Robert Weismantel
2015 arXiv   pre-print
We study 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  gave a polynomial time algorithm in fixeddimension to enumerate the vertices of the integer hull of a polyhedron.  ...

### Reverse Chvátal-Gomory rank [article]

Michele Conforti, Alberto Del Pia, Marco Di Summa, Yuri Faenza, and Roland Grappe
2014 arXiv   pre-print
We introduce 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  , that lead to Lemma 18.  ...

### Relaxations of mixed integer sets from lattice-free polyhedra

Alberto Del Pia, Robert Weismantel
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

Oleg German
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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