276 research outputs found
The Projected Faces Property and Polyhedral Relations
Margot (1994) in his doctoral dissertation studied extended formulations of
combinatorial polytopes that arise from "smaller" polytopes via some
composition rule. He introduced the "projected faces property" of a polytope
and showed that this property suffices to iteratively build extended
formulations of composed polytopes.
For the composed polytopes, we show that an extended formulation of the type
studied in this paper is always possible only if the smaller polytopes have the
projected faces property. Therefore, this produces a characterization of the
projected faces property.
Affinely generated polyhedral relations were introduced by Kaibel and
Pashkovich (2011) to construct extended formulations for the convex hull of the
images of a point under the action of some finite group of reflections. In this
paper we prove that the projected faces property and affinely generated
polyhedral relation are equivalent conditions
Optimality certificates for convex minimization and Helly numbers
We consider the problem of minimizing a convex function over a subset of R^n
that is not necessarily convex (minimization of a convex function over the
integer points in a polytope is a special case). We define a family of duals
for this problem and show that, under some natural conditions, strong duality
holds for a dual problem in this family that is more restrictive than
previously considered duals.Comment: 5 page
On the convergence of the affine hull of the Chv\'atal-Gomory closures
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 closure operator have to be
performed on Q to obtain a polyhedron contained in the affine hull of P. We
show that if P contains an integer point in its relative interior, then such a
number of iterations can be bounded by a function depending only on n. On the
other hand, we prove that if P is not full-dimensional and does not contain any
integer point in its relative interior, then no finite bound on the number of
iterations exists.Comment: 13 pages, 2 figures - the introduction has been extended and an extra
chapter has been adde
Ki-covers I: Complexity and polytopes
AbstractA Ki in a graph is a complete subgraph of size i. A Ki-cover of a graph G(V, E is a set C of Ki â 1's of G such that every Ki in G contains at least one Ki â 1 in C. Thus a K2-cover is a vertex cover. The problem of determining whether a graph has a Ki-cover (i â©Ÿ 2) of cardinality â©œk is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i â©Ÿ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the Ki-cover polytope. Further results of this type are also produced for the K3-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the Ki-cover polytope
Cut-generating functions and S-free sets
International audienceWe consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts
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