930 research outputs found
The Triangle Closure is a Polyhedron
Recently, cutting planes derived from maximal lattice-free convex sets have
been studied intensively by the integer programming community. An important
question in this research area has been to decide whether the closures
associated with certain families of lattice-free sets are polyhedra. For a long
time, the only result known was the celebrated theorem of Cook, Kannan and
Schrijver who showed that the split closure is a polyhedron. Although some
fairly general results were obtained by Andersen, Louveaux and Weismantel [ An
analysis of mixed integer linear sets based on lattice point free convex sets,
Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated
closures in the theory of cutting planes, Discrete Optimization 9 (2012), no.
4, 209--215], some basic questions have remained unresolved. For example,
maximal lattice-free triangles are the natural family to study beyond the
family of splits and it has been a standing open problem to decide whether the
triangle closure is a polyhedron. In this paper, we show that when the number
of integer variables the triangle closure is indeed a polyhedron and its
number of facets can be bounded by a polynomial in the size of the input data.
The techniques of this proof are also used to give a refinement of necessary
conditions for valid inequalities being facet-defining due to Cornu\'ejols and
Margot [On the facets of mixed integer programs with two integer variables and
two constraints, Mathematical Programming 120 (2009), 429--456] and obtain
polynomial complexity results about the mixed integer hull.Comment: 39 pages; made self-contained by merging material from
arXiv:1107.5068v
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
On dominating set polyhedra of circular interval graphs
Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. [18] take advantage of this relationship to propose a complete linear description of the stable set polytope on circular interval graphs. In this paper we follow similar ideas to obtain a complete description of the dominating set polytope on the same class of graphs. As in the packing case, our results are established for a larger class of covering polyhedra of the form Q ∗ (A, b) := conv {x ∈ Z n + : Ax ≥ b}, with A a circular matrix and b an integer vector. These results also provide linear descriptions of polyhedra associated with several variants of the dominating set problem on circular interval graphs.Fil: Bianchi, Silvia MarÃa. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, IngenierÃa y Agrimensura; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, IngenierÃa y Agrimensura; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario; ArgentinaFil: Tolomei, Paola Beatriz. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, IngenierÃa y Agrimensura; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario; ArgentinaFil: Torres, Luis Miguel. Escuela Politécnica Nacional; Ecuado
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