60 research outputs found
Finding Short Paths on Polytopes by the Shadow Vertex Algorithm
We show that the shadow vertex algorithm can be used to compute a short path
between a given pair of vertices of a polytope P = {x : Ax \leq b} along the
edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the
length of the path and the running time of the algorithm, are polynomial in m,
n, and a parameter 1/delta that is a measure for the flatness of the vertices
of P. For integer matrices A \in Z^{m \times n} we show a connection between
delta and the largest absolute value Delta of any sub-determinant of A,
yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This
bound is expressed in the same parameter Delta as the recent non-constructive
bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al.
For the special case of totally unimodular matrices, the length of the
computed path simplifies to O(m n^4), which significantly improves the
previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer
and Frieze
Quantum Computing with NMR
A review of progress in NMR quantum computing and a brief survey of the
literatureComment: Commissioned by Progress in NMR Spectroscopy (95 pages, no figures
Exponentially many perfect matchings in cubic graphs
We show that every cubic bridgeless graph G has at least 2^(|V(G)|/3656)
perfect matchings. This confirms an old conjecture of Lovasz and Plummer.
This version of the paper uses a different definition of a burl from the
journal version of the paper and a different proof of Lemma 18 is given. This
simplifies the exposition of our arguments throughout the whole paper
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to
George Dantzig. It states that the graph of a d-dimensional polytope with n
facets cannot have diameter greater than n - d.
Despite being one of the most fundamental, basic and old problems in polytope
theory, what we know is quite scarce. Most notably, no polynomial upper bound
is known for the diameters that are conjectured to be linear. In contrast, very
few polytopes are known where the bound is attained. This paper collects
known results and remarks both on the positive and on the negative side of the
conjecture. Some proofs are included, but only those that we hope are
accessible to a general mathematical audience without introducing too many
technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423
Hamiltonicity and combinatorial polyhedra
AbstractWe say that a polyhedron with 0–1 valued vertices is combinatorial if the midpoint of the line joining any pair of nonadjacent vertices is the midpoint of the line joining another pair of vertices. We show that the class of combinatorial polyhedra includes such well-known classes of polyhedra as matching polyhedra, matroid basis polyhedra, node packing or stable set polyhedra and permutation polyhedra. We show the graph of a combinatorial polyhedron is always either a hypercube (i.e., isomorphic to the convex hull of a k-dimension unit cube) or else is hamilton connected (every pair of nodes is the set of terminal nodes of a hamilton path). This implies several earlier results concerning special cases of combinatorial polyhedra
HALIN GRAPHS AND THE TRAVELLING SALESMAN PROBLEM
A Halin graph is obtained by imbedding a tree
having no degree two nodes in the plane, and then adding a cycle
to join the leaves of in such a way that the resulting
graph is planar. These graphs are edge minimal 3-connected, hamiltonian,
and in general have large numbers of hamilton cycles. We show that for
arbitrary real edge costs the travelling salesman problem can be
polynomially solved for such a graph, and we give an explicit
linear description of the travelling salesman polytope (the convex
hull of the incidence vectors of the hamilton cycles) for such
a graph.We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at [email protected]
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