171 research outputs found
A superlinear bound on the number of perfect matchings in cubic bridgeless graphs
Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs
have exponentially many perfect matchings. This conjecture has been verified
for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky
and Seymour in 2008, but in general only linear bounds are known. In this
paper, we provide the first superlinear bound in the general case.Comment: 54 pages v2: a short (missing) proof of Lemma 10 was adde
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
The stacker crane problem and the directed general routing problem
This is the peer reviewed version of the following article: Ávila, Thais , Corberán, Angel, Plana, Isaac, Sanchís Llopis, José María. (2015). The stacker crane problem and the directed general routing problem.Networks, 65, 1, 43-55. DOI: 10.1002/net.21591
, which has been published in final form at http://doi.org/10.1002/net.21591. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving[EN] This article deals with the polyhedral description and the resolution of the directed general routing problem (DGRP) and the stacker crane problem (SCP). The DGRP contains a large number of important arc and node routing problems as special cases, including the SCP. Large families of facet-defining inequalities for the DGRP are described and a branch-and-cut algorithm for these problems is presented. Extensive computational experiments over different sets of DGRP and SCP instances are included.Contract grant sponsor: Ministerio de Economía y Competitividad (project MTM2012-36163-C06-02) of Spain Contract grant sponsor: Generalitat Valenciana (project GVPROMETEO2013-049)Ávila, T.; Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2015). The stacker crane problem and the directed general routing problem. Networks. 65(1):43-55. https://doi.org/10.1002/net.21591S435565
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
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