705 research outputs found
The Salesman's Improved Tours for Fundamental Classes
Finding the exact integrality gap for the LP relaxation of the
metric Travelling Salesman Problem (TSP) has been an open problem for over
thirty years, with little progress made. It is known that , and a famous conjecture states . For this problem,
essentially two "fundamental" classes of instances have been proposed. This
fundamental property means that in order to show that the integrality gap is at
most for all instances of metric TSP, it is sufficient to show it only
for the instances in the fundamental class. However, despite the importance and
the simplicity of such classes, no apparent effort has been deployed for
improving the integrality gap bounds for them. In this paper we take a natural
first step in this endeavour, and consider the -integer points of one such
class. We successfully improve the upper bound for the integrality gap from
to for a superclass of these points, as well as prove a lower
bound of for the superclass. Our methods involve innovative applications
of tools from combinatorial optimization which have the potential to be more
broadly applied
General duality for abelian-group-valued statistical-mechanics models
We introduce a general class of statistical-mechanics models, taking values
in an abelian group, which includes examples of both spin and gauge models,
both ordered and disordered. The model is described by a set of ``variables''
and a set of ``interactions''. A Gibbs factor is associated to each variable
and to each interaction. We introduce a duality transformation for systems in
this class. The duality exchanges the abelian group with its dual, the Gibbs
factors with their Fourier transforms, and the interactions with the variables.
High (low) couplings in the interaction terms are mapped into low (high)
couplings in the one-body terms. The idea is that our class of systems extends
the one for which the classical procedure 'a la Kramers and Wannier holds, up
to include randomness into the pattern of interaction. We introduce and study
some physical examples: a random Gaussian Model, a random Potts-like model, and
a random variant of discrete scalar QED. We shortly describe the consequence of
duality for each example.Comment: 26 pages, 2 Postscript figure
Excluding Kuratowski graphs and their duals from binary matroids
We consider some applications of our characterisation of the internally
4-connected binary matroids with no M(K3,3)-minor. We characterise the
internally 4-connected binary matroids with no minor in some subset of
{M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We
also describe a practical algorithm for testing whether a binary matroid has a
minor in the subset. In addition we characterise the growth-rate of binary
matroids with no M(K3,3)-minor, and we show that a binary matroid with no
M(K3,3)-minor has critical exponent over GF(2) at most equal to four.Comment: Some small change
Quaternionic Geometry of Matroids
Building on a recent joint paper with Sturmfels, here we argue that the
combinatorics of matroids is intimately related to the geometry and topology of
toric hyperkaehler varieties. We show that just like toric varieties occupy a
central role in Stanley's proof for the necessity of McMullen's conjecture (or
g-inequalities) about the classification of face vectors of simplicial
polytopes, the topology of toric hyperkaehler varieties leads to new
restrictions on face vectors of matroid complexes. Namely in this paper we give
two proofs that the injectivity part of the Hard Lefschetz theorem survives for
toric hyperkaehler varieties. We explain how this implies the g-inequalities
for rationally representable matroids. We show how the geometrical intuition in
the first proof, coupled with results of Chari, leads to a proof of the
g-inequalities for general matroid complexes, which is a recent result of
Swartz. The geometrical idea in the second proof will show that a pure
O-sequence should satisfy the g-inequalities, thus showing that our result is
in fact a consequence of a long-standing conjecture of Stanley.Comment: 11 page
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