128,358 research outputs found
The gap between Gromov-vague and Gromov-Hausdorff-vague topology
In Athreya, L\"ohr, Winter (2016), an invariance principle is stated for a
class of strong Markov processes on tree-like metric measure spaces. It is
shown that if the underlying spaces converge Gromov vaguely, then the processes
converge in the sense of finite dimensional distributions. Further, if the
underlying spaces converge Gromov-Hausdorff vaguely, then the processes
converge weakly in path space. In this paper we systematically introduce and
study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of
equivalence classes of metric boundedly finite measure spaces. The latter
topology is closely related to the Gromov-Hausdorff-Prohorov metric which is
defined on different equivalence classes of metric measure spaces.
We explain the necessity of these two topologies via several examples, and
close the gap between them. That is, we show that convergence in Gromov-vague
topology implies convergence in Gromov-Hausdorff-vague topology if and only if
the so-called lower mass-bound property is satisfied. Furthermore, we prove and
disprove Polishness of several spaces of metric measure spaces in the
topologies mentioned above (summarized in Figure~1).
As an application, we consider the Galton-Watson tree with critical offspring
distribution of finite variance conditioned to not get extinct, and construct
the so-called Kallenberg-Kesten tree as the weak limit in
Gromov-Hausdorff-vague topology when the edge length are scaled down to go to
zero
A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces
We present an extension of the Gromov-Hausdorff metric on the set of compact
metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact
metric spaces endowed with a finite measure. We then extend it to the
non-compact case by describing a metric on the set of rooted complete locally
compact length spaces endowed with a locally finite measure. We prove that this
space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space.
This generalization is needed to define L\'evy trees, which are (possibly
unbounded) random real trees endowed with a locally finite measure
An extension of disjunctive programming and its impact for compact tree formulations
In the 1970's, Balas introduced the concept of disjunctive programming, which
is optimization over unions of polyhedra. One main result of his theory is
that, given linear descriptions for each of the polyhedra to be taken in the
union, one can easily derive an extended formulation of the convex hull of the
union of these polyhedra. In this paper, we give a generalization of this
result by extending the polyhedral structure of the variables coupling the
polyhedra taken in the union. Using this generalized concept, we derive
polynomial size linear programming formulations (compact formulations) for a
well-known spanning tree approximation of Steiner trees, for Gomory-Hu trees,
and, as a consequence, of the minimum -cut problem (but not for the
associated -cut polyhedron). Recently, Kaibel and Loos (2010) introduced a
more involved framework called {\em polyhedral branching systems} to derive
extended formulations. The most parts of our model can be expressed in terms of
their framework. The value of our model can be seen in the fact that it
completes their framework by an interesting algorithmic aspect.Comment: 17 page
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