42,963 research outputs found
Maximum Edge-Disjoint Paths in -sums of Graphs
We consider the approximability of the maximum edge-disjoint paths problem
(MEDP) in undirected graphs, and in particular, the integrality gap of the
natural multicommodity flow based relaxation for it. The integrality gap is
known to be even for planar graphs due to a simple
topological obstruction and a major focus, following earlier work, has been
understanding the gap if some constant congestion is allowed.
In this context, it is natural to ask for which classes of graphs does a
constant-factor constant-congestion property hold. It is easy to deduce that
for given constant bounds on the approximation and congestion, the class of
"nice" graphs is nor-closed. Is the converse true? Does every proper
minor-closed family of graphs exhibit a constant factor, constant congestion
bound relative to the LP relaxation? We conjecture that the answer is yes.
One stumbling block has been that such bounds were not known for bounded
treewidth graphs (or even treewidth 3). In this paper we give a polytime
algorithm which takes a fractional routing solution in a graph of bounded
treewidth and is able to integrally route a constant fraction of the LP
solution's value. Note that we do not incur any edge congestion. Previously
this was not known even for series parallel graphs which have treewidth 2. The
algorithm is based on a more general argument that applies to -sums of
graphs in some graph family, as long as the graph family has a constant factor,
constant congestion bound. We then use this to show that such bounds hold for
the class of -sums of bounded genus graphs
A Note on the Maximum Genus of Graphs with Diameter 4
Let G be a simple graph with diameter four, if G does not contain complete
subgraph K3 of order three
-invariant symplectic hypersurfaces in dimension and the Fano condition
We prove that any symplectic Fano -manifold with a Hamiltonian
-action is simply connected and satisfies . This is done by
showing that the fixed submanifold on which the
Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface,
a -sphere or a point. In the case when , we use the fact
that symplectic Fano -manifolds are symplectomorphic to del Pezzo surfaces.
The case when involves a study of -dimensional
Hamiltonian -manifolds with diffeomorphic to a surface of
positive genus. By exploiting an analogy with the algebro-geometric situation
we construct in each such -manifold an -invariant symplectic
hypersurface playing the role of a smooth fibre of a hypothetical
Mori fibration over . This relies upon applying Seiberg-Witten theory
to the resolution of symplectic -orbifolds occurring as the reduced spaces
of .Comment: Exposition improve
Embedding of metric graphs on hyperbolic surfaces
An embedding of a metric graph on a closed hyperbolic surface is
\emph{essential}, if each complementary region has a negative Euler
characteristic. We show, by construction, that given any metric graph, its
metric can be rescaled so that it admits an essential and isometric embedding
on a closed hyperbolic surface. The essential genus of is the
lowest genus of a surface on which such an embedding is possible. In the next
result, we establish a formula to compute . Furthermore, we show that
for every integer , admits such an embedding (possibly
after a rescaling of ) on a surface of genus .
Next, we study minimal embeddings where each complementary region has Euler
characteristic . The maximum essential genus of is
the largest genus of a surface on which the graph is minimally embedded.
Finally, we describe a method explicitly for an essential embedding of , where and are realized.Comment: Revised version, 11 pages, 3 figure
Refined floor diagrams from higher genera and lambda classes
We show that, after the change of variables , refined floor
diagrams for and Hirzebruch surfaces compute generating series
of higher genus relative Gromov-Witten invariants with insertion of a lambda
class. The proof uses an inductive application of the degeneration formula in
relative Gromov-Witten theory and an explicit result in relative Gromov-Witten
theory of . Combining this result with the similar looking
refined tropical correspondence theorem for log Gromov-Witten invariants, we
obtain some non-trivial relation between relative and log Gromov-Witten
invariants for and Hirzebruch surfaces. We also prove that the
Block-G\"ottsche invariants of and are related by
the Abramovich-Bertram formula.Comment: 44 pages, 8 figures, revised version, exposition greatly improved,
main results unchanged, published in Selecta Mathematic
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