22,862 research outputs found
Local symplectic field theory
Generalizing local Gromov-Witten theory, in this paper we define a local
version of symplectic field theory. When the symplectic manifold with
cylindrical ends is four-dimensional and the underlying simple curve is regular
by automatic transversality, we establish a transversality result for all its
multiple covers and discuss the resulting algebraic structures
Enumeration of paths and cycles and e-coefficients of incomparability graphs
We prove that the number of Hamiltonian paths on the complement of an acyclic
digraph is equal to the number of cycle covers. As an application, we obtain a
new expansion of the chromatic symmetric function of incomparability graphs in
terms of elementary symmetric functions. Analysis of some of the combinatorial
implications of this expansion leads to three bijections involving acyclic
orientations
Gravitational descendants in symplectic field theory
It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the
rich algebraic formalism of symplectic field theory leads to a natural
appearance of quantum and classical integrable systems, at least in the case
when the contact manifold is the prequantization space of a symplectic
manifold. In this paper we generalize the definition of gravitational
descendants in SFT from circle bundles in the Morse-Bott case to general
contact manifolds. After we have shown that for the basic examples of
holomorphic curves in SFT, that is, branched covers of cylinders over closed
Reeb orbits, the gravitational descendants have a geometric interpretation in
terms of branching conditions, we compute the corresponding sequences of
Poisson-commuting functions when the contact manifold is the unit cotangent
bundle of a Riemannian manifold.Comment: 44 pages, no figure
Approximation Algorithms for Multi-Criteria Traveling Salesman Problems
In multi-criteria optimization problems, several objective functions have to
be optimized. Since the different objective functions are usually in conflict
with each other, one cannot consider only one particular solution as the
optimal solution. Instead, the aim is to compute a so-called Pareto curve of
solutions. Since Pareto curves cannot be computed efficiently in general, we
have to be content with approximations to them.
We design a deterministic polynomial-time algorithm for multi-criteria
g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate
Pareto curves for all 1/2<=g<=1. In particular, we obtain a
(2+eps)-approximation for multi-criteria metric STSP. We also present two
randomized approximation algorithms for multi-criteria g-metric STSP that
achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1
+3g -4g^2) + eps, respectively.
Moreover, we present randomized approximation algorithms for multi-criteria
g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with
weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do
this, we design randomized approximation schemes for multi-criteria cycle cover
and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented
at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006
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