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Survey on the Tukey theory of ultrafilters
This article surveys results regarding the Tukey theory of ultrafilters on
countable base sets. The driving forces for this investigation are Isbell's
Problem and the question of how closely related the Rudin-Keisler and Tukey
reducibilities are. We review work on the possible structures of cofinal types
and conditions which guarantee that an ultrafilter is below the Tukey maximum.
The known canonical forms for cofinal maps on ultrafilters are reviewed, as
well as their applications to finding which structures embed into the Tukey
types of ultrafilters. With the addition of some Ramsey theory, fine analyses
of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page
Data Reductions and Combinatorial Bounds for Improved Approximation Algorithms
Kernelization algorithms in the context of Parameterized Complexity are often
based on a combination of reduction rules and combinatorial insights. We will
expose in this paper a similar strategy for obtaining polynomial-time
approximation algorithms. Our method features the use of
approximation-preserving reductions, akin to the notion of parameterized
reductions. We exemplify this method to obtain the currently best approximation
algorithms for \textsc{Harmless Set}, \textsc{Differential} and
\textsc{Multiple Nonblocker}, all of them can be considered in the context of
securing networks or information propagation
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