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Approximating Upper Degree-Constrained Partial Orientations
In the Upper Degree-Constrained Partial Orientation problem we are given an
undirected graph , together with two degree constraint functions
. The goal is to orient as many edges as possible,
in such a way that for each vertex the number of arcs entering is
at most , whereas the number of arcs leaving is at most .
This problem was introduced by Gabow [SODA'06], who proved it to be MAXSNP-hard
(and thus APX-hard). In the same paper Gabow presented an LP-based iterative
rounding -approximation algorithm.
Since the problem in question is a special case of the classic 3-Dimensional
Matching, which in turn is a special case of the -Set Packing problem, it is
reasonable to ask whether recent improvements in approximation algorithms for
the latter two problems [Cygan, FOCS'13; Sviridenko & Ward, ICALP'13] allow for
an improved approximation for Upper Degree-Constrained Partial Orientation. We
follow this line of reasoning and present a polynomial-time local search
algorithm with approximation ratio . Our algorithm uses a
combination of two types of rules: improving sets of bounded pathwidth from the
recent -approximation algorithm for 3-Set Packing [Cygan,
FOCS'13], and a simple rule tailor-made for the setting of partial
orientations. In particular, we exploit the fact that one can check in
polynomial time whether it is possible to orient all the edges of a given graph
[Gy\'arf\'as & Frank, Combinatorics'76].Comment: 12 pages, 1 figur
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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