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Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Dušan Knop (TU Berlin)

We study the Steiner Tree problem, in which a set of terminal vertices needs to
be connected in the cheapest possible way in an edge-weighted graph. This
problem has been extensively studied from the viewpoint of approximation and
also parametrization. In particular, on one hand Steiner Tree is known to be
APX-hard, and W[2]-hard on the other, if parameterized by the number of
non-terminals (Steiner vertices) in the optimum solution. In contrast to this
we give an efficient parameterized approximation scheme (EPAS), which
circumvents both hardness results. Moreover, our methods imply the existence of
a polynomial size approximate kernelization scheme (PSAKS) for the considered
parameter. We further study the parameterized approximability of other variants
of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither
of these an EPAS is likely to exist for the studied parameter: for Steiner
Forest an easy observation shows that the problem is APX-hard, even if the
input graph contains no Steiner vertices. For Directed Steiner Tree we prove
that approximating within any function of the studied parameter is W[1]-hard.
Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree,
but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for
Steiner Forest if in addition to the number of Steiner vertices, the number of
connected components of an optimal solution is considered to be a parameter. I
will further discuss some implementation details and experimental results.

The talk is based on a paper with the same title coauthored by Pavel Dvořák,
Andreas Emil Feldmann, Tomáš Masařík, Tomáš Toufar, and Pavel Veselý.
The practical part is based on an implementation submitted to PACE 2018
coauthored by Eduard Eiben, Radek HuSek, Tomáš Masařík, and Tomáš Toufar.



Dušan Knop
TEL 512

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