2,790 research outputs found
Towards optimal kernel for connected vertex cover in planar graphs
We study the parameterized complexity of the connected version of the vertex
cover problem, where the solution set has to induce a connected subgraph.
Although this problem does not admit a polynomial kernel for general graphs
(unless NP is a subset of coNP/poly), for planar graphs Guo and Niedermeier
[ICALP'08] showed a kernel with at most 14k vertices, subsequently improved by
Wang et al. [MFCS'11] to 4k. The constant 4 here is so small that a natural
question arises: could it be already an optimal value for this problem? In this
paper we answer this quesion in negative: we show a (11/3)k-vertex kernel for
Connected Vertex Cover in planar graphs. We believe that this result will
motivate further study in search for an optimal kernel
The Minimum Shared Edges Problem on Grid-like Graphs
We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide
whether it is possible to route paths from a start vertex to a target
vertex in a given graph while using at most edges more than once. We show
that MSE can be decided on bounded (i.e. finite) grids in linear time when both
dimensions are either small or large compared to the number of paths. On
the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids.
Finally, we study MSE from a parametrised complexity point of view. It is known
that MSE is fixed-parameter tractable with respect to the number of paths.
We show that, under standard complexity-theoretical assumptions, the problem
parametrised by the combined parameter , , maximum degree, diameter, and
treewidth does not admit a polynomial-size problem kernel, even when restricted
to planar graphs
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices
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.Comment: 23 pages, 6 figures An extended abstract appeared in proceedings of
STACS 201
Hitting forbidden minors: Approximation and Kernelization
We study a general class of problems called F-deletion problems. In an
F-deletion problem, we are asked whether a subset of at most vertices can
be deleted from a graph such that the resulting graph does not contain as a
minor any graph from the family F of forbidden minors.
We obtain a number of algorithmic results on the F-deletion problem when F
contains a planar graph. We give (1) a linear vertex kernel on graphs excluding
-claw , the star with leves, as an induced subgraph, where
is a fixed integer. (2) an approximation algorithm achieving an approximation
ratio of , where is the size of an optimal solution on
general undirected graphs. Finally, we obtain polynomial kernels for the case
when F contains graph as a minor for a fixed integer . The graph
consists of two vertices connected by parallel edges. Even
though this may appear to be a very restricted class of problems it already
encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback
Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is
based on a non-trivial application of protrusion techniques, previously used
only for problems on topological graph classes
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution
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