4,165 research outputs found
Algorithm for Finding -Vertex Out-trees and its Application to -Internal Out-branching Problem
An out-tree is an oriented tree with only one vertex of in-degree zero. A
vertex of is internal if its out-degree is positive. We design
randomized and deterministic algorithms for deciding whether an input digraph
contains a given out-tree with vertices. The algorithms are of runtime
and , respectively. We apply the
deterministic algorithm to obtain a deterministic algorithm of runtime
, where is a constant, for deciding whether an input digraph
contains a spanning out-tree with at least internal vertices. This answers
in affirmative a question of Gutin, Razgon and Kim (Proc. AAIM'08)
A -Vertex Kernel for Maximum Internal Spanning Tree
We consider the parameterized version of the maximum internal spanning tree
problem, which, given an -vertex graph and a parameter , asks for a
spanning tree with at least internal vertices. Fomin et al. [J. Comput.
System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a
simple application of this rule is sufficient to yield a -vertex kernel.
Here we propose a novel way to use the same reduction rule, resulting in an
improved -vertex kernel. Our algorithm applies first a greedy procedure
consisting of a sequence of local exchange operations, which ends with a
local-optimal spanning tree, and then uses this special tree to find a
reducible structure. As a corollary of our kernel, we obtain a deterministic
algorithm for the problem running in time
A Faster Exact Algorithm for the Directed Maximum Leaf Spanning Tree Problem
Given a directed graph , the Directed Maximum Leaf Spanning Tree
problem asks to compute a directed spanning tree (i.e., an out-branching) with
as many leaves as possible. By designing a Branch-and-Reduced algorithm
combined with the Measure & Conquer technique for running time analysis, we
show that the problem can be solved in time \Oh^*(1.9043^n) using polynomial
space. Hitherto, there have been only few examples. Provided exponential space
this run time upper bound can be lowered to \Oh^*(1.8139^n)
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
We develop two different methods to achieve subexponential time parameterized
algorithms for problems on sparse directed graphs. We exemplify our approaches
with two well studied problems.
For the first problem, {\sc -Leaf Out-Branching}, which is to find an
oriented spanning tree with at least leaves, we obtain an algorithm solving
the problem in time on directed graphs
whose underlying undirected graph excludes some fixed graph as a minor. For
the special case when the input directed graph is planar, the running time can
be improved to . The second example is a
generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc
-Internal Out-Branching}, which is to find an oriented spanning tree with at
least internal vertices. We obtain an algorithm solving the problem in time
on directed graphs whose underlying
undirected graph excludes some fixed apex graph as a minor. Finally, we
observe that for any , the {\sc -Directed Path} problem is
solvable in time , where is some
function of \ve.
Our methods are based on non-trivial combinations of obstruction theorems for
undirected graphs, kernelization, problem specific combinatorial structures and
a layering technique similar to the one employed by Baker to obtain PTAS for
planar graphs
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
Linear kernels for outbranching problems in sparse digraphs
In the -Leaf Out-Branching and -Internal Out-Branching problems we are
given a directed graph with a designated root and a nonnegative integer
. The question is to determine the existence of an outbranching rooted at
that has at least leaves, or at least internal vertices,
respectively. Both these problems were intensively studied from the points of
view of parameterized complexity and kernelization, and in particular for both
of them kernels with vertices are known on general graphs. In this
work we show that -Leaf Out-Branching admits a kernel with vertices
on -minor-free graphs, for any fixed family of graphs
, whereas -Internal Out-Branching admits a kernel with
vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page
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