1,331 research outputs found
Arc-Disjoint Paths and Trees in 2-Regular Digraphs
An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected
spanning subdigraph of D in which every vertex x != s has precisely one arc
entering (leaving) it and s has no arcs entering (leaving) it. We settle the
complexity of the following two problems:
1) Given a 2-regular digraph , decide if it contains two arc-disjoint
branchings B^+_u, B^-_v.
2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u
such that D remains connected after removing the arcs of B^+_u.
Both problems are NP-complete for general digraphs. We prove that the first
problem remains NP-complete for 2-regular digraphs, whereas the second problem
turns out to be polynomial when we do not prescribe the root in advance. We
also prove that, for 2-regular digraphs, the latter problem is in fact
equivalent to deciding if contains two arc-disjoint out-branchings. We
generalize this result to k-regular digraphs where we want to find a number of
pairwise arc-disjoint spanning trees and out-branchings such that there are k
in total, again without prescribing any roots.Comment: 9 pages, 7 figure
Parameterized Algorithms for Directed Maximum Leaf Problems
We prove that finding a rooted subtree with at least leaves in a digraph
is a fixed parameter tractable problem. A similar result holds for finding
rooted spanning trees with many leaves in digraphs from a wide family
that includes all strong and acyclic digraphs. This settles completely an open
question of Fellows and solves another one for digraphs in . Our
algorithms are based on the following combinatorial result which can be viewed
as a generalization of many results for a `spanning tree with many leaves' in
the undirected case, and which is interesting on its own: If a digraph of order with minimum in-degree at least 3 contains a rooted
spanning tree, then contains one with at least leaves
On Complexity of Minimum Leaf Out-branching Problem
Given a digraph , the Minimum Leaf Out-Branching problem (MinLOB) is the
problem of finding in an out-branching with the minimum possible number of
leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved
that MinLOB is polynomial time solvable for acyclic digraphs which are exactly
the digraphs of directed path-width (DAG-width, directed tree-width,
respectively) 0. We investigate how much one can extend this polynomiality
result. We prove that already for digraphs of directed path-width (directed
tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand,
we show that for digraphs of restricted directed tree-width (directed
path-width, DAG-width, respectively) and a fixed integer , the problem of
checking whether there is an out-branching with at most leaves is
polynomial time solvable
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
An FPT Algorithm for Directed Spanning k-Leaf
An out-branching of a directed graph is a rooted spanning tree with all arcs
directed outwards from the root. We consider the problem of deciding whether a
given directed graph D has an out-branching with at least k leaves (Directed
Spanning k-Leaf). We prove that this problem is fixed parameter tractable, when
k is chosen as the parameter. Previously this was only known for restricted
classes of directed graphs.
The main new ingredient in our approach is a lemma that shows that given a
locally optimal out-branching of a directed graph in which every arc is part of
at least one out-branching, either an out-branching with at least k leaves
exists, or a path decomposition with width O(k^3) can be found. This enables a
dynamic programming based algorithm of running time 2^{O(k^3 \log k)} n^{O(1)},
where n=|V(D)|.Comment: 17 pages, 8 figure
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