3,719 research outputs found
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
Minimum Cuts in Near-Linear Time
We significantly improve known time bounds for solving the minimum cut
problem on undirected graphs. We use a ``semi-duality'' between minimum cuts
and maximum spanning tree packings combined with our previously developed
random sampling techniques. We give a randomized algorithm that finds a minimum
cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We
also give a simpler randomized algorithm that finds all minimum cuts with high
probability in O(n^2 log n) time. This variant has an optimal RNC
parallelization. Both variants improve on the previous best time bound of O(n^2
log^3 n). Other applications of the tree-packing approach are new, nearly tight
bounds on the number of near minimum cuts a graph may have and a new data
structure for representing them in a space-efficient manner
Spanning directed trees with many leaves
The {\sc Directed Maximum Leaf Out-Branching} problem is to find an
out-branching (i.e. a rooted oriented spanning tree) in a given digraph with
the maximum number of leaves. In this paper, we obtain two combinatorial
results on the number of leaves in out-branchings. We show that
- every strongly connected -vertex digraph with minimum in-degree at
least 3 has an out-branching with at least leaves;
- if a strongly connected digraph does not contain an out-branching with
leaves, then the pathwidth of its underlying graph UG() is .
Moreover, if the digraph is acyclic, the pathwidth is at most .
The last result implies that it can be decided in time whether a strongly connected digraph on vertices has an
out-branching with at least leaves. On acyclic digraphs the running time of
our algorithm is
- …