5 research outputs found
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
Kernel(s) for Problems With no Kernel: On Out-Trees With Many Leaves
The {\sc -Leaf Out-Branching} problem is to find an out-branching (i.e. a
rooted oriented spanning tree) with at least leaves in a given digraph. The
problem has recently received much attention from the viewpoint of
parameterized algorithms {alonLNCS4596,AlonFGKS07fsttcs,BoDo2,KnLaRo}. In this
paper we step aside and take a kernelization based approach to the {\sc
-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc
Rooted -Leaf-Out-Branching}, a variant of {\sc -Leaf-Out-Branching} where
the root of the tree searched for is also a part of the input. Our kernel has
cubic size and is obtained using extremal combinatorics.
For the {\sc -Leaf-Out-Branching} problem we show that no polynomial
kernel is possible unless polynomial hierarchy collapses to third level
% by applying a recent breakthrough result by Bodlaender et al.
{BDFH08} in a non-trivial fashion. However our positive results for {\sc Rooted
-Leaf-Out-Branching} immediately imply that the seemingly intractable the
{\sc -Leaf-Out-Branching} problem admits a data reduction to independent
kernels. These two results, tractability and intractability side by
side, are the first separating {\it many-to-one kernelization} from {\it Turing
kernelization}. This answers affirmatively an open problem regarding "cheat
kernelization" raised in {IWPECOPEN08}
On the Approximability of Some Maximum Spanning Tree Problems
AbstractWe study the approximability of some problems which aim at finding spanning trees in undirected graphs which maximize, rather than minimize, a single objective function representing a form of benefit or usefulness of the tree. We prove that the problem of finding a spanning tree which maximizes the number of paths which connect pairs of vertices and pass through a common arc can be polynomially approximated within 98. It is known that this problem can be solved exactly in polynomial time if the graph is 2-connected; we extend this result to graphs having at most two articulation points. We leave open whether in the general case the problem admits a polynomial time approximation scheme or is MAX-SNP hard and therefore not polynomially approximable within 1 + ε, for any fixed ε > 0, unless P = NP. On the other hand, we show that the problems of finding a spanning tree which has maximum diameter, or maximum height with respect to a specified root, or maximum sum of the distances between all pairs of vertices, or maximum sum of the distances from a specified root to all remaining vertices, are not polynomially approximable within any constant factor, unless P = NP. The same result holds for the problem of finding a lineal spanning tree with maximum height, and this solves a problem which was left open in Fellows et al. (1988)