19,977 research outputs found
Max-Leaves Spanning Tree is APX-hard for Cubic Graphs
We consider the problem of finding a spanning tree with maximum number of
leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a
3/2-approximation algorithm when restricted to graphs where every vertex has
degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and
NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic
graphs. The APX-hardness of the related problem Minimum Connected Dominating
Set for cubic graphs follows
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)
Distributed Connectivity Decomposition
We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
(I) A decomposition of each undirected graph with vertex-connectivity
into (fractionally) vertex-disjoint weighted dominating trees with total weight
, in rounds.
(II) A decomposition of each undirected graph with edge-connectivity
into (fractionally) edge-disjoint weighted spanning trees with total
weight , in
rounds.
We also show round complexity lower bounds of
and
for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from
to near-optimal .
As corollaries, we also get distributed oblivious routing broadcast with
-competitive edge-congestion and -competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal -approximation of vertex connectivity: centralized
and distributed . The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation
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