3,345 research outputs found
Low-Degree Spanning Trees of Small Weight
The degree-d spanning tree problem asks for a minimum-weight spanning tree in
which the degree of each vertex is at most d. When d=2 the problem is TSP, and
in this case, the well-known Christofides algorithm provides a
1.5-approximation algorithm (assuming the edge weights satisfy the triangle
inequality).
In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of
finding an algorithm with performance guarantee less than 2 for Euclidean
graphs (points in R^n) and d > 2. This paper gives the first answer to that
challenge, presenting an algorithm to compute a degree-3 spanning tree of cost
at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2
and the algorithm can also find a degree-4 spanning tree of cost at most 5/4
times the MST.Comment: conference version in Symposium on Theory of Computing (1994
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints
We introduce a new structure for a set of points in the plane and an angle
, which is similar in flavor to a bounded-degree MST. We name this
structure -MST. Let be a set of points in the plane and let be an angle. An -ST of is a spanning tree of the
complete Euclidean graph induced by , with the additional property that for
each point , the smallest angle around containing all the edges
adjacent to is at most . An -MST of is then an
-ST of of minimum weight. For , an -ST does
not always exist, and, for , it always exists. In this paper,
we study the problem of computing an -MST for several common values of
.
Motivated by wireless networks, we formulate the problem in terms of
directional antennas. With each point , we associate a wedge of
angle and apex . The goal is to assign an orientation and a radius
to each wedge , such that the resulting graph is connected and its
MST is an -MST. (We draw an edge between and if , , and .) Unsurprisingly, the problem of computing an
-MST is NP-hard, at least for and . We
present constant-factor approximation algorithms for .
One of our major results is a surprising theorem for ,
which, besides being interesting from a geometric point of view, has important
applications. For example, the theorem guarantees that given any set of
points in the plane and any partitioning of the points into triplets,
one can orient the wedges of each triplet {\em independently}, such that the
graph induced by is connected. We apply the theorem to the {\em antenna
conversion} problem
Approximation Algorithms for Generalized MST and TSP in Grid Clusters
We consider a special case of the generalized minimum spanning tree problem
(GMST) and the generalized travelling salesman problem (GTSP) where we are
given a set of points inside the integer grid (in Euclidean plane) where each
grid cell is . In the MST version of the problem, the goal is to
find a minimum tree that contains exactly one point from each non-empty grid
cell (cluster). Similarly, in the TSP version of the problem, the goal is to
find a minimum weight cycle containing one point from each non-empty grid cell.
We give a and -approximation
algorithm for these two problems in the described setting, respectively.
Our motivation is based on the problem posed in [7] for a constant
approximation algorithm. The authors designed a PTAS for the more special case
of the GMST where non-empty cells are connected end dense enough. However,
their algorithm heavily relies on this connectivity restriction and is
unpractical. Our results develop the topic further
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