5,195 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
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
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
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme
The Traveling Salesman Problem (TSP) is among the most famous NP-hard
optimization problems. We design for this problem a randomized polynomial-time
algorithm that computes a (1+eps)-approximation to the optimal tour, for any
fixed eps>0, in TSP instances that form an arbitrary metric space with bounded
intrinsic dimension.
The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the
above result holds in the special case of TSP in a fixed-dimensional Euclidean
space. Thus, our algorithm demonstrates that the algorithmic tractability of
metric TSP depends on the dimensionality of the space and not on its specific
geometry. This result resolves a problem that has been open since the
quasi-polynomial time algorithm of Talwar (T-04)
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