Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Approximating bottleneck spanning trees on partitioned tuples of points   We present approximation algorithms for the following NP-hard optimization
problems related to bottleneck spanning trees in metric spaces.
  1. The disjoint bottleneck spanning tree problem: Given $n$ pairs of points
in a metric space, find two disjoint trees each containing exactly one point
from each pair and minimize the largest edge length (over all edges of both
trees). It is known that approximating this problem by a factor better than 2
is NP-hard. We present a 4-approximation algorithm for this problem. This
improves upon the previous best known approximation ratio of $9$. Our algorithm
extends to a $(3k-2)$-approximation for a more general case where points are
partitioned into $k$-tuples and we seek $k$ disjoint trees.
  2. The generalized bottleneck spanning tree problem: Given $n$ points in some
metric space that are partitioned into clusters of size at most 2, find a tree
that contains exactly one point from each cluster and minimizes the largest
edge length. We show that it is NP-hard to approximate this problem by a factor
better than 2, and present a 3-approximation algorithm.
  3. The partitioned bottleneck spanning tree problem: Given $kn$ points in
some metric space, find $k$ trees each containing exactly $n$ points and
minimize the largest edge length (over all edges of the $k$ trees). We show
that it is NP-hard to approximate this problem by a factor better than 2 for
any $k\ge 2$. We present an $\alpha$-approximation algorithm for this problem
where $\alpha=2$ for $k=2,3$ and $\alpha=3$ for $k\ge 4$. Towards obtaining
these approximation ratios we present tight upper bounds on the edge lengths of
$k$ equal-size disjoint trees that can be obtained from the nodes of a given
tree. This result is of independent interest.
  Our hardness proofs imply that it is NP-hard to approximate the non-metric
version of the above problems within any constant factor.

16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex