Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: FPT-algorithms for computing Gromov-Hausdorff and interleaving distances
  between trees   Gromov-Hausdorff (GH) distance is a natural way to measure the distortion
between two metric spaces. However, there has been only limited algorithmic
development to compute or approximate this distance. We focus on computing the
Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric
tree is a metric space that can be realized by the shortest path metric on a
tree. Previously, Agarwal et al. showed that even for trees with unit edge
length, it is NP hard to approximate the GH distance between them within a
factor of 3. In this paper, we present a fixed-parameter tractable (FPT)
algorithm that can approximate the GH distance between two general metric trees
within a factor of 14.
  Interestingly, the development of our algorithm is made possible by a
connection between the GH distance for metric trees and the interleaving
distance for the so-called merge trees. The merge trees arise in practice
naturally as a simple yet meaningful topological summary, and are of
independent interest. It turns out that an exact or approximation algorithm for
the interleaving distance leads to an approximation algorithm for the
Gromov-Hausdorff distance. One of the key contributions of our work is that we
re-define the interleaving distance in a way that makes it easier to develop
dynamic programming approaches to compute it. We then present a FPT algorithm
to compute the interleaving distance between two merge trees exactly, which
ultimately leads to an FPT-algorithm to approximate the GH distance between two
metric trees. This exact FPT-algorithm to compute the interleaving distance
between merge trees is of interest itself, as it is known that it is NP-hard to
approximate it within a factor of 3, and previously the best known algorithm
has an approximation factor of $O(\sqrt{n})$ even for trees with unit edge
length.

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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th