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Documents authored by Großwendt, Anna

Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.18
Upper and Lower Bounds for Complete Linkage in General Metric Spaces

Authors: Anna Arutyunova, Anna Großwendt, Heiko Röglin, Melanie Schmidt, and Julian Wargalla

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
In a hierarchical clustering problem the task is to compute a series of mutually compatible clusterings of a finite metric space (P,dist). Starting with the clustering where every point forms its own cluster, one iteratively merges two clusters until only one cluster remains. Complete linkage is a well-known and popular algorithm to compute such clusterings: in every step it merges the two clusters whose union has the smallest radius (or diameter) among all currently possible merges. We prove that the radius (or diameter) of every k-clustering computed by complete linkage is at most by factor O(k) (or O(k²)) worse than an optimal k-clustering minimizing the radius (or diameter). Furthermore we give a negative answer to the question proposed by Dasgupta and Long [Sanjoy Dasgupta and Philip M. Long, 2005], who show a lower bound of Ω(log(k)) and ask if the approximation guarantee is in fact Θ(log(k)). We present instances where complete linkage performs poorly in the sense that the k-clustering computed by complete linkage is off by a factor of Ω(k) from an optimal solution for radius and diameter. We conclude that in general metric spaces complete linkage does not perform asymptotically better than single linkage, merging the two clusters with smallest inter-cluster distance, for which we prove an approximation guarantee of O(k).
Cite as

Anna Arutyunova, Anna Großwendt, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Upper and Lower Bounds for Complete Linkage in General Metric Spaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arutyunova_et_al:LIPIcs.APPROX/RANDOM.2021.18,
  author =	{Arutyunova, Anna and Gro{\ss}wendt, Anna and R\"{o}glin, Heiko and Schmidt, Melanie and Wargalla, Julian},
  title =	{{Upper and Lower Bounds for Complete Linkage in General Metric Spaces}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{18:1--18:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.18},
  URN =		{urn:nbn:de:0030-drops-147115},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.18},
  annote =	{Keywords: Hierarchical Clustering, Complete Linkage, agglomerative Clustering, k-Center}
}
Document
DOI: 10.4230/LIPIcs.STACS.2015.171
Solving Totally Unimodular LPs with the Shadow Vertex Algorithm

Authors: Tobias Brunsch, Anna Großwendt, and Heiko Röglin

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number n of variables, the number m of constraints, and 1/\delta, where \delta is a parameter that measures the flatness of the vertices of the polyhedron. This extends our recent result that the shadow vertex algorithm finds paths of polynomial length (w.r.t. n, m, and 1/delta) between two given vertices of a polyhedron [4]. Our result also complements a recent result due to Eisenbrand and Vempala [6] who have shown that a certain version of the random edge pivot rule solves linear programs with a running time that is strongly polynomial in the number of variables n and 1/\delta, but independent of the number m of constraints. Even though the running time of our algorithm depends on m, it is significantly faster for the important special case of totally unimodular linear programs, for which 1/delta\le n and which have only O(n^2) constraints.
Cite as

Tobias Brunsch, Anna Großwendt, and Heiko Röglin. Solving Totally Unimodular LPs with the Shadow Vertex Algorithm. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 171-183, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{brunsch_et_al:LIPIcs.STACS.2015.171,
  author =	{Brunsch, Tobias and Gro{\ss}wendt, Anna and R\"{o}glin, Heiko},
  title =	{{Solving Totally Unimodular LPs with the Shadow Vertex Algorithm}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{171--183},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.171},
  URN =		{urn:nbn:de:0030-drops-49125},
  doi =		{10.4230/LIPIcs.STACS.2015.171},
  annote =	{Keywords: linear optimization, simplex algorithm, shadow vertex method}
}
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