License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.21
URN: urn:nbn:de:0030-drops-87344
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8734/
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Carpenter, Timothy ; Fomin, Fedor V. ; Lokshtanov, Daniel ; Saurabh, Saket ; Sidiropoulos, Anastasios

Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces

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LIPIcs-SoCG-2018-21.pdf (0.4 MB)

Abstract

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs G, H and integer c, is it possible to embed G with distortion c into a graph homeomorphic to H? Then embedding into the line is the special case H=K_2, and embedding into the cycle is the case H=K_3, where K_k denotes the complete graph on k vertices. For this problem we give
- an approximation algorithm, which in time f(H)* poly (n), for some function f, either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion poly(c);
- an exact algorithm, which in time f'(H, c)* poly (n), for some function f', either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion c. Prior to our work, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

BibTeX - Entry

@InProceedings{carpenter_et_al:LIPIcs:2018:8734,
  author =	{Timothy Carpenter and Fedor V. Fomin and Daniel Lokshtanov and Saket Saurabh and Anastasios Sidiropoulos},
  title =	{{Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8734},
  URN =		{urn:nbn:de:0030-drops-87344},
  doi =		{10.4230/LIPIcs.SoCG.2018.21},
  annote =	{Keywords: Metric embeddings, minimum-distortion embeddings, 1-dimensional simplicial complex, Fixed-parameter tractable algorithms, Approximation algorithms}
}

Keywords: Metric embeddings, minimum-distortion embeddings, 1-dimensional simplicial complex, Fixed-parameter tractable algorithms, Approximation algorithms
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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