Carpenter, Timothy ;
Fomin, Fedor V. ;
Lokshtanov, Daniel ;
Saurabh, Saket ;
Sidiropoulos, Anastasios
Algorithms for LowDistortion Embeddings into Arbitrary 1Dimensional Spaces
Abstract
We study the problem of finding a minimumdistortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric X. Computing such an embedding (exactly or approximately) is a nontrivial task even when X is the metric induced by a path, or, equivalently, the real line. In this paper we give approximation and fixedparameter tractable (FPT) algorithms for minimumdistortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1dimensional 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 LowDistortion Embeddings into Arbitrary 1Dimensional Spaces}},
booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)},
pages = {21:121:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770668},
ISSN = {18688969},
year = {2018},
volume = {99},
editor = {Bettina Speckmann and Csaba D. T{\'o}th},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8734},
URN = {urn:nbn:de:0030drops87344},
doi = {10.4230/LIPIcs.SoCG.2018.21},
annote = {Keywords: Metric embeddings, minimumdistortion embeddings, 1dimensional simplicial complex, Fixedparameter tractable algorithms, Approximation algorithms}
}
08.06.2018
Keywords: 

Metric embeddings, minimumdistortion embeddings, 1dimensional simplicial complex, Fixedparameter tractable algorithms, Approximation algorithms 
Seminar: 

34th International Symposium on Computational Geometry (SoCG 2018)

Issue date: 

2018 
Date of publication: 

08.06.2018 