Cevallos, Alfonso ;
Eisenbrand, Friedrich ;
Morell, Sarah
Diversity Maximization in Doubling Metrics
Abstract
Diversity maximization is an important geometric optimization problem with many applications in recommender systems, machine learning or search engines among others. A typical diversification problem is as follows: Given a finite metric space (X,d) and a parameter k in N, find a subset of k elements of X that has maximum diversity. There are many functions that measure diversity. One of the most popular measures, called remoteclique, is the sum of the pairwise distances of the chosen elements. In this paper, we present novel results on three widely used diversity measures: Remoteclique, remotestar and remotebipartition.
Our main result are polynomial time approximation schemes for these three diversification problems under the assumption that the metric space is doubling. This setting has been discussed in the recent literature. The existence of such a PTAS however was left open.
Our results also hold in the setting where the distances are raised to a fixed power q >= 1, giving rise to more variants of diversity functions, similar in spirit to the variations of clustering problems depending on the power applied to the pairwise distances. Finally, we provide a proof of NPhardness for remoteclique with squared distances in doubling metric spaces.
BibTeX  Entry
@InProceedings{cevallos_et_al:LIPIcs:2018:9981,
author = {Alfonso Cevallos and Friedrich Eisenbrand and Sarah Morell},
title = {{Diversity Maximization in Doubling Metrics}},
booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)},
pages = {33:133:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770941},
ISSN = {18688969},
year = {2018},
volume = {123},
editor = {WenLian Hsu and DerTsai Lee and ChungShou Liao},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9981},
URN = {urn:nbn:de:0030drops99818},
doi = {10.4230/LIPIcs.ISAAC.2018.33},
annote = {Keywords: Remoteclique, remotestar, remotebipartition, doubling dimension, grid rounding, epsilonnets, polynomial time approximation scheme, facility locati}
}
2018
Keywords: 

Remoteclique, remotestar, remotebipartition, doubling dimension, grid rounding, epsilonnets, polynomial time approximation scheme, facility locati 
Seminar: 

29th International Symposium on Algorithms and Computation (ISAAC 2018)

Issue date: 

2018 
Date of publication: 

2018 