2026-07-24T12:06:15Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9981 2024-03-06T10:45:15Z ddc:004 open_access

Diversity Maximization in Doubling Metrics Cevallos, Alfonso Eisenbrand, Friedrich Morell, Sarah Remote-clique remote-star remote-bipartition doubling dimension grid rounding epsilon-nets polynomial time approximation scheme facility location information retrieval 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 remote-clique, is the sum of the pairwise distances of the chosen elements. In this paper, we present novel results on three widely used diversity measures: Remote-clique, remote-star and remote-bipartition. 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 NP-hardness for remote-clique with squared distances in doubling metric spaces. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Alfonso Cevallos and Friedrich Eisenbrand and Sarah Morell 2018 Is Part Of LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2018.33 urn:nbn:de:0030-drops-99818 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.33 eng https://creativecommons.org/licenses/by/3.0/legalcode