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Diversity Maximization in Doubling Metrics

Authors Alfonso Cevallos , Friedrich Eisenbrand , Sarah Morell

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Author Details

Alfonso Cevallos
  • Swiss Federal Institute of Technology (ETH), Switzerland
Friedrich Eisenbrand
  • École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Sarah Morell
  • Technische Universität Berlin (TU Berlin), Germany

Funding

  • Eisenbrand, Friedrich: The second author acknowledges support from the Swiss National Science Foundation grant 163071, "Convexity, geometry of numbers, and the complexity of integer programming".
  • Morell, Sarah: Work conducted while the third author was affiliated to EPFL, Switzerland.

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Alfonso Cevallos, Friedrich Eisenbrand, and Sarah Morell. Diversity Maximization in Doubling Metrics. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 33:1-33:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ISAAC.2018.33

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • Remote-clique
  • remote-star
  • remote-bipartition
  • doubling dimension
  • grid rounding
  • epsilon-nets
  • polynomial time approximation scheme
  • facility location
  • information retrieval

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