4 research outputs found
Diversity Maximization in Doubling Metrics
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
Fully dynamic clustering and diversity maximization in doubling metrics
We present approximation algorithms for some variants of center-based
clustering and related problems in the fully dynamic setting, where the
pointset evolves through an arbitrary sequence of insertions and deletions.
Specifically, we target the following problems: -center (with and without
outliers), matroid-center, and diversity maximization. All algorithms employ a
coreset-based strategy and rely on the use of the cover tree data structure,
which we crucially augment to maintain, at any time, some additional
information enabling the efficient extraction of the solution for the specific
problem. For all of the aforementioned problems our algorithms yield
-approximations, where is the best known
approximation attainable in polynomial time in the standard off-line setting
(except for -center with outliers where but we get a
-approximation) and is a user-provided
accuracy parameter. The analysis of the algorithms is performed in terms of the
doubling dimension of the underlying metric. Remarkably, and unlike previous
works, the data structure and the running times of the insertion and deletion
procedures do not depend in any way on the accuracy parameter
and, for the two -center variants, on the parameter . For spaces of
bounded doubling dimension, the running times are dramatically smaller than
those that would be required to compute solutions on the entire pointset from
scratch. To the best of our knowledge, ours are the first solutions for the
matroid-center and diversity maximization problems in the fully dynamic
setting