However, the most common algorithm used for this problem in practical applications is a natural greedy algorithm.

Given a set of $n$ vectors in $\mathbb{R}^d$, the goal of the \emph{determinant maximization} problem is to pick $k$ vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP) and has recently received considerable attention for modeling diversity.

Large-scale clustering of data points in metric spaces is an important problem in mining big data sets. For many applications, we face explicit or implicit size constraints for each cluster which leads to the problem of clustering under capacity constraints or the "balanced clustering" problem. Although the balanced clustering problem has been widely studied, developing a theoretically sound distributed algorithm remains an open problem. In this paper we develop a new framework based on "mapping coresets" to tackle this issue. Our technique results in first distributed approximation algorithms for balanced clustering problems for a wide range of clustering objective functions such as k-center, k-median, and k-means.

Given a set of n vectors in \mathbb{R} d, the goal of the \emph{determinant maximization} problem is to pick k vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP) and has recently received considerable attention for modeling diversity. This is tight up to the additive constant 1 . Finally, our experiments show that the local optimality of the greedy algorithm is even lower than the theoretical bound on real data sets.

Large-scale clustering of data points in metric spaces is an important problem in mining big data sets. For many applications, we face explicit or implicit size constraints for each cluster which leads to the problem of clustering under capacity constraints or the "balanced clustering" problem. Although the balanced clustering problem has been widely studied, developing a theoretically sound distributed algorithm remains an open problem. In this paper we develop a new framework based on "mapping coresets" to tackle this issue. Our technique results in first distributed approximation algorithms for balanced clustering problems for a wide range of clustering objective functions such as k-center, k-median, and k-means.

Given a set of $n$ vectors in $\mathbb{R}^d$, the goal of the \emph{determinant maximization} problem is to pick $k$ vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP) and has recently received considerable attention for modeling diversity. As most applications for the problem use large amounts of data, this problem has been studied in the relevant \textit{composable coreset} setting. In particular, [Indyk-Mahabadi-OveisGharan-Rezaei--SODA'20, ICML'19] showed that one can get composable coresets with optimal approximation factor of $\tilde O(k)^k$ for the problem, and that a local search algorithm achieves an almost optimal approximation guarantee of $O(k)^{2k}$. In this work, we show that the widely-used Greedy algorithm also provides composable coresets with an almost optimal approximation factor of $O(k)^{3k}$, which improves over the previously known guarantee of $C^{k^2}$, and supports the prior experimental results showing the practicality of the greedy algorithm as a coreset. Our main result follows by showing a local optimality property for Greedy: swapping a single point from the greedy solution with a vector that was not picked by the greedy algorithm can increase the volume by a factor of at most $(1+\sqrt{k})$. This is tight up to the additive constant $1$. Finally, our experiments show that the local optimality of the greedy algorithm is even lower than the theoretical bound on real data sets.

Large-scale clustering of data points in metric spaces is an important problem in mining big data sets. For many applications, we face explicit or implicit size constraints for each cluster which leads to the problem of clustering under capacity constraints or the ``balanced clustering'' problem. Although the balanced clustering problem has been widely studied, developing a theoretically sound distributed algorithm remains an open problem. In the present paper we develop a general framework based on ``mapping coresets'' to tackle this issue. For a wide range of clustering objective functions such as k-center, k-median, and k-means, our techniques give distributed algorithms for balanced clustering that match the best known single machine approximation ratios.