Given an $n$-point metric space $(\mathcal{X},d)$ where each point belongs to one of $m=O(1)$ different categories or groups and a set of integers $k_1, \ldots, k_m$, the fair Max-Min diversification problem is to select $k_i$ points belonging to category $i\in [m]$, such that the minimum pairwise distance between selected points is maximized. The problem was introduced by Moumoulidou et al. [ICDT 2021] and is motivated by the need to down-sample large data sets in various applications so that the derived sample achieves a balance over diversity, i.e., the minimum distance between a pair of selected points, and fairness, i.e., ensuring enough points of each category are included. We prove the following results:
1. We first consider general metric spaces. We present a randomized polynomial time algorithm that returns a factor $2$-approximation to the diversity but only satisfies the fairness constraints in expectation. Building upon this result, we present a $6$-approximation that is guaranteed to satisfy the fairness constraints up to a factor $1-ε$ for any constant $ε$. We also present a linear time algorithm returning an $m+1$ approximation with exact fairness. The best previous result was a $3m-1$ approximation.
2. We then focus on Euclidean metrics. We first show that the problem can be solved exactly in one dimension. For constant dimensions, categories and any constant $ε>0$, we present a $1+ε$ approximation algorithm that runs in $O(nk) + 2^{O(k)}$ time where $k=k_1+\ldots+k_m$. We can improve the running time to $O(nk)+ poly(k)$ at the expense of only picking $(1-ε) k_i$ points from category $i\in [m]$.
Finally, we present algorithms suitable to processing massive data sets including single-pass data stream algorithms and composable coresets for the distributed processing.
A Framework for the Design of Efficient Diversification Algorithms for Packing and Covering Problems
Universidad de Concepción（康塞普西翁大学）；Queens College of CUNY（纽约市立大学皇后学院）；University of Chile（智利大学）；CUNY Graduate Center（纽约市立大学研究生院）；Academia Sinica（中央研究院）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）
