发表机构
Sharif University of Technology（谢里夫理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对公平多样性最大化问题，提出局部搜索框架，实现与组数无关的常数因子近似（任意常数组成4近似，两组达2近似且最优），改进此前线性依赖。
AI 中文摘要
多样性最大化是一个基础优化问题，在机器学习、数据摘要、信息检索和推荐系统中有着广泛应用。在许多此类应用中，数据被划分为多个组，所选子集必须满足规定的组配额。我们研究公平多样性最大化问题：给定一个度量空间中划分为$m$组的点集，目标是从每组$i$中恰好选择$k_i$个点，同时最大化所选点之间的最小成对距离。此前已知的最佳近似保证为$m+1$，该保证随组数线性增长。我们证明这种对$m$的依赖并非本质性的。我们提出一个新的局部搜索框架，对于任意常数数量的组，该框架可实现$4$近似，且对度量空间或所选集合的大小没有任何限制。据我们所知，这是在此一般设置中首个近似保证独立于组数的常数因子近似算法。我们的框架在逐步消除多样性目标违规的同时，精确维持所有组配额。我们进一步为两组情况开发了一种专门算法，可实现$2$近似，将此前最佳因子$3$加以改进。该因子是最优的：除非$\text{P}=\text{NP}$，否则即使对于无约束情形，也没有多项式时间算法能够实现严格优于$2$的近似因子。
英文摘要
Diversity maximization is a fundamental optimization problem with applications in machine learning, data summarization, information retrieval, and recommendation systems. In many such applications, the data are partitioned into groups, and the selected subset must satisfy prescribed group quotas. We study Fair Diversity Maximization: given a set of points in a metric space partitioned into $m$ groups, the goal is to select exactly $k_i$ points from each group $i$ while maximizing the minimum pairwise distance among the selected points. The best previously known approximation guarantee is $m+1$, which grows linearly with the number of groups. We show that this dependence on $m$ is not fundamental. We present a new local-search framework that yields a $4$-approximation for any constant number of groups, with no restrictions on the metric space or on the size of the selected set. To the best of our knowledge, this is the first constant-factor approximation whose guarantee is independent of the number of groups in this general setting. Our framework maintains all group quotas exactly while progressively eliminating violations of the diversity objective. We further develop a specialized algorithm for two groups that achieves a $2$-approximation, improving the previous best factor of $3$. This factor is optimal: unless $\mathrm{P}=\mathrm{NP}$, no polynomial-time algorithm can achieve an approximation factor strictly better than $2$, even for the unconstrained case.
