Let $\mathcal{D}$ be a set family that is the solution domain of some combinatorial problem. The \emph{max-min diversification problem on $\mathcal{D}$} is the problem to select $k$ sets from $\mathcal{D}$ such that the Hamming distance between any two selected sets is at least $d$. FPT algorithms parameterized by $k,l:=\max_{D\in \mathcal{D}}|D|$ and $k,d$ have been actively studied recently for several specific domains. This paper provides unified algorithmic frameworks to solve this problem. Specifically, for each parameterization $k,l$ and $k,d$, we provide an FPT oracle algorithm for the max-min diversification problem using oracles related to $\mathcal{D}$. We then demonstrate that our frameworks generalize most of the existing domain-specific tractability results and provide the first FPT algorithms for several domains. Our main technical breakthrough is introducing the notion of \emph{max-distance sparsifier} of $\mathcal{D}$, a domain on which the max-min diversification problem is equivalent to the same problem on the original domain $\mathcal{D}$. The core of our framework is to design FPT oracle algorithms that construct a constant-size max-distance sparsifier of $\mathcal{D}$. Using max-distance sparsifiers, we provide FPT algorithms for the max-min and max-sum diversification problems on $\mathcal{D}$, as well as $k$-center and $k$-sum-of-radii clustering problems on $\mathcal{D}$, which are also natural problems in the context of diversification and have their own interests.

翻译：令 $\mathcal{D}$ 为某组合问题解空间的集合族。\emph{$\mathcal{D}$ 上的最大-最小多样化问题} 指从 $\mathcal{D}$ 中选取 $k$ 个集合，使得任意两个被选集合之间的汉明距离至少为 $d$。近年来，针对若干特定解域，以 $k,l:=\max_{D\in \mathcal{D}}|D|$ 和 $k,d$ 为参数的 FPT 算法已得到广泛研究。本文提出了解决该问题的统一算法框架。具体而言，针对 $k,l$ 和 $k,d$ 两种参数化方案，我们分别设计了利用 $\mathcal{D}$ 相关预言机的最大-最小多样化问题 FPT 预言算法。我们证明该框架可推广现有大多数领域特定的可解性结果，并为若干新领域首次提供 FPT 算法。主要技术突破在于引入 \emph{$\mathcal{D}$ 的最大距离稀疏化器} 概念——在该解域上，最大-最小多样化问题与原解域 $\mathcal{D}$ 上的问题等价。本框架的核心是设计能构造 $\mathcal{D}$ 的常数规模最大距离稀疏化器的 FPT 预言算法。借助最大距离稀疏化器，我们为 $\mathcal{D}$ 上的最大-最小与最大-和多样化问题，以及 $k$-中心与 $k$-半径和聚类问题提供了 FPT 算法，这些在多样化背景下均为自然问题且具有独立研究价值。