Selection of a group of representatives satisfying certain fairness constraints, is a commonly occurring scenario. Motivated by this, we initiate a systematic algorithmic study of a \emph{fair} version of \textsc{Hitting Set}. In the classical \textsc{Hitting Set} problem, the input is a universe $\mathcal{U}$, a family $\mathcal{F}$ of subsets of $\mathcal{U}$, and a non-negative integer $k$. The goal is to determine whether there exists a subset $S \subseteq \mathcal{U}$ of size $k$ that \emph{hits} (i.e., intersects) every set in $\mathcal{F}$. Inspired by several recent works, we formulate a fair version of this problem, as follows. The input additionally contains a family $\mathcal{B}$ of subsets of $\mathcal{U}$, where each subset in $\mathcal{B}$ can be thought of as the group of elements of the same \emph{type}. We want to find a set $S \subseteq \mathcal{U}$ of size $k$ that (i) hits all sets of $\mathcal{F}$, and (ii) does not contain \emph{too many} elements of each type. We call this problem \textsc{Fair Hitting Set}, and chart out its tractability boundary from both classical as well as multivariate perspective. Our results use a multitude of techniques from parameterized complexity including classical to advanced tools, such as, methods of representative sets for matroids, FO model checking, and a generalization of best known kernels for \textsc{Hitting Set}.

翻译：选取一组满足特定公平约束条件的代表，是常见场景。受此启发，我们针对公平版本的击集问题启动系统性算法研究。经典击集问题中，输入是全集$\mathcal{U}$、$\mathcal{U}$的子集族$\mathcal{F}$以及非负整数$k$，目标是判断是否存在大小为$k$的子集$S \subseteq \mathcal{U}$，使得$S$击中（即相交于）$\mathcal{F}$中的每个集合。结合近期多项工作，我们提出该问题的公平版本：输入额外包含$\mathcal{U}$的子集族$\mathcal{B}$，其中$\mathcal{B}$中的每个子集可视为同类型的元素组。我们需要找到大小为$k$的集合$S \subseteq \mathcal{U}$，满足：(i) 击中$\mathcal{F}$中所有集合；(ii) 每种类型的元素不超过给定阈值。我们将此问题命名为公平击集问题，并从经典与多变量双重视角刻画其可处理性边界。我们的结果运用了参数化复杂性的多种技术，涵盖经典到前沿工具，包括拟阵代表集方法、一阶模型检验以及击集问题最优核的泛化。