In this paper, we study two generalizations of Vertex Cover and Edge Cover, namely Colorful Vertex Cover and Colorful Edge Cover. In the Colorful Vertex Cover problem, given an $n$-vertex edge-colored graph $G$ with colors from $\{1, \ldots, \omega\}$ and coverage requirements $r_1, r_2, \ldots, r_\omega$, the goal is to find a minimum-sized set of vertices that are incident on at least $r_i$ edges of color $i$, for each $1 \le i \le \omega$, i.e., we need to cover at least $r_i$ edges of color $i$. Colorful Edge Cover is similar to Colorful Vertex Cover, except here we are given a vertex-colored graph and the goal is to cover at least $r_i$ vertices of color $i$, for each $1 \le i \le \omega$, by a minimum-sized set of edges. These problems have several applications in fair covering and hitting of geometric set systems involving points and lines that are divided into multiple groups. Here, fairness ensures that the coverage (resp. hitting) requirement of every group is fully satisfied. We obtain a $(2+\epsilon)$-approximation for the Colorful Vertex Cover problem in time $n^{O(\omega/\epsilon)}$. Thus, for a constant number of colors, the problem admits a $(2+\epsilon)$-approximation in polynomial time. Next, for the Colorful Edge Cover problem, we design an $O(\omega n^3)$ time exact algorithm, via a chain of reductions to a matching problem. For all intermediate problems in this chain of reductions, we design polynomial-time algorithms, which might be of independent interest.

翻译：本文研究了顶点覆盖和边覆盖的两种推广形式，即多彩顶点覆盖与多彩边覆盖。在多彩顶点覆盖问题中，给定一个具有$n$个顶点、边带颜色的图$G$，其颜色取自集合$\{1, \ldots, \omega\}$，以及覆盖需求$r_1, r_2, \ldots, r_\omega$，目标是为每个$1 \le i \le \omega$找到一个最小规模的顶点集合，使得该集合关联到至少$r_i$条颜色为$i$的边，即需要覆盖至少$r_i$条颜色为$i$的边。多彩边覆盖与多彩顶点覆盖类似，区别在于给定的是顶点带颜色的图，目标是通过一个最小规模的边集合，为每个$1 \le i \le \omega$覆盖至少$r_i$个颜色为$i$的顶点。这些问题在涉及被划分为多个组的点与线等几何集合系统的公平覆盖与击打问题中具有多种应用。其中，公平性确保每个组的覆盖（或击打）需求得到完全满足。我们在时间复杂度为$n^{O(\omega/\epsilon)}$内得到了多彩顶点覆盖问题的$(2+\epsilon)$近似算法。因此，对于常数种颜色，该问题可在多项式时间内实现$(2+\epsilon)$近似。接下来，针对多彩边覆盖问题，我们通过一系列归约将其转化为匹配问题，设计了一个运行时间为$O(\omega n^3)$的精确算法。在这一归约链中的所有中间问题中，我们都设计了多项式时间算法，这些算法可能具有独立的研究价值。