In this paper, we introduce the problem of Matroid-Constrained Vertex Cover: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied max $k$-vertex cover problem, in which the matroid is the simple uniform matroid, and it is also a special case of the problem of maximizing a monotone submodular function under a matroid constraint. First, we give a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if $k$ is the rank of the matroid, we obtain $(1 - \varepsilon)$ approximation using $(1/\varepsilon)^{O(k)}n^{O(1)}$ time for partition and laminar matroids and using $(1/\varepsilon+k)^{O(k)}n^{O(1)}$ time for transversal matroids. This extends a result of Manurangsi for uniform matroids [Manurangsi, 2018]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Besides, our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics. In the second part, we consider general matroids. We propose a simple local search algorithm that guarantees $2/3 \approx 0.66$ approximation. For the more general problem where two matroids are imposed on the vertices and a feasible solution must be a common independent set, we show that a local search algorithm gives a $2/3 \cdot (1 - 1/(p+1))$ approximation in $n^{O(p)}$ time, for any integer $p$. We also provide some evidence to show that with the constraint of one or two matroids, the approximation ratio of $2/3$ is likely the best possible, using the currently known techniques of local search.

翻译：本文提出了拟阵约束顶点覆盖问题：给定一个边赋权图及顶点上定义的拟阵，目标是选择拟阵中独立的一个顶点子集，以最大化被覆盖边的总权重。该问题是广泛研究的最大$k$-顶点覆盖问题的推广（其中拟阵为简单均匀拟阵），同时也是拟阵约束下单调子模函数最大化问题的特例。首先，我们针对分割拟阵、分层拟阵或横贯拟阵给出了固定参数可追踪近似方案（FPT-AS）。具体而言，若$k$为拟阵的秩，对于分割拟阵和分层拟阵，我们在$(1/\varepsilon)^{O(k)}n^{O(1)}$时间内获得$(1 - \varepsilon)$近似比；对于横贯拟阵，则在$(1/\varepsilon+k)^{O(k)}n^{O(1)}$时间内实现。这推广了Manurangsi关于均匀拟阵的结果[Manurangsi, 2018]。我们还证明这些思想可应用于（单遍）流算法。此外，我们的FPT-AS引入了基于拟阵并的新技术，该技术可能在极值组合学中具有独立意义。在第二部分，我们考虑一般拟阵，提出一个保证$2/3 \approx 0.66$近似比的简单局部搜索算法。针对顶点上施加两个拟阵且可行解必须为公共独立集的一般化问题，我们证明对于任意整数$p$，局部搜索算法可在$n^{O(p)}$时间内实现$2/3 \cdot (1 - 1/(p+1))$近似比。我们同时提供证据表明，在单拟阵或双拟阵约束下，基于现有局部搜索技术，$2/3$近似比很可能是最佳可能结果。