Matroid-constrained vertex cover

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