Matroid-Constrained Vertex Cover
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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 - ε) approximation using (1/ε)^O(k)n^O(1) time for partition and laminar matroids and using (1/ε+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 ≈ 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 · (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.
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