Parameterized complexity of computing maximum minimal blocking and hitting sets

A blocking set in a graph $G$ is a subset of vertices that intersects every maximum independent set of $G$. Let ${\sf mmbs}(G)$ be the size of a maximum (inclusion-wise) minimal blocking set of $G$. This parameter has recently played an important role in the kernelization of Vertex Cover parameterized by the distance to a graph class ${\cal F}$. Indeed, it turns out that the existence of a polynomial kernel for this problem is closely related to the property that ${\sf mmbs}({\cal F})=\sup_{G \in {\cal F}}{\sf mmbs}(G)$ is bounded by a constant, and thus several recent results focused on determining ${\sf mmbs}({\cal F})$ for different classes ${\cal F}$. We consider the parameterized complexity of computing ${\sf mmbs}$ under various parameterizations, such as the size of a maximum independent set of the input graph and the natural parameter. We provide a panorama of the complexity of computing both ${\sf mmbs}$ and ${\sf mmhs}$, which is the size of a maximum minimal hitting set of a hypergraph, a closely related parameter. Finally, we consider the problem of computing ${\sf mmbs}$ parameterized by treewidth, especially relevant in the context of kernelization. Given the "counting" nature of ${\sf mmbs}$, it does not seem to be expressible in monadic second-order logic, hence its tractability does not follow from Courcelle's theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem.

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