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.

中文翻译：

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参数化的计算复杂度，最大程度地减少了阻塞和打击集

图$ G $中的阻塞集是顶点的子集，该子集与$ G $的每个最大独立集相交。令$ {\ sf mmbs}（G）$为$ G $的最大（包含）最小阻塞集的大小。该参数最近在“顶点覆盖”的内核化中起着重要作用，该顶点覆盖由到图形类$ {\ cal F} $的距离来参数化。确实，事实证明，针对此问题的多项式内核的存在与$ {\ sf mmbs}（{\ cal F}）= \ sup_ {G \ in {\ cal F}} {\ sf mmbs}（G）$受常数限制，因此最近的一些结果集中于确定不同类$ {\ cal F} $的$ {\ sf mmbs}（{\ cal F}）$。我们考虑在各种参数化下计算$ {\ sf mmbs} $的参数化复杂性，例如输入图的最大独立集的大小和自然参数。我们提供了同时计算$ {\ sf mmbs} $和$ {\ sf mmhs} $的复杂性的全景图，这是超图的最大最小匹配集的大小，而超图是一个紧密相关的参数。最后，我们考虑计算以树宽为参数的$ {\ sf mmbs} $的问题，这在内核化的情况下尤其重要。考虑到$ {\ sf mmbs} $的“计数”性质，它似乎无法在单子二阶逻辑中表达，因此其可处理性并不遵循库尔切勒定理。我们的主要技术贡献是针对此问题的固定参数可处理算法。我们考虑计算以树宽为参数的$ {\ sf mmbs} $的问题，特别是在内核化的情况下。考虑到$ {\ sf mmbs} $的“计数”性质，它似乎无法在单子二阶逻辑中表达，因此其可处理性并不遵循库尔切勒定理。我们的主要技术贡献是针对此问题的固定参数可处理算法。我们考虑计算以树宽为参数的$ {\ sf mmbs} $的问题，特别是在内核化的情况下。考虑到$ {\ sf mmbs} $的“计数”性质，它似乎无法在单子二阶逻辑中表达，因此其可处理性并不遵循库尔切勒定理。我们的主要技术贡献是针对此问题的固定参数可处理算法。