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In the first chapter of "Parameterized Complexity Theory" by Flum and Grohe, an example is presented to find a hitting set of minimal cardinality.

In Fig. 1.3, the author says a black colored leaf is the minimal with respect to set inclusion and a hitting set consists of the vertices labeling the edges on the path from the root to the leaf. However, both grey and black colored hitting set consists of 3 vertices. Why grey colored leaf is not minimal in this case?

Related question: Given a set of sets, find the smallest set(s) containing at least one element from each set

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It seems you are confusing the terms minimal and minimum. The Hitting set problem is to find a hitting set of minimum cardinality, not a minimal set.

A set $S$ is called minimal with respect to some property (in this case, being a hitting set) if there exists no strict subset $T$ of $S$ (i.e. $T\subsetneq S$) that also satisfies that property.

On the other hand, a set $S$ is said to have minimum size or cardinality with respect to some property if there exists no set $T$ of strictly smaller size (i.e. $|T|< |S|$) that also satisfies this property.

A minimum size set must be minimal (why?), but a minimal set need not be of minimum size (why?).

In the example in the book, the set $\{b,c,d\}$ is not a minimal hitting set, because its strict subset $\{c,d\}$ is also a hitting set. The subset $\{b,d,f\}$ is a minimal hitting set, because it has no strict subset that is a hitting set. It is not a minimum hitting set, because there is a hitting set of size $2$ ($\{c,d\}$).

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  • $\begingroup$ Thanks, I now understand. $\endgroup$ – Domination Sep 3 at 1:22

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