# W[t]-containment of partial covering problems

I would like to know more about the W[t]-containment of partial covering problems. Especially, I am interested in the question whether Partial Set Cover (Problem Definition at the end of the question) with respect to the size of a solution $$k$$ is in W[2].

Because Set Cover is the special case of Partial Set Cover with $$p=|\mathcal U|$$, we conclude that Partial Set Cover wrt $$k$$ is W[2]-hard. It remains open whether Partial Set Cover is also contained in W[2]. A natural way to prove this would be to reduce from Partial Vertex Cover to Weighted Circuit Satisfiability. But in my reduction, I struggle in how to "count" the number of covered items.

For Partial Vertex Cover a surprising thing happens. While Vertex cover is FPT wrt $$k$$, Partial Vertex Cover is W[1]-complete wrt $$k$$. Guo, Niedermaier and Wernicke: Parameterized complexity of vertex cover variants https://link.springer.com/content/pdf/10.1007/s00224-007-1309-3.pdf

Guo, Niedermaier and Wernicke did not reduce to Weighted Circuit Satisfiability in order to show W[1]-containment. However, I would still like how such a reduction could look like because also here we need to "count" the number of covered edges.

Partial Set Cover is defined as follows. Given a universe $$\mathcal U$$, a family of sets $$\mathcal F$$ over $$\mathcal U$$ and two integers $$k$$ and $$p$$. It is asked whether there are sets $$F_1,\dots,F_k\in\mathcal F$$ such that the union $$\bigcup_{i=1}^k F_i \subseteq \mathcal U$$ contains at least $$p$$ items.
Partial Vertex Cover is defined as follows. Given an undirected graph $$G=(V,E)$$ and two integers $$k$$ and $$p$$. It is asked whether there is a set of vertices $$X$$ such that at least $$p$$ edges are incident with at least one vertex of $$X$$.