  In Maximum $k$-Vertex Cover (Max $k$-VC), the input is an edge-weighted graph
$G$ and an integer $k$, and the goal is to find a subset $S$ of $k$ vertices
that maximizes the total weight of edges covered by $S$. Here we say that an
edge is covered by $S$ iff at least one of its endpoints lies in $S$.
  We present an FPT approximation scheme (FPT-AS) that runs in
$(1/\epsilon)^{O(k)} poly(n)$ time for the problem, which improves upon Gupta
et al.'s $(k/\epsilon)^{O(k)} poly(n)$-time FPT-AS [SODA'18, FOCS'18]. Our
algorithm is simple: just use brute force to find the best $k$-vertex subset
among the $O(k/\epsilon)$ vertices with maximum weighted degrees.
  Our algorithm naturally yields an efficient approximate kernelization scheme
of $O(k/\epsilon)$ vertices; previously, an $O(k^5/\epsilon^2)$-vertex
approximate kernel is only known for the unweighted version of Max $k$-VC
[Lokshtanov et al., STOC'17]. Interestingly, this has an application outside of
parameterized complexity: using our approximate kernelization as a
preprocessing step, we can directly apply Raghavendra and Tan's SDP-based
algorithm for 2SAT with cardinality constraint [SODA'12] to give an
$0.92$-approximation for Max $k$-VC in polynomial time. This improves upon
Feige and Langberg's algorithm [J. Algorithms'01] which yields $(0.75 +
\delta)$-approximation for some (unspecified) constant $\delta > 0$.
  We also consider the minimization version (Min $k$-VC), where the goal is to
minimize the total weight of edges covered by $S$. We provide an FPT-AS for Min
$k$-VC with similar running time of $(1/\epsilon)^{O(k)} poly(n)$, which again
improves on a $(k/\epsilon)^{O(k)} poly(n)$-time FPT-AS of Gupta et al. On the
other hand, we show that there is unlikely a polynomial size approximate
kernelization for Min $k$-VC for any factor less than two.
