  In the past decade, many parameterized algorithms were developed for packing
problems. Our goal is to obtain tradeoffs that improve the running times of
these algorithms at the cost of computing approximate solutions. Consider a
packing problem for which there is no known algorithm with approximation ratio
$\alpha$, and a parameter $k$. If the value of an optimal solution is at least
$k$, we seek a solution of value at least $\alpha k$; otherwise, we seek an
arbitrary solution. Clearly, if the best known parameterized algorithm that
finds a solution of value $t$ runs in time $O^*(f(t))$ for some function $f$,
we are interested in running times better than $O^*(f(\alpha k))$. We present
tradeoffs between running times and approximation ratios for the $P_2$-Packing,
$3$-Set $k$-Packing and $3$-Dimensional $k$-Matching problems. Our tradeoffs
are based on combinations of several known results, as well as a computation of
"approximate lopsided universal sets."
