  We investigate the parameterized complexity of GENERALIZED RED BLUE SET COVER
(Gen-RBSC), a generalization of the classic SET COVER problem and the more
recently studied RED BLUE SET COVER problem. Given a universe $U$ containing
$b$ blue elements and $r$ red elements, positive integers $k_\ell$ and $k_r$,
and a family $\F$ of $\ell$ sets over $U$, the \srbsc\ problem is to decide
whether there is a subfamily $\F'\subseteq \F$ of size at most $k_\ell$ that
covers all blue elements, but at most $k_r$ of the red elements. This
generalizes SET COVER and thus in full generality it is intractable in the
parameterized setting. In this paper, we study a geometric version of this
problem, called Gen-RBSC-lines, where the elements are points in the plane and
sets are defined by lines. We study this problem for an array of parameters,
namely, $k_\ell, k_r, r, b$, and $\ell$, and all possible combinations of them.
For all these cases, we either prove that the problem is W-hard or show that
the problem is fixed parameter tractable (FPT). In particular, on the
algorithmic side, our study shows that a combination of $k_\ell$ and $k_r$
gives rise to a nontrivial algorithm for Gen-RBSC-lines. On the hardness side,
we show that the problem is para-NP-hard when parameterized by $k_r$, and
W[1]-hard when parameterized by $k_\ell$. Finally, for the combination of
parameters for which Gen-RBSC-lines admits FPT algorithms, we ask for the
existence of polynomial kernels. We are able to provide a complete
kernelization dichotomy by either showing that the problem admits a polynomial
kernel or that it does not contain a polynomial kernel unless $\CoNP \subseteq
\NP/\mbox{poly}$.
