  We study the Generalized Red-Blue Annulus Cover problem for two sets of
points, red ($R$) and blue ($B$), where each point $p \in R\cup B$ is
associated with a positive penalty ${\cal P}(p)$. The red points have
non-covering penalties, and the blue points have covering penalties. The
objective is to compute a circular annulus ${\cal A}$ such that the value of
the function ${\cal P}({R}^{out})$ + ${\cal P}({ B}^{in})$ is minimum, where
${R}^{out} \subseteq {R}$ is the set of red points not covered by ${\cal A}$
and ${B}^{in} \subseteq {B}$ is the set of blue points covered by $\cal A$. We
also study another version of this problem, where all the red points in $R$ and
the minimum number of points in $B$ are covered by the circular annulus in two
dimensions. We design polynomial-time algorithms for all such circular annulus
problems.
