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.

翻译：我们研究了面向红点集(R)和蓝点集(B)的广义红蓝圆环覆盖问题，其中每个点p∈R∪B关联一个正惩罚值P(p)。红点具有未覆盖惩罚，蓝点具有覆盖惩罚。目标是计算一个圆环A使得函数P(R^{out})+P(B^{in})的值最小化，其中R^{out}⊆R表示未被A覆盖的红点集合，B^{in}⊆B表示被A覆盖的蓝点集合。我们还研究了该问题的另一个变体，即要求圆环在二维空间中覆盖R中所有红点及B中最少数量的点。我们为所有此类圆环问题设计了多项式时间算法。