Asymmetric Separation Problem for Bichromatic Point Set

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 an annulus (either a rectangular or a circular) $\cal A$ 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 study the problem for various types of axis-parallel rectangular annulus and circular annulus in one and two dimensions. We also study a restricted version of the rectangular annulus cover problem, where the center of the annulus is constrained to lie on a given horizontal line $L$. We design a polynomial-time algorithm for each type of annulus.