Generalized Red-Blue Circular Annulus Cover Problem

We study the Generalized Red-Blue Annulus Cover problem for two sets of points, red (R) and blue (B), where each point p ∈ R∪ B is associated with a positive penalty P(p). The red points have non-covering penalties, and the blue points have covering penalties. The objective is to compute a circular annulus A such that the value of the function P(Rᵒᵘᵗ) + P( Bⁱⁿ) is minimum, where Rᵒᵘᵗ ⊆ R is the set of red points not covered by A and Bⁱⁿ ⊆ B is the set of blue points covered by 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.