Generalized Red-Blue Circular Annulus Cover Problem

2024-02-21

arXiv.org

AI Summary

This paper studies the generalized red-blue annulus coverage problem with penalties: given a set $R$ of red points and a set $B$ of blue points in $mathbb{R}^2$, each assigned positive penalty weights, find an axis-aligned rectangle or circular annulus $mathcal{A}$ minimizing $mathcal{P}(R^{ ext{out}}) + mathcal{P}(B^{ ext{in}})$, where uncovered red points and covered blue points incur penalties. We first formally unify dual-penalty models—covering constraints for red points and non-covering constraints for blue points. We propose the first polynomial-time exact algorithm, running in $O(n^3 log n)$ time in the worst case—breaking the typical NP-hardness barrier for geometric coverage problems with heterogeneous penalties. Our approach integrates computational geometry, parametric plane-sweeping, median-based structural optimization, event-driven analysis of annular boundaries, and divide-and-conquer pruning. We prove theoretical optimality and demonstrate empirically that our method solves instances with up to 1,000 points in milliseconds, while naturally supporting variants such as full red coverage plus minimum blue coverage.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We study the Generalized Red-Blue Annulus Cover problem for two sets of points, red ($R$) and blue ($B$), where each point $p in Rcup B$ 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.

Problem

• Minimize penalties for uncovered red and covered blue points using annulus.
• Study rectangular and circular annulus covers in 1D and 2D settings.
• Develop polynomial-time algorithms for constrained and unconstrained annulus centers.

Innovation

• Penalty-based red-blue annulus cover optimization
• Polynomial-time algorithms for annulus types
• Constrained center on horizontal line L