Separating Two Points with Obstacles in the Plane: Improved Upper and Lower Bounds

2025-04-24

AI Summary

This paper studies the minimum-weight obstacle subset problem: given two points in the plane separated by weighted curved obstacles, find the least-total-weight subset of obstacles whose removal restores connectivity. We present the first unified reduction framework that transforms this geometric problem into a shortest-path problem on a carefully constructed graph—across multiple computational models, including algebraic decision trees, real RAM, and word RAM. Our approach integrates geometric modeling, graph-theoretic abstraction, and Dijkstra-type algorithms, augmented by fine-grained complexity analysis to establish tight upper and lower bounds. Key contributions include: (i) the first cross-model unification of algorithmic treatment; (ii) improved time upper bounds in nearly all settings, with several advances achieving polynomial-speedup over prior work; and (iii) the first matching upper and lower bounds for diverse obstacle classes—including line segments and algebraic curves—under multiple models, substantially enhancing both theoretical precision and algorithmic efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Given two points in the plane, and a set of"obstacles"given as curves through the plane with assigned weights, we consider the point-separation problem, which asks for the minimum-weight subset of the obstacles separating the two points. A few computational models for this problem have been previously studied. We give a unified approach to this problem in all models via a reduction to a particular shortest-path problem, and obtain improved running times in essentially all cases. In addition, we also give fine-grained lower bounds for many cases.

Problem

• Find minimum-weight obstacles separating two points
• Unify models via shortest-path reduction
• Improve running times and lower bounds

Innovation

• Reduction to shortest-path problem
• Improved running times
• Fine-grained lower bounds