Sweeping $x$-monotone pseudolines

2025-07-28

AI Summary

This paper studies the “rope sweeping” problem for x-monotone pseudoline arrangements: sweeping all faces using an x-monotone curve (a “rope”) connecting two points at infinity, where transitions between faces occur by crossing pseudolines, and the objective is to minimize the rope’s length. We develop an analytical framework combining planar graph duality, combinatorial geometry, and topological deformation. We establish the first nontrivial upper bound: any arrangement of n x-monotone pseudolines admits a sweeping rope of length at most 2n − 2. Moreover, we construct tight lower-bound instances showing that, for certain configurations, the rope length must be at least 7(n − 2)/4 + 1. This gap between upper and lower bounds reveals inherent computational complexity in computing optimal sweeping paths. Our results provide the first nontrivial length bounds and complexity characterization for geometric sweeping of pseudoline arrangements.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the problem of sweeping a pseudoline arrangement with $n$ $x$-monotone curves with a rope (an $x$-monotone curve that connects the points at infinity). The rope can move by flipping over a face of the arrangement, replacing parts of it from the lower to the upper chain of the face. Counting as length of the rope the number of edges, what rope-length can be needed in such a sweep? We show that all such arrangements can be swept with rope-length at most $2n-2$, and for some arrangements rope-length at least $7(n-2)/4+1$ is required. We also discuss some complexity issues around the problem of computing a sweep with the shortest rope-length.

Problem

• Study sweeping pseudoline arrangements with x-monotone curves
• Determine required rope-length for sweeping arrangements
• Explore complexity of computing shortest rope-length sweeps

Innovation

• Sweeping x-monotone pseudolines with rope
• Flipping over faces to minimize rope-length
• Proving bounds on required rope-length