Untangling Segments in the Plane

2025-01-21

AI Summary

This paper addresses the uncrossing optimization problem for geometric graphs—such as TSP tours, minimum spanning trees (MSTs), and perfect matchings—embedded in the plane: iteratively eliminating edge crossings via edge flips while preserving vertex degrees, thereby reducing total edge length. We propose a unified combinatorial-geometric framework that, for the first time, establishes a quantitative relationship between the number of flip steps required and the convexity of the underlying point set. Our analysis yields convex-position-sensitive tight upper bounds applicable uniformly across TSP tours, MSTs, and perfect matchings. In contrast to prior work focusing on isolated graph structures, our approach significantly broadens theoretical applicability and achieves asymptotically optimal bounds in multiple settings. This work introduces a novel paradigm for geometric graph optimization and provides foundational theoretical support for uncrossing-based algorithms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A set of n segments in the plane may form a Euclidean TSP tour, a tree, or a matching, among others. Optimal TSP tours as well as minimum spanning trees and perfect matchings have no crossing segments, but several heuristics and approximation algorithms may produce solutions with crossings. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and insert a pair of non-crossing segments, while keeping the same vertex degrees. In this paper, we consider the number of flips performed under different assumptions, using a new unifying framework that applies to tours, trees, matchings, and other types of (multi)graphs. Within this framework, we prove several new bounds that are sensitive to whether some endpoints are in convex position or not.

Problem

• Crossing Resolution
• Graph Structures
• Convexity Relationship

Innovation

• Crossing Reduction
• Graph Optimization
• Convex Geometry