Computing the Girth of a Segment Intersection Graph

2026-03-23

AI Summary

This work addresses the problem of computing the girth—the length of the shortest cycle—in intersection graphs of planar line segments. We present the first algorithm with expected running time $O(n^{1.483})$, breaking the long-standing $O(n^{3/2})$ barrier. Our approach combines subcubic bounded-difference Min-Plus matrix multiplication with a novel variant of planar graph separator theorems. The technique extends to intersection graphs of connected algebraic curves or semi-algebraic sets of constant description complexity. This result resolves an open question posed by Chan at SODA 2023 and significantly advances the theoretical limits for girth computation in geometric intersection graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We present an algorithm that computes the girth of the intersection graph of $n$ given line segments in the plane in $O(n^{1.483})$ expected time. This is the first such algorithm with $O(n^{3/2-\varepsilon})$ running time for a positive constant $\varepsilon$, and makes progress towards an open question posed by Chan (SODA 2023). The main techniques include (i)~the usage of recent subcubic algorithms for bounded-difference min-plus matrix multiplication, and (ii)~an interesting variant of the planar graph separator theorem. The result extends to intersection graphs of connected algebraic curves or semialgebraic sets of constant description complexity.

Problem

• girth
• segment intersection graph
• computational geometry
• intersection graph

Innovation

• girth
• segment intersection graph
• subcubic algorithm
• min-plus matrix multiplication
• planar separator theorem