Computing the Girth of a Segment Intersection Graph

TL;DR

This paper introduces an efficient algorithm to compute the girth of segment intersection graphs in the plane, advancing the computational complexity bounds for this problem and extending to algebraic curves.

Contribution

It presents the first subquadratic expected-time algorithm for girth computation in segment intersection graphs, utilizing advanced matrix multiplication and separator techniques.

Findings

Achieved $O(n^{1.483})$ expected time complexity.

Extended the approach to algebraic curves and semialgebraic sets.

Progressed towards an open problem in computational geometry.

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.