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Computing the Girth of a Segment Intersection Graph

Timothy M. Chan, Yuancheng Yu

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.

Computing the Girth of a Segment Intersection Graph

Abstract

We present an algorithm that computes the girth of the intersection graph of given line segments in the plane in expected time. This is the first such algorithm with running time for a positive constant , 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.
Paper Structure (13 sections, 10 theorems, 1 equation, 4 figures)

This paper contains 13 sections, 10 theorems, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. New result
  3. Why is the problem challenging?
  4. Our approach
  5. Ingredient 2: A variant of the planar-graph separator theorem.
  6. Ingredient 3: Ensuring simplicity of walks.
  7. Preliminaries
  8. Bounded-difference min-plus product
  9. A variant of the planar separator theorem
  10. Ensuring simplicity of walks
  11. Multiple Distances in Clustered Planar Graphs
  12. Girth for Line Segments
  13. Extension to Algebraic Curves and Semialgebraic Sets

Key Result

Lemma 1

Given two integer $n\times n$ matrices $A$ and $B$ where each column of $A$ satisfies the bounded-difference property (i.e., $|A(i,j)-A(i+1,j)|\le O(1)$ for every $i,j$), we can compute their min-plus product $A\star B$ in $\widetilde{O}(n^{(3+\omega)/2})\le O(n^{2.686})$ expected time. The same hol

Figures (4)

  • Figure 1: An example of the shortest cycle of a segment intersection graph (segments not in the shortest cycle are dotted). The girth is $6$.
  • Figure 2: Proof of Lemma \ref{['lem-sep']}. Solid curves consist of edges in $G$, whereas dashed curves may contain edges in the triangulation but not in $G$.
  • Figure 3: The clustered planar graph ${\widehat{H}}$. The dotted edges and the dashed edges have weight 0, while the solid edges have weight 1. Edges with weight 2 are omitted.
  • Figure 4: These are not simple cycles in the intersection graph!

Theorems & Definitions (11)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Corollary 9
  • Lemma 10
  • ...and 1 more