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Better Diameter Algorithms for Bounded VC-dimension Graphs and Geometric Intersection Graphs

Lech Duraj, Filip Konieczny, Krzysztof Potępa

TL;DR

The paper tackles subquadratic diameter computation for graphs with bounded distance VC-dimension, a broad class that includes minor-free and geometric intersection graphs. It introduces a unified framework based on low-difference vertex orders and eps-net ideas, enabling efficient neighborhood encodings and NSDS-based diameter testing. The main results are a randomized Las Vegas algorithm that decides k-Diameter in $\tilde{O}(k \cdot m \cdot n^{1-1/d})$ time for explicit graphs and a subquadratic $\tilde{O}(k n^{2-1/d})$ time framework for implicit graphs; specialized to unit-square intersection graphs, it achieves $\tilde{O}(k \cdot n^{7/4})$ time, with analogous bounds for convex polygons. These advances sharpen subquadratic diameter computations in geometric settings and open questions for unit-disk graphs and tight lower bounds in related graph classes.

Abstract

We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in $\mathcal{O}(k \cdot n^{1-1/d} \cdot m \cdot \mathrm{polylog}(n))$ time complexity for the graph on $n$ vertices and $m$ edges, where $k$ is the diameter and $d$ is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an $\mathcal{O}(n^{7/4} \cdot \mathrm{polylog}(n))$ parameterized diameter algorithm for unit square intersection graph of size $n$, as well as a more general algorithm for convex polygon intersection graphs.

Better Diameter Algorithms for Bounded VC-dimension Graphs and Geometric Intersection Graphs

TL;DR

The paper tackles subquadratic diameter computation for graphs with bounded distance VC-dimension, a broad class that includes minor-free and geometric intersection graphs. It introduces a unified framework based on low-difference vertex orders and eps-net ideas, enabling efficient neighborhood encodings and NSDS-based diameter testing. The main results are a randomized Las Vegas algorithm that decides k-Diameter in time for explicit graphs and a subquadratic time framework for implicit graphs; specialized to unit-square intersection graphs, it achieves time, with analogous bounds for convex polygons. These advances sharpen subquadratic diameter computations in geometric settings and open questions for unit-disk graphs and tight lower bounds in related graph classes.

Abstract

We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in time complexity for the graph on vertices and edges, where is the diameter and is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an parameterized diameter algorithm for unit square intersection graph of size , as well as a more general algorithm for convex polygon intersection graphs.
Paper Structure (26 sections, 27 theorems, 2 equations, 3 figures, 5 algorithms)

This paper contains 26 sections, 27 theorems, 2 equations, 3 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution and paper structure
  3. Preliminaries
  4. Graphs, neighbourhoods and diameters
  5. Hypergraphs and VC-dimension
  6. Geometric intersection graphs
  7. eps-nets
  8. General algorithm framework
  9. Orders on hypergraphs
  10. Algorithm for general sparse graphs
  11. Diameter testing for implicit graphs
  12. Algorithm outline
  13. Ball expansion
  14. Polygon intersection graphs
  15. Conclusion and open problems
  16. ...and 11 more sections

Key Result

Theorem 2

Let $(X,\mathcal{R})$ be a hypergraph where $|X| = n$, and let its VC-dimension be bounded by $d$. Then:

Figures (3)

  • Figure 1: The top and bottom areas in a node.
  • Figure 4: Boundaries implied by squares overlapping with the stripe. The red and blue lines mark top and bottom boundaries respectively. The boundaries can intersect.
  • Figure 5: Information about the boundaries stored by a segment tree's node. The top_hash is the hash of the points in the red area. The bot_hash is the hash of the points in the blue area. The hash is the hash of the points in the union of red and blue area.

Theorems & Definitions (31)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Lemma 4
  • Definition 5
  • Definition 6
  • Lemma 6
  • Lemma 6
  • Definition 7
  • ...and 21 more