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Approximating Densest Subgraph in Geometric Intersection Graphs

Sariel Har-Peled, Rahul Saladi

TL;DR

This work tackles the densest subgraph problem on implicit geometric intersection graphs, focusing on disks in the plane with arbitrary radii. It introduces a near-linear-time framework by bridging range-reporting and approximate counting/sampling, enabling highly efficient solutions when edges are not explicitly provided. The authors present two complementary algorithms: a $(2+\varepsilon)$-approximation with $O_\varepsilon(n \log^3 n)$ time and a $(1+\varepsilon)$-approximation with $O_\varepsilon(n \log^2 n \log \log n)$ time, relying on a combination of low-degree pruning and uniform edge sampling respectively. The methods leverage a data-structure that can sample uniformly from intersecting disks and estimate intersection counts, offering practical pathways for large-scale geometric networks and applications such as network coverage and pollution mapping.

Abstract

$ \newcommand{\cardin}[1]{\left| {#1} \right|}% \newcommand{\Graph}{\Mh{\mathsf{G}}}% \providecommand{\G}{\Graph}% \renewcommand{\G}{\Graph}% \providecommand{\GA}{\Mh{H}}% \renewcommand{\GA}{\Mh{H}}% \newcommand{\VV}{\Mh{\mathsf{V}}}% \newcommand{\VX}[1]{\VV\pth{#1}}% \providecommand{\EE}{\Mh{\mathsf{E}}}% \renewcommand{\EE}{\Mh{\mathsf{E}}}% \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s}$ For an undirected graph $\mathsf{G}=(\mathsf{V}, \mathsf{E})$, with $n$ vertices and $m$ edges, the \emph{densest subgraph} problem, is to compute a subset $S \subseteq \mathsf{V}$ which maximizes the ratio $|\mathsf{E}_S| / |S|$, where $\mathsf{E}_S \subseteq \mathsf{E}$ is the set of all edges of $\mathsf{G}$ with endpoints in $S$. The densest subgraph problem is a well studied problem in computer science. Existing exact and approximation algorithms for computing the densest subgraph require $Ω(m)$ time. We present near-linear time (in $n$) approximation algorithms for the densest subgraph problem on \emph{implicit} geometric intersection graphs, where the vertices are explicitly given but not the edges. As a concrete example, we consider $n$ disks in the plane with arbitrary radii and present two different approximation algorithms.

Approximating Densest Subgraph in Geometric Intersection Graphs

TL;DR

This work tackles the densest subgraph problem on implicit geometric intersection graphs, focusing on disks in the plane with arbitrary radii. It introduces a near-linear-time framework by bridging range-reporting and approximate counting/sampling, enabling highly efficient solutions when edges are not explicitly provided. The authors present two complementary algorithms: a -approximation with time and a -approximation with time, relying on a combination of low-degree pruning and uniform edge sampling respectively. The methods leverage a data-structure that can sample uniformly from intersecting disks and estimate intersection counts, offering practical pathways for large-scale geometric networks and applications such as network coverage and pollution mapping.

Abstract

For an undirected graph , with vertices and edges, the \emph{densest subgraph} problem, is to compute a subset which maximizes the ratio , where is the set of all edges of with endpoints in . The densest subgraph problem is a well studied problem in computer science. Existing exact and approximation algorithms for computing the densest subgraph require time. We present near-linear time (in ) approximation algorithms for the densest subgraph problem on \emph{implicit} geometric intersection graphs, where the vertices are explicitly given but not the edges. As a concrete example, we consider disks in the plane with arbitrary radii and present two different approximation algorithms.
Paper Structure (34 sections, 20 theorems, 20 equations, 3 figures)

This paper contains 34 sections, 20 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Problem statement and our results
  4. From reporting to approximate counting/sampling.
  5. The application.
  6. A $(2+\varepsilon)$-approximation.
  7. A $(1+\varepsilon)$-approximation.
  8. Preliminaries
  9. Definitions
  10. Reporting all intersecting pairs of disks
  11. From reporting to approximate sampling/counting
  12. The data-structure
  13. The given reporting data-structure.
  14. Data-structure construction.
  15. Answering a query.
  16. ...and 19 more sections

Key Result

Lemma 2.6

Let $\mathcal{O} \subseteq \mathcal{D}$ be the densest subset, and let $\nabla = {\nabla}\mleft({\mathcal{O}}\mright)$. Then, for any object $u \in \mathcal{O}$, we have $d_{\mathcal{O}}\mleft({u}\mright) = \left| { u \sqcap (\mathcal{O}-u)} \right| \geq \nabla$, where $u \sqcap (\mathcal{O}-u) = \l

Figures (3)

  • Figure 1.1: Five disks and their corresponding graph. The densest subgraph is $\{a,b,c,d\}$ with density $5/4$.
  • Figure 1.2: Our results.
  • Figure 2.1: Vertical decomposition of four disks.

Theorems & Definitions (25)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Example 3.1
  • Lemma 3.2
  • Lemma 3.4
  • Theorem 3.5
  • ...and 15 more