Approximating Densest Subgraph in Geometric Intersection Graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866913367139024896 |
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| author | Har-Peled, Sariel Saladi, Rahul |
| author_facet | Har-Peled, Sariel Saladi, Rahul |
| contents | $ \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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximating Densest Subgraph in Geometric Intersection Graphs Har-Peled, Sariel Saladi, Rahul Computational Geometry $ \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. |
| title | Approximating Densest Subgraph in Geometric Intersection Graphs |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2405.18337 |