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Approximating Densest Subgraph in Geometric Intersection Graphs

Authors Sariel Har-Peled , Saladi Rahul

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Author Details

Sariel Har-Peled
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Saladi Rahul
  • Indian Institute of Science (IISc), Bangalore, India

Funding

Research of Saladi Rahul is funded by Walmart Center for Tech Excellence.

Cite As Get BibTex

Sariel Har-Peled and Saladi Rahul. Approximating Densest Subgraph in Geometric Intersection Graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.43

Abstract

For an undirected graph 𝖦 = (𝖡, 𝖀), with n vertices and m edges, the densest subgraph problem, is to compute a subset S βŠ† 𝖡 which maximizes the ratio |𝖀_S|/|S|, where 𝖀_S βŠ† 𝖀 is the set of all edges of 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 Ξ©(m) time. We present near-linear time (in n) approximation algorithms for the densest subgraph problem on 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.
As a by-product, we show a reduction from (shallow) range-reporting to approximate counting/sampling which seems to be new and is useful for other problems such as independent query sampling.

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Computational geometry
Keywords
  • Geometric intersection graphs
  • Densest subgraph
  • Range searching
  • Approximation algorithms

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