LIPIcs.STACS.2025.43.pdf
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For an undirected graph π¦ = (π΅, π€), with n vertices and m edges, the densest subgraph problem, is to compute a subset S β π΅ which maximizes the ratio |π€_S|/|S|, where π€_S β π€ is the set of all edges of π¦ 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.
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