Finding large and small dense subgraphs
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We consider two optimization problems related to finding dense subgraphs. The densest at-least-k-subgraph problem (DalkS) is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (DamkS) is defined similarly. These problems are related to the well-known densest k-subgraph problem (DkS), which is to find the densest subgraph on exactly k vertices. We show that DalkS can be approximated efficiently, while DamkS is nearly as hard to approximate as the densest k-subgraph problem.
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Cited by 2 Pith papers
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Fast and Simple Densest Subgraph with Predictions
With a reasonably accurate predictor for nodes in the solution, simple linear-time algorithms achieve (1-ε) approximation for densest subgraph and its densest at-most-k variant.
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A Note on Approximability of Densest At-Least-k-Subgraph
A reduction from DkS establishes (3/2-ε) inapproximability for DALkS under constant-factor hardness of DkS, with (2-ε) hardness under stronger assumptions and W[1]-hardness for exact DALkS parameterized by k.
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