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arxiv cs/0702032 v1 pith:R3DNZHJT submitted 2007-02-05 cs.DS

Finding large and small dense subgraphs

classification cs.DS
keywords densestproblemsubgraphsdalksdamksdensefindfinding
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Fast and Simple Densest Subgraph with Predictions

    cs.DS 2025-05 unverdicted novelty 7.0

    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.

  2. A Note on Approximability of Densest At-Least-k-Subgraph

    cs.DS 2026-05 unverdicted novelty 6.0

    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.