research-article

Detecting high log-densities: an O(n^¼) approximation for densest k-subgraph

Authors:

Aditya Bhaskara,

Moses Charikar,

Aravindan VijayaraghavanAuthors Info & Claims

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

Pages 201 - 210

https://doi.org/10.1145/1806689.1806719

Published: 05 June 2010 Publication History

Abstract

In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges. There is a significant gap between the best known upper and lower bounds for this problem. It is NP-hard, and does not have a PTAS unless NP has subexponential time algorithms. On the other hand, the current best known algorithm of Feige, Kortsarz and Peleg, gives an approximation ratio of n^{1/3 - c} for some fixed c>0 (later estimated at around c= 1/90).

We present an algorithm that for every ε> 0 approximates the Densest k-Subgraph problem within a ratio of n^{¼ + ε} in time n^O(1/ε). If allowed to run for time n^{O(log n)}, the algorithm achieves an approximation ratio of O(n^¼). Our algorithm is inspired by studying an average-case version of the problem where the goal is to distinguish random graphs from random graphs with planted dense subgraphs -- the approximation ratio we achieve for the general case matches the "distinguishing ratio" we obtain for this planted problem.

At a high level, our algorithms involve cleverly counting appropriately defined trees of constant size in G, and using these counts to identify the vertices of the dense subgraph. We say that a graph G(V,E) has log-density α if its average degree is Θ(|V|^α). The algorithmic core of our result is a procedure to output a k-subgraph of 'nontrivial' density whenever the log-density of the densest k-subgraph is larger than the log-density of the host graph.

We outline an extension to our approximation algorithm which achieves an O(n^{¼ -ε})-approximation in O(2^{n^O(ε)}) time. We also show that, for certain parameter ranges, eigenvalue and SDP based techniques can outperform our basic distinguishing algorithm for random instances (in polynomial time), though without improving upon the O(n^¼) guarantee overall.

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Cited By

Liu ZChen WLi FQi KWang J(2024)Algorithms for Densest Subgraphs of Vertex-Weighted GraphsMathematics10.3390/math1214220612:14(2206)Online publication date: 14-Jul-2024
https://doi.org/10.3390/math12142206
Balalau OBonchi FChan TGullo FSozio MXie H(2024)Finding Subgraphs with Maximum Total Density and Limited Overlap in Weighted HypergraphsACM Transactions on Knowledge Discovery from Data10.1145/363941018:4(1-21)Online publication date: 12-Feb-2024
https://dl.acm.org/doi/10.1145/3639410
Anand ALee ELi JSaranurak TMohar BShinkar IO'Donnell R(2024)Approximating Small Sparse CutsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649747(319-330)Online publication date: 10-Jun-2024
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Index Terms

Detecting high log-densities: an O(n^¼) approximation for densest k-subgraph
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

June 2010

812 pages

ISBN:9781450300506

DOI:10.1145/1806689

General Chair:
Michael Mitzenmacher
Harvard
,
Program Chair:
Leonard J. Schulman
Caltech

Copyright © 2010 ACM.

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Published: 05 June 2010

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STOC'10: Symposium on Theory of Computing

June 5 - 8, 2010

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https://doi.org/10.3390/math12142206
Balalau OBonchi FChan TGullo FSozio MXie H(2024)Finding Subgraphs with Maximum Total Density and Limited Overlap in Weighted HypergraphsACM Transactions on Knowledge Discovery from Data10.1145/363941018:4(1-21)Online publication date: 12-Feb-2024
https://dl.acm.org/doi/10.1145/3639410
Anand ALee ELi JSaranurak TMohar BShinkar IO'Donnell R(2024)Approximating Small Sparse CutsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649747(319-330)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649747
Oettershagen LWang HGionis AChua TNgo CKa-Wei Lee RKumar RLauw H(2024)Finding Densest Subgraphs with Edge-Color ConstraintsProceedings of the ACM Web Conference 202410.1145/3589334.3645647(936-947)Online publication date: 13-May-2024
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