Detecting high log-densities: an O(n¼) approximation for densest k-subgraph

  • Authors:

  • Aditya Bhaskara;Moses Charikar;Eden Chlamtac;Uriel Feige;Aravindan Vijayaraghavan

  • Affiliations:

  • Princeton University, Princeton, NJ, USA;Princeton University, Princeton, NJ, USA;Weizmann Institute of Science, Rehovot, Israel;Weizmann Institute of Science, Rehovot, Israel;Princeton University, Princeton, NJ, USA

  • Venue:
  • Proceedings of the forty-second ACM symposium on Theory of computing
  • Year:
  • 2010

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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 n1/3 - c for some fixed c0 (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 n¼ + ε in time nO(1/ε). If allowed to run for time nO(log n), the algorithm achieves an approximation ratio of O(n¼). 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(n¼ -ε)-approximation in O(2nO(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(n¼) guarantee overall.