Detecting high log-densities - An O(n^1/4) approximation for densest k-subgraph

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 - ε} for some fixed ε>0 (later estimated at around ε= 1/90). We present an algorithm that for every ε> 0 approximates the Densest k-Subgraph problem within a ratio of n^1/4+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^1/4). 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^1/4-ε)-approximation in O(2^nO(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 ^1/4) guarantee overall.