We present a local algorithm for finding dense subgraphs of bipartite graphs, according to the definition of density proposed by Kannan and Vinay. Our algorithm takes as input a bipartite graph with a specified starting vertex, and attempts to find a dense subgraph near that vertex. We prove that for any subgraph S with k vertices and density theta, there are a significant number of starting vertices within S for which our algorithm produces a subgraph S' with density theta O(log n) on at most O(D k^2) vertices, where D is the maximum degree. The running time of the algorithm is O(D k^2), independent of the number of vertices in the graph.
This paper considers the problem of computing the dense k -vertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n δ ) , for some δ < 1 3 . Given an n-vertex graph G and a parameter k, we are to find a k-vertex subgraph with the maximum number of edges. This problem is NP-hard. We show that the problem remains NP-hard even when the maximum degree in G is three. When G contains a k-clique, we give an algorithm that for any e sub sub<) e). We study the applicability of semidefinite programming for approximating the dense k-subgraph problem. Our main result in this respect is negative, showing that for k @ n1 3, semidefinite programs fail to distinguish between graphs that contain k-cliques and graphs in which the densest k-vertex subgraph has average degree below logn.