A Local Algorithm for Finding Dense Subgraphs

A local graph algorithm is one that searches for an approximation of the best solution near a specified starting vertex, and has a running time independent of the size of the graph. Recently, local algorithms have been developed for graph partitioning and clustering. In this paper, we present a local algorithm for finding dense subgraphs of bipartite graphs, according to the measure of density proposed by Karman and Vinay. The algorithm takes as input a bipartite graph with a. specified starting vertex, and attempts to find a dense subgraph near that vertex. We prove the following local approximation guarantee for the algorithm. For any subgraph S with k vertices and density theta, there is a large set of starting vertices within S for which the algorithm produces a subgraph with density Omega(theta/log Delta), where Delta is the maximum degree. The running time of the algorithm is O(Delta k(2)), independent of the number of vertices in the graph.