Finding Dense Subgraphs via Low-Rank Bilinear Optimization.

Given a graph, the Densest k -Subgraph (DkS) problem asks for the subgraph on k vertices that contains the largest number of edges. In this work, we develop a new algorithm for DkS that searches a low-dimensional space for provably dense subgraphs. Our algorithm comes with novel performance bounds that depend on the graph spectrum. Our graph-dependent bounds are surprisingly tight for real-world graphs where we find subgraphs with density provably within 70% of the optimum. These guarantees are significantly tighter than the best available worst case a priori bounds. Our algorithm runs in nearly linear time, under spectral assumptions satisfied by most graphs found in applications. Moreover, it is highly scalable and parallelizable. We demonstrate this by implementing it in MapReduce and executing numerous experiments on massive real-world graphs that have up to billions of edges. We empirically show that our algorithm can find subgraphs of significantly higher density compared to the previous state of the art.