A Local Algorithm for Finding Dense Bipartite-Like Subgraphs

Abstract

We give a local algorithm to extract dense bipartite-like subgraphs which characterize cyber-communities in the Web [13]. We use the bipartiteness ratio of a set as the quality measure that was introduced by Trevisan [20]. Our algorithm, denoted as FindDenseBipartite (v,s,θ), takes as input a starting vertex v, a volume target s and a bipartiteness ratio parameter θ and outputs an induced subgraph of G. It is guaranteed to have the following approximation performance: for any subgraph S with bipartiteness ratio θ, there exists a subset S _θ  ⊆ S such that $\textrm{vol}(S_\theta)\geq \textrm{vol}(S)/9$ and that if the starting vertex v ∈ S _θ and $s\geq \textrm{vol}(S)$, the algorithm FindDenseBipartite (v,s,θ) outputs a subgraph (X,Y) with bipartiteness ratio $O(\sqrt{\theta})$. The running time of the algorithm is O(s ²(Δ + logs)), where Δ is the maximum degree of G, independent of the size of G.

The author is partially supported by the Grand Project “Network Algorithms and Digital Information” of the Institute of Software, Chinese Academy of Sciences.