A local algorithm for finding dense subgraphs

Abstract

We describe a local algorithm for finding subgraphs with high density, according to a measure of density introduced by Kannan and Vinay [1999]. The algorithm takes as input a bipartite graph G , a starting vertex v , and a parameter k , and outputs an induced subgraph of G . It is local in the sense that it does not examine the entire input graph; instead, it adaptively explores a region of the graph near the starting vertex. The running time of the algorithm is bounded by O (Δ k 2 ), which depends on the maximum degree Δ, but is otherwise independent of the graph. We prove the following approximation guarantee: for any subgraph S with k′ vertices and density θ, there exists a set S ′ ⊆ S for which the algorithm outputs a subgraph with density Ω(θ/log Δ) whenever v ∈ S ′ and k ≥ k ′. We prove that S ′ contains at least half of the edges in S .