A local algorithm for finding dense subgraphs

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 <i>G</i>, a starting vertex <i>v</i>, and a parameter <i>k</i>, and outputs an induced subgraph of <i>G</i>. 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 <i>O</i>(Δ <i>k</i><sup>2</sup>), which depends on the maximum degree Δ, but is otherwise independent of the graph. We prove the following approximation guarantee: for any subgraph <i>S</i> with <i>k′</i> vertices and density θ, there exists a set <i>S</i>′ ⊆ <i>S</i> for which the algorithm outputs a subgraph with density Ω(θ/log Δ) whenever <i>v</i> ∈ <i>S</i>′ and <i>k</i> ≥ <i>k</i>′. We prove that <i>S</i>′ contains at least half of the edges in <i>S</i>.

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