Distributed Dense Subgraph Detection and Low Outdegree Orientation

The densest subgraph problem, introduced in the 80s by Picard and Queyranne as well as Goldberg, is a classic problem in combinatorial optimization with a wide range of applications. The lowest outdegree orientation problem is known to be its dual problem. We study both the problem of finding dense subgraphs and the problem of computing a low outdegree orientation in the distributed settings. Suppose $G=(V,E)$ is the underlying network as well as the input graph. Let $D$ denote the density of the maximum density subgraph of $G$. Our main results are as follows. Given a value $\tilde{D} \leq D$ and $0 < \epsilon < 1$, we show that a subgraph with density at least $(1-\epsilon)\tilde{D}$ can be identified deterministically in $O((\log n) / \epsilon)$ rounds in the LOCAL model. We also present a lower bound showing that our result for the LOCAL model is tight up to an $O(\log n)$ factor. In the CONGEST model, we show that such a subgraph can be identified in $O((\log³ n) / \epsilon^3)$ rounds with high probability. Our techniques also lead to an $O(diameter + (\log⁴ n)/\epsilon^4)$-round algorithm that yields a $1-\epsilon$ approximation to the densest subgraph. This improves upon the previous $O(diameter /\epsilon \cdot \log n)$-round algorithm by Das Sarma et al. [DISC 2012] that only yields a $1/2-\epsilon$ approximation. Given an integer $\tilde{D} \geq D$ and $\Omega(1/\tilde{D}) < \epsilon < 1/4$, we give a deterministic, $\tilde{O}((\log² n) /\epsilon^2)$-round algorithm in the CONGEST model that computes an orientation where the outdegree of every vertex is upper bounded by $(1+\epsilon)\tilde{D}$. Previously, the best deterministic algorithm and randomized algorithm by Harris [FOCS 2019] run in $\tilde{O}((\log⁶ n)/ \epsilon^4)$ rounds and $\tilde{O}((\log³ n) /\epsilon^3)$ rounds respectively and only work in the LOCAL model.