Distributed Dense Subgraph Detection and Low Outdegree Orientation.

The maximum density 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 $G$. We give the following results: Given a value $\tilde{D} \leq D$ and $0 < \epsilon < 1$, we show that a subgraph of density at least $(1-\epsilon)\tilde{D}$ can be identified deterministically in $O((\log n) / \epsilon)$ rounds in the \textsf{LOCAL} model. Using randomization, we show that such subgraph can be identified in $O((\log^3 n) / \epsilon^3)$ rounds in the \textsf{CONGEST} model with high probability. We also give a $\Omega(1/\epsilon)$-round lower bound which shows that our result for the \textsf{LOCAL} model is tight up to a $O(\log n)$ factor. Moreover, our result can be extended to solve the directed version of the problem introduced by Kannan and Vinay \cite{KV99}. Given an integer $\tilde{D} \geq D$ and $\Omega(1/\tilde{D}) < \epsilon < 1/4$, we give an $O(\log^2 n \cdot (\log^{2.71} \Delta) /\epsilon^2)$-round deterministic algorithm in the \textsf{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 for this problem is by Harris \cite{Harris19} that runs in $\tilde{O}((\log^6 n) / \epsilon^4)$ rounds in the \local model.