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# Distributed Dense Subgraph Detection and Low Outdegree Orientation

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LIPIcs.DISC.2020.15.pdf
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## Acknowledgements

We thank David Harris for pointing out Lemma 12, which can be used to improve the running time of the dual rounding algorithm in our previous version.

## Cite As

Hsin-Hao Su and Hoa T. Vu. Distributed Dense Subgraph Detection and Low Outdegree Orientation. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.DISC.2020.15

## Abstract

The densest subgraph problem, introduced in the 80s by Picard and Queyranne [Networks 1982] as well as Goldberg [Tech. Report 1984], 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 D̃ ≤ D and 0 < ε < 1, we show that a subgraph with density at least (1-ε)D̃ can be identified deterministically in O((log n) / ε) 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) / ε³) rounds with high probability. Our techniques also lead to an O(diameter + (log⁴ n)/ε⁴)-round algorithm that yields a 1-ε approximation to the densest subgraph. This improves upon the previous O(diameter /ε ⋅ log n)-round algorithm by Das Sarma et al. [DISC 2012] that only yields a 1/2-ε approximation. - Given an integer D̃ ≥ D and Ω(1/D̃) < ε < 1/4, we give a deterministic, Õ((log² n) /ε²)-round algorithm in the CONGEST~ model that computes an orientation where the outdegree of every vertex is upper bounded by (1+ε)D̃. Previously, the best deterministic algorithm and randomized algorithm by Harris [FOCS 2019] run in Õ((log⁶ n)/ ε⁴) rounds and Õ((log³ n) /ε³) rounds respectively and only work in the LOCAL model.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Distributed algorithms
• Networks → Network algorithms
• Mathematics of computing → Graph algorithms
##### Keywords
• Distributed Algorithms
• Network Algorithms

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