2026-08-18T12:42:53Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13093 2024-03-06T10:51:33Z ddc:004 open_access

Distributed Dense Subgraph Detection and Low Outdegree Orientation Su, Hsin-Hao Vu, Hoa T. Distributed Algorithms Network Algorithms 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Hsin-Hao Su and Hoa T. Vu 2020 Is Part Of LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.DISC.2020.15 urn:nbn:de:0030-drops-130938 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.15 eng https://creativecommons.org/licenses/by/3.0/legalcode