2026-09-08T02:21:58Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:26522 2026-09-05T19:46:19Z ddc:004 open_access

Partially-Dynamic Maximum Flow in Dense Graphs Kravchenko, Egor Probst Gutenberg, Maximilian Maximum Flow Dynamic Graph Algorithm Data Structure We give the first algorithms that, with high probability, maintain (1-ε)-approximate s-t maximum flow in an n-vertex undirected, capacitated graph undergoing either only edge insertions or only edge deletions in total update time Õ_ε(n²). For dense graphs, this yields polylogarithmic amortized update time, which was previously only obtained for the special case of uncapacitated graphs undergoing edge insertions. We develop the following two algorithms: - For graphs undergoing deletions, we generalize the congestion-balancing framework from [Aaron Bernstein et al., 2020], which was developed for maximum matching. We then show that this framework can be simulated on cut sparsifiers, which yields significant speed-ups. - For graphs undergoing insertions, we show that the sparsification techniques by Eppstein et al. [Eppstein et al., 1997] can be combined more directly with the techniques from Henzinger and Goranci [Goranci and Henzinger, 2023]. We thereby bypass the need to dynamize the more involved residual graph sparsification approach by Levin and Karger [Karger and Levine, 2015] suggested in [Goranci et al., 2025], and extend their result to capacitated graphs. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Egor Kravchenko and Maximilian Probst Gutenberg 2026 Is Part Of LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2026.133 urn:nbn:de:0030-drops-265226 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.133 eng https://creativecommons.org/licenses/by/4.0/legalcode