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Documents authored by Probst Gutenberg, Maximilian

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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.133
Partially-Dynamic Maximum Flow in Dense Graphs

Authors: Egor Kravchenko and Maximilian Probst Gutenberg

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract
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.
Cite as

Egor Kravchenko and Maximilian Probst Gutenberg. Partially-Dynamic Maximum Flow in Dense Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 133:1-133:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kravchenko_et_al:LIPIcs.ICALP.2026.133,
  author =	{Kravchenko, Egor and Probst Gutenberg, Maximilian},
  title =	{{Partially-Dynamic Maximum Flow in Dense Graphs}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{133:1--133:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.133},
  URN =		{urn:nbn:de:0030-drops-265226},
  doi =		{10.4230/LIPIcs.ICALP.2026.133},
  annote =	{Keywords: Maximum Flow, Dynamic Graph Algorithm, Data Structure}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.20
Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary

Authors: Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Designing efficient dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms and has witnessed many exciting recent developments in, e.g., dynamic matching (Wajc STOC'20) and decremental shortest paths (Chuzhoy and Khanna STOC'19). Compared to other graph primitives (e.g. spanning trees and matchings), designing such algorithms for graph spanners and (more broadly) graph sparsifiers poses a unique challenge since there is no fast deterministic algorithm known for static computation and the lack of a way to adjust the output slowly (known as "small recourse/replacements"). This paper presents the first non-trivial efficient adaptive algorithms for maintaining many sparsifiers against an adaptive adversary. Specifically, we present algorithms that maintain 1) a polylog(n)-spanner of size Õ(n) in polylog(n) amortized update time, 2) an O(k)-approximate cut sparsifier of size Õ(n) in Õ(n^{1/k}) amortized update time, and 3) a polylog(n)-approximate spectral sparsifier in polylog(n) amortized update time. Our bounds are the first non-trivial ones even when only the recourse is concerned. Our results hold even against a stronger adversary, who can access the random bits previously used by the algorithms and the amortized update time of all algorithms can be made worst-case by paying sub-polynomial factors. Our spanner result resolves an open question by Ahmed et al. (2019) and our results and techniques imply additional improvements over existing results, including (i) answering open questions about decremental single-source shortest paths by Chuzhoy and Khanna (STOC'19) and Gutenberg and Wulff-Nilsen (SODA'20), implying a nearly-quadratic time algorithm for approximating minimum-cost unit-capacity flow and (ii) de-amortizing a result of Abraham et al. (FOCS'16) for dynamic spectral sparsifiers. Our results are based on two novel techniques. The first technique is a generic black-box reduction that allows us to assume that the graph is initially an expander with almost uniform-degree and, more importantly, stays as an almost uniform-degree expander while undergoing only edge deletions. The second technique is called proactive resampling: here we constantly re-sample parts of the input graph so that, independent of an adversary’s computational power, a desired structure of the underlying graph can be always maintained. Despite its simplicity, the analysis of this sampling scheme is far from trivial, because the adversary can potentially create dependencies between the random choices used by the algorithm. We believe these two techniques could be useful for developing other adaptive algorithms.
Cite as

Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun. Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bernstein_et_al:LIPIcs.ICALP.2022.20,
  author =	{Bernstein, Aaron and van den Brand, Jan and Probst Gutenberg, Maximilian and Nanongkai, Danupon and Saranurak, Thatchaphol and Sidford, Aaron and Sun, He},
  title =	{{Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.20},
  URN =		{urn:nbn:de:0030-drops-163611},
  doi =		{10.4230/LIPIcs.ICALP.2022.20},
  annote =	{Keywords: dynamic graph algorithm, adaptive adversary, spanner, sparsifier}
}
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