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Documents authored by Gutenberg, Maximilian Probst

Document
DOI: 10.4230/LIPIcs.ESA.2024.61
Practical Expander Decomposition

Authors: Lars Gottesbüren, Nikos Parotsidis, and Maximilian Probst Gutenberg

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
The expander decomposition of a graph decomposes the set of vertices into clusters such that the induced subgraph of each cluster is a subgraph with high conductance, and there is only a small number of inter-cluster edges. Expander decompositions are at the forefront of recent theoretical developments in the area of efficient graph algorithms and act as a central component in several state-of-the-art graph algorithms for fundamental problems like maximum flow, min-cost flow, Gomory-Hu trees, global min-cut, and more. Despite this crucial role and the existence of theoretically efficient expander decomposition algorithms, little is known on their behavior in practice. In this paper we explore the engineering design space in implementations for computing expander decompositions. We base our implementation on the near-linear time algorithm of Saranurak and Wang [SODA'19], and enhance it with practical optimizations that accelerate its running time in practice and at the same time preserve the theoretical runtime and approximation guarantees. We evaluate our algorithm on real-world graphs with up to tens of millions of edges. We demonstrate significant speedups of up to two orders of magnitude over the only prior implementation. To the best of our knowledge, our implementation is the first to compute expander decompositions at this scale within reasonable time.
Cite as

Lars Gottesbüren, Nikos Parotsidis, and Maximilian Probst Gutenberg. Practical Expander Decomposition. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 61:1-61:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gottesburen_et_al:LIPIcs.ESA.2024.61,
  author =	{Gottesb\"{u}ren, Lars and Parotsidis, Nikos and Gutenberg, Maximilian Probst},
  title =	{{Practical Expander Decomposition}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{61:1--61:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.61},
  URN =		{urn:nbn:de:0030-drops-211323},
  doi =		{10.4230/LIPIcs.ESA.2024.61},
  annote =	{Keywords: Expander Decomposition, Clustering, Graph Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.65
Optimal Electrical Oblivious Routing on Expanders

Authors: Cella Florescu, Rasmus Kyng, Maximilian Probst Gutenberg, and Sushant Sachdeva

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In this paper, we investigate the question of whether the electrical flow routing is a good oblivious routing scheme on an m-edge graph G = (V, E) that is a Φ-expander, i.e. where |∂ S| ≥ Φ ⋅ vol(S) for every S ⊆ V, vol(S) ≤ vol(V)/2. Beyond its simplicity and structural importance, this question is well-motivated by the current state-of-the-art of fast algorithms for 𝓁_∞ oblivious routings that reduce to the expander-case which is in turn solved by electrical flow routing. Our main result proves that the electrical routing is an O(Φ^{-1} log m)-competitive oblivious routing in the 𝓁₁- and 𝓁_∞-norms. We further observe that the oblivious routing is O(log² m)-competitive in the 𝓁₂-norm and, in fact, O(log m)-competitive if 𝓁₂-localization is O(log m) which is widely believed. Using these three upper bounds, we can smoothly interpolate to obtain upper bounds for every p ∈ [2, ∞] and q given by 1/p + 1/q = 1. Assuming 𝓁₂-localization in O(log m), we obtain that in 𝓁_p and 𝓁_q, the electrical oblivious routing is O(Φ^{-(1-2/p)}log m) competitive. Using the currently known result for 𝓁₂-localization, this ratio deteriorates by at most a sublogarithmic factor for every p, q ≠ 2. We complement our upper bounds with lower bounds that show that the electrical routing for any such p and q is Ω(Φ^{-(1-2/p)} log m)-competitive. This renders our results in 𝓁₁ and 𝓁_∞ unconditionally tight up to constants, and the result in any 𝓁_p- and 𝓁_q-norm to be tight in case of 𝓁₂-localization in O(log m).
Cite as

Cella Florescu, Rasmus Kyng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Optimal Electrical Oblivious Routing on Expanders. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 65:1-65:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{florescu_et_al:LIPIcs.ICALP.2024.65,
  author =	{Florescu, Cella and Kyng, Rasmus and Gutenberg, Maximilian Probst and Sachdeva, Sushant},
  title =	{{Optimal Electrical Oblivious Routing on Expanders}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{65:1--65:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.65},
  URN =		{urn:nbn:de:0030-drops-202083},
  doi =		{10.4230/LIPIcs.ICALP.2024.65},
  annote =	{Keywords: Expanders, Oblivious routing for 𝓁\underlinep, Electrical flow routing}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.53
Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets

Authors: Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, and Peng Zhang

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

Abstract
We study linear equations in combinatorial Laplacians of k-dimensional simplicial complexes (k-complexes), a natural generalization of graph Laplacians. Combinatorial Laplacians play a crucial role in homology and are a central tool in topology. Beyond this, they have various applications in data analysis and physical modeling problems. It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes. This paper shows that linear equations in combinatorial Laplacians of 2-complexes are as hard to solve as general linear equations. More precisely, for any constant c ≥ 1, if we can solve linear equations in combinatorial Laplacians of 2-complexes up to high accuracy in time Õ((# of nonzero coefficients)^c), then we can solve general linear equations with polynomially bounded integer coefficients and condition numbers up to high accuracy in time Õ((# of nonzero coefficients)^c). We prove this by a nearly-linear time reduction from general linear equations to combinatorial Laplacians of 2-complexes. Our reduction preserves the sparsity of the problem instances up to poly-logarithmic factors.
Cite as

Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, and Peng Zhang. Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 53:1-53:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ding_et_al:LIPIcs.ICALP.2022.53,
  author =	{Ding, Ming and Kyng, Rasmus and Gutenberg, Maximilian Probst and Zhang, Peng},
  title =	{{Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{53:1--53:19},
  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.2022.53},
  URN =		{urn:nbn:de:0030-drops-163945},
  doi =		{10.4230/LIPIcs.ICALP.2022.53},
  annote =	{Keywords: Simplicial Complexes, Combinatorial Laplacians, Linear Equations, Fine-Grained Complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.64
Decremental APSP in Unweighted Digraphs Versus an Adaptive Adversary

Authors: Jacob Evald, Viktor Fredslund-Hansen, Maximilian Probst Gutenberg, and Christian Wulff-Nilsen

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Given an unweighted digraph G = (V,E), undergoing a sequence of edge deletions, with m = |E|, n = |V|, we consider the problem of maintaining all-pairs shortest paths (APSP). Whilst this problem has been studied in a long line of research [ACM'81, FOCS'99, FOCS'01, STOC'02, STOC'03, SWAT'04, STOC'13] and the problem of (1+ε)-approximate, weighted APSP was solved to near-optimal update time Õ(mn) by Bernstein [STOC'13], the problem has mainly been studied in the context of an oblivious adversary which fixes the update sequence before the algorithm is started. In this paper, we make significant progress on the problem for an adaptive adversary which can perform updates based on answers to previous queries: - We first present a deterministic data structure that maintains the exact distances with total update time Õ(n³). - We also present a deterministic data structure that maintains (1+ε)-approximate distance estimates with total update time Õ(√m n²/ε) which for sparse graphs is Õ(n^{2+1/2}/ε). - Finally, we present a randomized (1+ε)-approximate data structure which works against an adaptive adversary; its total update time is Õ(m^{2/3}n^{5/3} + n^{8/3}/(m^{1/3}ε²)) which for sparse graphs is Õ(n^{2+1/3}/ε²). Our exact data structure matches the total update time of the best randomized data structure by Baswana et al. [STOC'02] and maintains the distance matrix in near-optimal time. Our approximate data structures improve upon the best data structures against an adaptive adversary which have Õ(mn²) total update time [JACM'81, STOC'03].
Cite as

Jacob Evald, Viktor Fredslund-Hansen, Maximilian Probst Gutenberg, and Christian Wulff-Nilsen. Decremental APSP in Unweighted Digraphs Versus an Adaptive Adversary. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 64:1-64:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{evald_et_al:LIPIcs.ICALP.2021.64,
  author =	{Evald, Jacob and Fredslund-Hansen, Viktor and Gutenberg, Maximilian Probst and Wulff-Nilsen, Christian},
  title =	{{Decremental APSP in Unweighted Digraphs Versus an Adaptive Adversary}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{64:1--64:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.64},
  URN =		{urn:nbn:de:0030-drops-141337},
  doi =		{10.4230/LIPIcs.ICALP.2021.64},
  annote =	{Keywords: Dynamic Graph Algorithm, Data Structure, Shortest Paths}
}
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