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Documents authored by Neumann, Stefan


Document
Dynamic Maintenance of Monotone Dynamic Programs and Applications

Authors: Monika Henzinger, Stefan Neumann, Harald Räcke, and Stefan Schmid

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)


Abstract
Dynamic programming (DP) is one of the fundamental paradigms in algorithm design. However, many DP algorithms have to fill in large DP tables, represented by two-dimensional arrays, which causes at least quadratic running times and space usages. This has led to the development of improved algorithms for special cases when the DPs satisfy additional properties like, e.g., the Monge property or total monotonicity. In this paper, we consider a new condition which assumes (among some other technical assumptions) that the rows of the DP table are monotone. Under this assumption, we introduce a novel data structure for computing (1+ε)-approximate DP solutions in near-linear time and space in the static setting, and with polylogarithmic update times when the DP entries change dynamically. To the best of our knowledge, our new condition is incomparable to previous conditions and is the first which allows to derive dynamic algorithms based on existing DPs. Instead of using two-dimensional arrays to store the DP tables, we store the rows of the DP tables using monotone piecewise constant functions. This allows us to store length-n DP table rows with entries in [0,W] using only polylog(n,W) bits, and to perform operations, such as (min,+)-convolution or rounding, on these functions in polylogarithmic time. We further present several applications of our data structure. For bicriteria versions of k-balanced graph partitioning and simultaneous source location, we obtain the first dynamic algorithms with subpolynomial update times, as well as the first static algorithms using only near-linear time and space. Additionally, we obtain the currently fastest algorithm for fully dynamic knapsack.

Cite as

Monika Henzinger, Stefan Neumann, Harald Räcke, and Stefan Schmid. Dynamic Maintenance of Monotone Dynamic Programs and Applications. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 36:1-36:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{henzinger_et_al:LIPIcs.STACS.2023.36,
  author =	{Henzinger, Monika and Neumann, Stefan and R\"{a}cke, Harald and Schmid, Stefan},
  title =	{{Dynamic Maintenance of Monotone Dynamic Programs and Applications}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.36},
  URN =		{urn:nbn:de:0030-drops-176889},
  doi =		{10.4230/LIPIcs.STACS.2023.36},
  annote =	{Keywords: Dynamic programming, dynamic algorithms, data structures}
}
Document
Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles

Authors: Monika Henzinger, Stefan Neumann, and Andreas Wiese

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)


Abstract
Independent set is a fundamental problem in combinatorial optimization. While in general graphs the problem is essentially inapproximable, for many important graph classes there are approximation algorithms known in the offline setting. These graph classes include interval graphs and geometric intersection graphs, where vertices correspond to intervals/geometric objects and an edge indicates that the two corresponding objects intersect. We present dynamic approximation algorithms for independent set of intervals, hypercubes and hyperrectangles in d dimensions. They work in the fully dynamic model where each update inserts or deletes a geometric object. All our algorithms are deterministic and have worst-case update times that are polylogarithmic for constant d and ε>0, assuming that the coordinates of all input objects are in [0, N]^d and each of their edges has length at least 1. We obtain the following results: - For weighted intervals, we maintain a (1+ε)-approximate solution. - For d-dimensional hypercubes we maintain a (1+ε)2^d-approximate solution in the unweighted case and a O(2^d)-approximate solution in the weighted case. Also, we show that for maintaining an unweighted (1+ε)-approximate solution one needs polynomial update time for d ≥ 2 if the ETH holds. - For weighted d-dimensional hyperrectangles we present a dynamic algorithm with approximation ratio (1+ε)log^{d-1}N.

Cite as

Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 51:1-51:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{henzinger_et_al:LIPIcs.SoCG.2020.51,
  author =	{Henzinger, Monika and Neumann, Stefan and Wiese, Andreas},
  title =	{{Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{51:1--51:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.51},
  URN =		{urn:nbn:de:0030-drops-122094},
  doi =		{10.4230/LIPIcs.SoCG.2020.51},
  annote =	{Keywords: Dynamic algorithms, independent set, approximation algorithms, interval graphs, geometric intersection graphs}
}
Document
Conditional Hardness for Sensitivity Problems

Authors: Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)


Abstract
In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity. In this paper we prove conditional lower bounds for these and additional problems in a sensitivity setting. For example, we show that under the Boolean Matrix Multiplication (BMM) conjecture combinatorial algorithms cannot compute the (4/3-\varepsilon)-approximate diameter of an undirected unweighted dense graph with truly subcubic preprocessing time and truly subquadratic update/query time. This result is surprising since in the static setting it is not clear whether a reduction from BMM to diameter is possible. We further show under the BMM conjecture that many problems, such as reachability or approximate shortest paths, cannot be solved faster than by recomputation from scratch even after only one or two edge insertions. We extend our reduction from BMM to Diameter to give a reduction from All Pairs Shortest Paths to Diameter under one deletion in weighted graphs. This is intriguing, as in the static setting it is a big open problem whether Diameter is as hard as APSP. We further get a nearly tight lower bound for shortest paths after two edge deletions based on the APSP conjecture. We give more lower bounds under the Strong Exponential Time Hypothesis. Many of our lower bounds also hold for static oracle data structures where no sensitivity is required. Finally, we give the first algorithm for the (1+\varepsilon)-approximate radius, diameter, and eccentricity problems in directed or undirected unweighted graphs in case of single edges failures. The algorithm has a truly subcubic running time for graphs with a truly subquadratic number of edges; it is tight w.r.t. the conditional lower bounds we obtain.

Cite as

Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams. Conditional Hardness for Sensitivity Problems. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 26:1-26:31, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{henzinger_et_al:LIPIcs.ITCS.2017.26,
  author =	{Henzinger, Monika and Lincoln, Andrea and Neumann, Stefan and Vassilevska Williams, Virginia},
  title =	{{Conditional Hardness for Sensitivity Problems}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{26:1--26:31},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.26},
  URN =		{urn:nbn:de:0030-drops-81783},
  doi =		{10.4230/LIPIcs.ITCS.2017.26},
  annote =	{Keywords: sensitivity, conditional lower bounds, data structures, dynamic graph algorithms}
}
Document
Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning

Authors: Monika Henzinger and Stefan Neumann

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)


Abstract
During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size d; after the update we have to answer queries. In this paper, we consider the dynamic subgraph connectivity problem with sensitivity d: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to $d$ vertices are changed. Then we get a sequence of connectivity queries in the subgraph of activated vertices. We present the first fully dynamic algorithm for this problem which has an update and query time only slightly worse than the best decremental algorithm. In addition, we present the first incremental algorithm which is tight with respect to the best known conditional lower bound; moreover, the algorithm is simple and we believe it is implementable and efficient in practice.

Cite as

Monika Henzinger and Stefan Neumann. Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 48:1-48:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{henzinger_et_al:LIPIcs.ESA.2016.48,
  author =	{Henzinger, Monika and Neumann, Stefan},
  title =	{{Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{48:1--48:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.48},
  URN =		{urn:nbn:de:0030-drops-63607},
  doi =		{10.4230/LIPIcs.ESA.2016.48},
  annote =	{Keywords: connectivity, emergency planning, sensitivity}
}
Document
This House Proves That Debating is Harder Than Soccer

Authors: Stefan Neumann and Andreas Wiese

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)


Abstract
During the last twenty years, a lot of research was conducted on the sport elimination problem: Given a sports league and its remaining matches, we have to decide whether a given team can still possibly win the competition, i.e., place first in the league at the end. Previously, the computational complexity of this problem was investigated only for games with two participating teams per game. In this paper we consider Debating Tournaments and Debating Leagues in the British Parliamentary format, where four teams are participating in each game. We prove that it is NP-hard to decide whether a given team can win a Debating League, even if at most two matches are remaining for each team. This contrasts settings like football where two teams play in each game since there this case is still polynomial time solvable. We prove our result even for a fictitious restricted setting with only three teams per game. On the other hand, for the common setting of Debating Tournaments we show that this problem is fixed parameter tractable if the parameter is the number of remaining rounds k. This also holds for the practically very important question of whether a team can still qualify for the knock-out phase of the tournament and the combined parameter k+b where b denotes the threshold rank for qualifying. Finally, we show that the latter problem is polynomial time solvable for any constant k and arbitrary values b that are part of the input.

Cite as

Stefan Neumann and Andreas Wiese. This House Proves That Debating is Harder Than Soccer. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 25:1-25:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{neumann_et_al:LIPIcs.FUN.2016.25,
  author =	{Neumann, Stefan and Wiese, Andreas},
  title =	{{This House Proves That Debating is Harder Than Soccer}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.25},
  URN =		{urn:nbn:de:0030-drops-58716},
  doi =		{10.4230/LIPIcs.FUN.2016.25},
  annote =	{Keywords: complexity, elimination games, soccer, debating}
}
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