3 Search Results for "Biermeier, Felix"

Document
DOI: 10.4230/LIPIcs.DISC.2025.27
On the h-Majority Dynamics with Many Opinions

Authors: Francesco d'Amore, Niccolò D'Archivio, George Giakkoupis, and Emanuele Natale

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract
We present the first upper bound on the convergence time to consensus of the well-known h-majority dynamics with k opinions, in the synchronous setting, for h and k that are both non-constant values. We suppose that, at the beginning of the process, there is some initial additive bias towards some plurality opinion, that is, there is an opinion that is supported by x nodes while any other opinion is supported by strictly fewer nodes. We prove that, with high probability, if the bias is ω(√x) and the initial plurality opinion is supported by at least x = ω(log n) nodes, then the process converges to plurality consensus in O(log n) rounds whenever h = ω(n log n / x). A main corollary is the following: if k = o(n / log n) and the process starts from an almost-balanced configuration with an initial bias of magnitude ω(√{n/k}) towards the initial plurality opinion, then any function h = ω(k log n) suffices to guarantee convergence to consensus in O(log n) rounds, with high probability. Our upper bound shows that the lower bound of Ω(k / h²) rounds to reach consensus given by Becchetti et al. (2017) cannot be pushed further than Ω̃(k / h). Moreover, the bias we require is asymptotically smaller than the Ω(√{nlog n}) bias that guarantees plurality consensus in the 3-majority dynamics: in our case, the required bias is at most any (arbitrarily small) function in ω(√x) for any value of k ≥ 2.
Cite as

Francesco d'Amore, Niccolò D'Archivio, George Giakkoupis, and Emanuele Natale. On the h-Majority Dynamics with Many Opinions. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 27:1-27:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{damore_et_al:LIPIcs.DISC.2025.27,
  author =	{d'Amore, Francesco and D'Archivio, Niccol\`{o} and Giakkoupis, George and Natale, Emanuele},
  title =	{{On the h-Majority Dynamics with Many Opinions}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{27:1--27:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.27},
  URN =		{urn:nbn:de:0030-drops-248448},
  doi =		{10.4230/LIPIcs.DISC.2025.27},
  annote =	{Keywords: Distributed Algorithms, Randomized Algorithms, Markov Chains, Consensus Problem, Opinion dynamics, Plurality Consensus}
}
Document
DOI: 10.4230/LIPIcs.SAND.2022.7
Loosely-Stabilizing Phase Clocks and The Adaptive Majority Problem

Authors: Petra Berenbrink, Felix Biermeier, Christopher Hahn, and Dominik Kaaser

Published in: LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

Abstract
We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of n identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires O(log n) states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within O(n log n) interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly(n) parallel time w.h.p. We use our clock to design a loosely-stabilizing protocol that solves the adaptive variant of the majority problem. We assume that the agents have either opinion A or B or they are undecided and agents can change their opinion at a rate of 1/n. The goal is to keep track which of the two opinions is (momentarily) the majority. We show that if the majority has a support of at least Ω(log n) agents and a sufficiently large bias is present, then the protocol converges to a correct output within O(n log n) interactions and stays in a correct configuration for poly(n) interactions, w.h.p.
Cite as

Petra Berenbrink, Felix Biermeier, Christopher Hahn, and Dominik Kaaser. Loosely-Stabilizing Phase Clocks and The Adaptive Majority Problem. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berenbrink_et_al:LIPIcs.SAND.2022.7,
  author =	{Berenbrink, Petra and Biermeier, Felix and Hahn, Christopher and Kaaser, Dominik},
  title =	{{Loosely-Stabilizing Phase Clocks and The Adaptive Majority Problem}},
  booktitle =	{1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-224-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{221},
  editor =	{Aspnes, James and Michail, Othon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.7},
  URN =		{urn:nbn:de:0030-drops-159493},
  doi =		{10.4230/LIPIcs.SAND.2022.7},
  annote =	{Keywords: Population Protocols, Phase Clocks, Loose Self-stabilization, Clock Synchronization, Majority, Adaptive}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.8
On the Complexity of Anchored Rectangle Packing

Authors: Antonios Antoniadis, Felix Biermeier, Andrés Cristi, Christoph Damerius, Ruben Hoeksma, Dominik Kaaser, Peter Kling, and Lukas Nölke

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
In the Anchored Rectangle Packing (ARP) problem, we are given a set of points P in the unit square [0,1]^2 and seek a maximum-area set of axis-aligned interior-disjoint rectangles S, each of which is anchored at a point p in P. In the most prominent variant - Lower-Left-Anchored Rectangle Packing (LLARP) - rectangles are anchored in their lower-left corner. Freedman [W. T. Tutte (Ed.), 1969] conjectured in 1969 that, if (0,0) in P, then there is a LLARP that covers an area of at least 0.5. Somewhat surprisingly, this conjecture remains open to this day, with the best known result covering an area of 0.091 [Dumitrescu and Tóth, 2015]. Maybe even more surprisingly, it is not known whether LLARP - or any ARP-problem with only one anchor - is NP-hard. In this work, we first study the Center-Anchored Rectangle Packing (CARP) problem, where rectangles are anchored in their center. We prove NP-hardness and provide a PTAS. In fact, our PTAS applies to any ARP problem where the anchor lies in the interior of the rectangles. Afterwards, we turn to the LLARP problem and investigate two different resource-augmentation settings: In the first we allow an epsilon-perturbation of the input P, whereas in the second we permit an epsilon-overlap between rectangles. For the former setting, we give an algorithm that covers at least as much area as an optimal solution of the original problem. For the latter, we give an (1 - epsilon)-approximation.
Cite as

Antonios Antoniadis, Felix Biermeier, Andrés Cristi, Christoph Damerius, Ruben Hoeksma, Dominik Kaaser, Peter Kling, and Lukas Nölke. On the Complexity of Anchored Rectangle Packing. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{antoniadis_et_al:LIPIcs.ESA.2019.8,
  author =	{Antoniadis, Antonios and Biermeier, Felix and Cristi, Andr\'{e}s and Damerius, Christoph and Hoeksma, Ruben and Kaaser, Dominik and Kling, Peter and N\"{o}lke, Lukas},
  title =	{{On the Complexity of Anchored Rectangle Packing}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.8},
  URN =		{urn:nbn:de:0030-drops-111297},
  doi =		{10.4230/LIPIcs.ESA.2019.8},
  annote =	{Keywords: anchored rectangle, rectangle packing, resource augmentation, PTAS, NP, hardness}
}
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