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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and deterministic sequences of matchings covering the whole graph. We bound the discrepancy, which is defined as the difference between the maximum and the minimum load. Our results cover a broad range of graph classes and, to the best of our knowledge, our analysis is the first result for discrete and dynamic averaging load balancing processes. As our main technical contribution we develop a drift result that allows us to apply techniques based on the effective resistance in an electrical network to the setting of dynamic load balancing.

Petra Berenbrink, Lukas Hintze, Hamed Hosseinpour, Dominik Kaaser, and Malin Rau. Dynamic Averaging Load Balancing on Arbitrary Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.ICALP.2023.18, author = {Berenbrink, Petra and Hintze, Lukas and Hosseinpour, Hamed and Kaaser, Dominik and Rau, Malin}, title = {{Dynamic Averaging Load Balancing on Arbitrary Graphs}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {18:1--18:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.18}, URN = {urn:nbn:de:0030-drops-180707}, doi = {10.4230/LIPIcs.ICALP.2023.18}, annote = {Keywords: Dynamic Load Balancing, Distributed Computing, Randomized Algorithms, Drift Analysis} }

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**Published in:** LIPIcs, Volume 253, 26th International Conference on Principles of Distributed Systems (OPODIS 2022)

We study the consensus problem among n agents, defined as follows. Initially, each agent holds one of two possible opinions. The goal is to reach a consensus configuration in which every agent shares the same opinion. To this end, agents randomly sample other agents and update their opinion according to a simple update function depending on the sampled opinions.
We consider two communication models: the gossip model and a variant of the population model. In the gossip model, agents are activated in parallel, synchronous rounds. In the population model, one agent is activated after the other in a sequence of discrete time steps. For both models we analyze the following natural family of majority processes called j-Majority: when activated, every agent samples j other agents uniformly at random (with replacement) and adopts the majority opinion among the sample (breaking ties uniformly at random). As our main result we show a hierarchy among majority protocols: (j+1)-Majority (for j > 1) converges stochastically faster than j-Majority for any initial opinion configuration. In our analysis we use Strassen’s Theorem to prove the existence of a coupling. This gives an affirmative answer for the case of two opinions to an open question asked by Berenbrink et al. [PODC 2017].

Petra Berenbrink, Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Dominik Kaaser, and Malin Rau. On the Hierarchy of Distributed Majority Protocols. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.OPODIS.2022.23, author = {Berenbrink, Petra and Coja-Oghlan, Amin and Gebhard, Oliver and Hahn-Klimroth, Max and Kaaser, Dominik and Rau, Malin}, title = {{On the Hierarchy of Distributed Majority Protocols}}, booktitle = {26th International Conference on Principles of Distributed Systems (OPODIS 2022)}, pages = {23:1--23:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-265-5}, ISSN = {1868-8969}, year = {2023}, volume = {253}, editor = {Hillel, Eshcar and Palmieri, Roberto and Rivi\`{e}re, Etienne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2022.23}, URN = {urn:nbn:de:0030-drops-176434}, doi = {10.4230/LIPIcs.OPODIS.2022.23}, annote = {Keywords: Consensus, Majority, Hierarchy, Stochastic Dominance, Population Protocols, Gossip Model, Strassen’s Theorem} }

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

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.

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-dev.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} }

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Track A: Algorithms, Complexity and Games

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

Consider a set P of points in the unit square U = [1,0), one of them being the origin. For each point p ∈ P you may draw an axis-aligned rectangle in U with its lower-left corner being p. What is the maximum area such rectangles can cover without overlapping each other?
Freedman posed this problem in 1969, asking whether one can always cover at least 50% of U. Over 40 years later, Dumitrescu and Tóth [Adrian Dumitrescu and Csaba D. Tóth, 2015] achieved the first constant coverage of 9.1%; since then, no significant progress was made. While 9.1% might seem low, the authors could not find any instance where their algorithm covers less than 50%, nourishing the hope to eventually prove a 50% bound. While we indeed significantly raise the algorithm’s coverage to 39%, we extinguish the hope of reaching 50% by giving points for which its coverage stays below 43.3%.
Our analysis studies the algorithm’s average and worst-case density of so-called tiles, which represent the staircase polygons in which a point can freely choose its maximum-area rectangle. Our approach is comparatively general and may potentially help in analyzing related algorithms.

Christoph Damerius, Dominik Kaaser, Peter Kling, and Florian Schneider. On Greedily Packing Anchored Rectangles. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 61:1-61:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{damerius_et_al:LIPIcs.ICALP.2021.61, author = {Damerius, Christoph and Kaaser, Dominik and Kling, Peter and Schneider, Florian}, title = {{On Greedily Packing Anchored Rectangles}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {61:1--61:20}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.61}, URN = {urn:nbn:de:0030-drops-141306}, doi = {10.4230/LIPIcs.ICALP.2021.61}, annote = {Keywords: lower-left anchored rectangle packing, greedy algorithm, charging scheme} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

We consider the efficient simulation of population protocols. In the population model, we are given a system of n agents modeled as identical finite-state machines. In each step, two agents are selected uniformly at random to interact by updating their states according to a common transition function. We empirically and analytically analyze two classes of simulators for this model. First, we consider sequential simulators executing one interaction after the other. Key to the performance of these simulators is the data structure storing the agents' states. For our analysis, we consider plain arrays, binary search trees, and a novel Dynamic Alias Table data structure. Secondly, we consider batch processing to efficiently update the states of multiple independent agents in one step. For many protocols considered in literature, our simulator requires amortized sub-constant time per interaction and is fast in practice: given a fixed time budget, the implementation of our batched simulator is able to simulate population protocols several orders of magnitude larger compared to the sequential competitors, and can carry out 2^50 interactions among the same number of agents in less than 400s.

Petra Berenbrink, David Hammer, Dominik Kaaser, Ulrich Meyer, Manuel Penschuck, and Hung Tran. Simulating Population Protocols in Sub-Constant Time per Interaction. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{berenbrink_et_al:LIPIcs.ESA.2020.16, author = {Berenbrink, Petra and Hammer, David and Kaaser, Dominik and Meyer, Ulrich and Penschuck, Manuel and Tran, Hung}, title = {{Simulating Population Protocols in Sub-Constant Time per Interaction}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {16:1--16:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.16}, URN = {urn:nbn:de:0030-drops-128827}, doi = {10.4230/LIPIcs.ESA.2020.16}, annote = {Keywords: Population Protocols, Simulation, Random Sampling, Dynamic Alias Table} }

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

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.

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-dev.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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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

A population protocol is a sequence of pairwise interactions of n agents. During one interaction, two randomly selected agents update their states by applying a deterministic transition function. The goal is to stabilize the system at a desired output property. The main performance objectives in designing such protocols are small number of states per agent and fast stabilization time.
We present a fast population protocol for the exact-majority problem, which uses Theta(log n) states (per agent) and stabilizes in O(log^{5/3} n) parallel time (i.e., in O(n log^{5/3} n) interactions) in expectation and with high probability. Alistarh et al. [SODA 2018] showed that exact-majority protocols which stabilize in expected O(n^{1-Omega(1)}) parallel time and have the properties of monotonicity and output dominance require Omega(log n) states. Note that the properties mentioned above are satisfied by all known population protocols for exact majority, including ours. They also showed an O(log^2 n)-time exact-majority protocol with O(log n) states, which, prior to our work, was the fastest exact-majority protocol with polylogarithmic number of states. The standard design framework for majority protocols is based on O(log n) phases and requires that all agents are well synchronized within each phase, leading naturally to upper bounds of the order of log^2 n because of Theta(log n) synchronization time per phase. We show how this framework can be tightened with weak synchronization to break the O(log^2 n) upper bound of previous protocols.

Petra Berenbrink, Robert Elsässer, Tom Friedetzky, Dominik Kaaser, Peter Kling, and Tomasz Radzik. A Population Protocol for Exact Majority with O(log5/3 n) Stabilization Time and Theta(log n) States. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{berenbrink_et_al:LIPIcs.DISC.2018.10, author = {Berenbrink, Petra and Els\"{a}sser, Robert and Friedetzky, Tom and Kaaser, Dominik and Kling, Peter and Radzik, Tomasz}, title = {{A Population Protocol for Exact Majority with O(log5/3 n) Stabilization Time and Theta(log n) States}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.10}, URN = {urn:nbn:de:0030-drops-97999}, doi = {10.4230/LIPIcs.DISC.2018.10}, annote = {Keywords: Population Protocols, Randomized Algorithms, Majority} }

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**Published in:** OASIcs, Volume 61, 1st Symposium on Simplicity in Algorithms (SOSA 2018)

We provide a simple and efficient population protocol for leader election that uses O(log n) states and elects exactly one leader in O(n (log n)^2) interactions with high probability and in expectation. Our analysis is simple and based on fundamental stochastic arguments. Our protocol combines the tournament based leader elimination by Alistarh and Gelashvili, ICALP'15, with the synthetic coin introduced by Alistarh et al., SODA'17.

Petra Berenbrink, Dominik Kaaser, Peter Kling, and Lena Otterbach. Simple and Efficient Leader Election. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{berenbrink_et_al:OASIcs.SOSA.2018.9, author = {Berenbrink, Petra and Kaaser, Dominik and Kling, Peter and Otterbach, Lena}, title = {{Simple and Efficient Leader Election}}, booktitle = {1st Symposium on Simplicity in Algorithms (SOSA 2018)}, pages = {9:1--9:11}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-064-4}, ISSN = {2190-6807}, year = {2018}, volume = {61}, editor = {Seidel, Raimund}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2018.9}, URN = {urn:nbn:de:0030-drops-83029}, doi = {10.4230/OASIcs.SOSA.2018.9}, annote = {Keywords: population protocols, leader election, distributed, randomized} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

In the deterministic binary majority process we are given a simple graph where each node has one out of two initial opinions. In every round, each node adopts the majority opinion among its neighbors. It is known that this process always converges in O(|E|) rounds to a two-periodic state in which every node either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the O(|E|) bound on the convergence time of the deterministic binary majority process is even for dense graphs tight. However, in many graphs such as the complete graph the process converges in just
a constant number of rounds from any initial opinion assignment.
We show that it is NP-hard to decide whether there exists an initial opinion assignment for which it takes more than k rounds to converge to the two-periodic stable state, for a given integer k. We then give a new upper bound on the voting time of the deterministic binary majority process. Our bound can be computed in linear time by carefully exploiting the structure of the potential function by Goles and Olivos. We identify certain modules of a graph G to obtain a new graph G^Delta. This new graph G^Delta has the property that the worst-case convergence time of G^Delta is an upper bound on that of G. Our new bounds asymptotically improve the best known bounds for various graph classes.

Dominik Kaaser, Frederik Mallmann-Trenn, and Emanuele Natale. On the Voting Time of the Deterministic Majority Process. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kaaser_et_al:LIPIcs.MFCS.2016.55, author = {Kaaser, Dominik and Mallmann-Trenn, Frederik and Natale, Emanuele}, title = {{On the Voting Time of the Deterministic Majority Process}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {55:1--55:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.55}, URN = {urn:nbn:de:0030-drops-64675}, doi = {10.4230/LIPIcs.MFCS.2016.55}, annote = {Keywords: distributed voting, majority rule} }

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