11 Search Results for "Rybicki, Joel"


Document
Brief Announcement
Brief Announcement: Temporal Locality in Online Algorithms

Authors: Maciej Pacut, Mahmoud Parham, Joel Rybicki, Stefan Schmid, Jukka Suomela, and Aleksandr Tereshchenko

Published in: LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)


Abstract
Online algorithms make decisions based on past inputs, with the goal of being competitive against an algorithm that sees also future inputs. In this work, we introduce time-local online algorithms; these are online algorithms in which the output at any given time is a function of only T latest inputs. Our main observation is that time-local online algorithms are closely connected to local distributed graph algorithms: distributed algorithms make decisions based on the local information in the spatial dimension, while time-local online algorithms make decisions based on the local information in the temporal dimension. We formalize this connection, and show how we can directly use the tools developed to study distributed approximability of graph optimization problems to prove upper and lower bounds on the competitive ratio achieved with time-local online algorithms. Moreover, we show how to use computational techniques to synthesize optimal time-local algorithms.

Cite as

Maciej Pacut, Mahmoud Parham, Joel Rybicki, Stefan Schmid, Jukka Suomela, and Aleksandr Tereshchenko. Brief Announcement: Temporal Locality in Online Algorithms. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 52:1-52:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{pacut_et_al:LIPIcs.DISC.2022.52,
  author =	{Pacut, Maciej and Parham, Mahmoud and Rybicki, Joel and Schmid, Stefan and Suomela, Jukka and Tereshchenko, Aleksandr},
  title =	{{Brief Announcement: Temporal Locality in Online Algorithms}},
  booktitle =	{36th International Symposium on Distributed Computing (DISC 2022)},
  pages =	{52:1--52:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-255-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{246},
  editor =	{Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.52},
  URN =		{urn:nbn:de:0030-drops-172431},
  doi =		{10.4230/LIPIcs.DISC.2022.52},
  annote =	{Keywords: Online algorithms, distributed algorithms}
}
Document
Fast Graphical Population Protocols

Authors: Dan Alistarh, Rati Gelashvili, and Joel Rybicki

Published in: LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)


Abstract
Let G be a graph on n nodes. In the stochastic population protocol model, a collection of n indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other’s states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when G is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in n polylog n pairwise interactions, with high probability, using at most polylog n states per node. In this work, we consider the more general setting where G is an arbitrary regular graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As a sample application, we show that, in any regular graph with conductance φ, both leader election and exact majority can be solved in φ^{-2} ⋅ n polylog n pairwise interactions, with high probability, using at most φ^{-2} ⋅ polylog n states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties. We believe our results will prove generally useful, as they allow efficient technology transfer between the well-mixed (clique) case, and the under-explored spatial setting.

Cite as

Dan Alistarh, Rati Gelashvili, and Joel Rybicki. Fast Graphical Population Protocols. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 14:1-14:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{alistarh_et_al:LIPIcs.OPODIS.2021.14,
  author =	{Alistarh, Dan and Gelashvili, Rati and Rybicki, Joel},
  title =	{{Fast Graphical Population Protocols}},
  booktitle =	{25th International Conference on Principles of Distributed Systems (OPODIS 2021)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-219-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{217},
  editor =	{Bramas, Quentin and Gramoli, Vincent and Milani, Alessia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.14},
  URN =		{urn:nbn:de:0030-drops-157897},
  doi =		{10.4230/LIPIcs.OPODIS.2021.14},
  annote =	{Keywords: population protocols, leader election, exact majority, graphs}
}
Document
Brief Announcement
Brief Announcement: Fast Graphical Population Protocols

Authors: Dan Alistarh, Rati Gelashvili, and Joel Rybicki

Published in: LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)


Abstract
Let G be a graph on n nodes. In the stochastic population protocol model, a collection of n indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other’s states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when G is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in n polylog n pairwise interactions, with high probability, using at most polylog n states per node. In this work, we consider the more general setting where G is an arbitrary graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As an example, this implies that, in any regular graph with conductance φ, both leader election and exact majority can be solved in φ^{-2} ⋅ n polylog n pairwise interactions, with high probability, using at most φ^{-2} ⋅ polylog n states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties.

Cite as

Dan Alistarh, Rati Gelashvili, and Joel Rybicki. Brief Announcement: Fast Graphical Population Protocols. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 43:1-43:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{alistarh_et_al:LIPIcs.DISC.2021.43,
  author =	{Alistarh, Dan and Gelashvili, Rati and Rybicki, Joel},
  title =	{{Brief Announcement: Fast Graphical Population Protocols}},
  booktitle =	{35th International Symposium on Distributed Computing (DISC 2021)},
  pages =	{43:1--43:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-210-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{209},
  editor =	{Gilbert, Seth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.43},
  URN =		{urn:nbn:de:0030-drops-148451},
  doi =		{10.4230/LIPIcs.DISC.2021.43},
  annote =	{Keywords: population protocols, leader election, majority}
}
Document
Brief Announcement
Brief Announcement: Sinkless Orientation Is Hard Also in the Supported LOCAL Model

Authors: Janne H. Korhonen, Ami Paz, Joel Rybicki, Stefan Schmid, and Jukka Suomela

Published in: LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)


Abstract
We show that any algorithm that solves the sinkless orientation problem in the supported LOCAL model requires Ω(log n) rounds, and this is tight. The supported LOCAL is at least as strong as the usual LOCAL model, and as a corollary this also gives a new, short and elementary proof that shows that the round complexity of the sinkless orientation problem in the deterministic LOCAL model is Ω(log n).

Cite as

Janne H. Korhonen, Ami Paz, Joel Rybicki, Stefan Schmid, and Jukka Suomela. Brief Announcement: Sinkless Orientation Is Hard Also in the Supported LOCAL Model. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 58:1-58:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{korhonen_et_al:LIPIcs.DISC.2021.58,
  author =	{Korhonen, Janne H. and Paz, Ami and Rybicki, Joel and Schmid, Stefan and Suomela, Jukka},
  title =	{{Brief Announcement: Sinkless Orientation Is Hard Also in the Supported LOCAL Model}},
  booktitle =	{35th International Symposium on Distributed Computing (DISC 2021)},
  pages =	{58:1--58:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-210-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{209},
  editor =	{Gilbert, Seth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.58},
  URN =		{urn:nbn:de:0030-drops-148609},
  doi =		{10.4230/LIPIcs.DISC.2021.58},
  annote =	{Keywords: Supported LOCAL model, sinkless orientation, round elimination}
}
Document
Distributed Computation with Continual Population Growth

Authors: Da-Jung Cho, Matthias Függer, Corbin Hopper, Manish Kushwaha, Thomas Nowak, and Quentin Soubeyran

Published in: LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)


Abstract
Computing with synthetically engineered bacteria is a vibrant and active field with numerous applications in bio-production, bio-sensing, and medicine. Motivated by the lack of robustness and by resource limitation inside single cells, distributed approaches with communication among bacteria have recently gained in interest. In this paper, we focus on the problem of population growth happening concurrently, and possibly interfering, with the desired bio-computation. Specifically, we present a fast protocol in systems with continuous population growth for the majority consensus problem and prove that it correctly identifies the initial majority among two inputs with high probability if the initial difference is Ω(√{nlog n}) where n is the total initial population. We also present a fast protocol that correctly computes the NAND of two inputs with high probability. We demonstrate that combining the NAND gate protocol with the continuous-growth majority consensus protocol, using the latter as an amplifier, it is possible to implement circuits computing arbitrary Boolean functions.

Cite as

Da-Jung Cho, Matthias Függer, Corbin Hopper, Manish Kushwaha, Thomas Nowak, and Quentin Soubeyran. Distributed Computation with Continual Population Growth. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 7:1-7:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{cho_et_al:LIPIcs.DISC.2020.7,
  author =	{Cho, Da-Jung and F\"{u}gger, Matthias and Hopper, Corbin and Kushwaha, Manish and Nowak, Thomas and Soubeyran, Quentin},
  title =	{{Distributed Computation with Continual Population Growth}},
  booktitle =	{34th International Symposium on Distributed Computing (DISC 2020)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-168-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{179},
  editor =	{Attiya, Hagit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.7},
  URN =		{urn:nbn:de:0030-drops-130856},
  doi =		{10.4230/LIPIcs.DISC.2020.7},
  annote =	{Keywords: microbiological circuits, majority consensus, birth-death processes}
}
Document
Brief Announcement
Brief Announcement: Efficient Load-Balancing Through Distributed Token Dropping

Authors: Sebastian Brandt, Barbara Keller, Joel Rybicki, Jukka Suomela, and Jara Uitto

Published in: LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)


Abstract
We introduce a new graph problem, the token dropping game, and we show how to solve it efficiently in a distributed setting. We use the token dropping game as a tool to design an efficient distributed algorithm for the stable orientation problem, which is a special case of the more general locally optimal semi-matching problem. The prior work by Czygrinow et al. (DISC 2012) finds a locally optimal semi-matching in O(Δ⁵) rounds in graphs of maximum degree Δ, which directly implies an algorithm with the same runtime for stable orientations. We improve the runtime to O(Δ⁴) for stable orientations and prove a lower bound of Ω(Δ) rounds.

Cite as

Sebastian Brandt, Barbara Keller, Joel Rybicki, Jukka Suomela, and Jara Uitto. Brief Announcement: Efficient Load-Balancing Through Distributed Token Dropping. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 40:1-40:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{brandt_et_al:LIPIcs.DISC.2020.40,
  author =	{Brandt, Sebastian and Keller, Barbara and Rybicki, Joel and Suomela, Jukka and Uitto, Jara},
  title =	{{Brief Announcement: Efficient Load-Balancing Through Distributed Token Dropping}},
  booktitle =	{34th International Symposium on Distributed Computing (DISC 2020)},
  pages =	{40:1--40:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-168-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{179},
  editor =	{Attiya, Hagit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.40},
  URN =		{urn:nbn:de:0030-drops-131182},
  doi =		{10.4230/LIPIcs.DISC.2020.40},
  annote =	{Keywords: distributed algorithms, graph problems, semi-matching}
}
Document
Byzantine Approximate Agreement on Graphs

Authors: Thomas Nowak and Joel Rybicki

Published in: LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)


Abstract
Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors' input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.

Cite as

Thomas Nowak and Joel Rybicki. Byzantine Approximate Agreement on Graphs. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 29:1-29:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{nowak_et_al:LIPIcs.DISC.2019.29,
  author =	{Nowak, Thomas and Rybicki, Joel},
  title =	{{Byzantine Approximate Agreement on Graphs}},
  booktitle =	{33rd International Symposium on Distributed Computing (DISC 2019)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-126-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{146},
  editor =	{Suomela, Jukka},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.29},
  URN =		{urn:nbn:de:0030-drops-113363},
  doi =		{10.4230/LIPIcs.DISC.2019.29},
  annote =	{Keywords: consensus, approximate agreement, Byzantine faults, chordal graphs, lattice agreement}
}
Document
Deterministic Subgraph Detection in Broadcast CONGEST

Authors: Janne H. Korhonen and Joel Rybicki

Published in: LIPIcs, Volume 95, 21st International Conference on Principles of Distributed Systems (OPODIS 2017)


Abstract
We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: - For any constant k, detecting k-paths and trees on k nodes can be done in O(1) rounds. - For any constant k, detecting k-cycles and pseudotrees on k nodes can be done in O(n) rounds. - On d-degenerate graphs, cliques and 4-cycles can be enumerated in O(d + log n) rounds, and 5-cycles in O(d2 + log n) rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for d-degenerate graphs can be improved to O(d/logn) and O(d2/logn), respect- ively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique.

Cite as

Janne H. Korhonen and Joel Rybicki. Deterministic Subgraph Detection in Broadcast CONGEST. In 21st International Conference on Principles of Distributed Systems (OPODIS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 95, pp. 4:1-4:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{korhonen_et_al:LIPIcs.OPODIS.2017.4,
  author =	{Korhonen, Janne H. and Rybicki, Joel},
  title =	{{Deterministic Subgraph Detection in Broadcast CONGEST}},
  booktitle =	{21st International Conference on Principles of Distributed Systems (OPODIS 2017)},
  pages =	{4:1--4:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-061-3},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{95},
  editor =	{Aspnes, James and Bessani, Alysson and Felber, Pascal and Leit\~{a}o, Jo\~{a}o},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2017.4},
  URN =		{urn:nbn:de:0030-drops-86252},
  doi =		{10.4230/LIPIcs.OPODIS.2017.4},
  annote =	{Keywords: distributed computing, subgraph detection, CONGEST model, lower bounds}
}
Document
Self-Stabilising Byzantine Clock Synchronisation is Almost as Easy as Consensus

Authors: Christoph Lenzen and Joel Rybicki

Published in: LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)


Abstract
We give fault-tolerant algorithms for establishing synchrony in distributed systems in which each of the n nodes has its own clock. Our algorithms operate in a very strong fault model: we require self-stabilisation, i.e., the initial state of the system may be arbitrary, and there can be up to f<n/3 ongoing Byzantine faults, i.e., nodes that deviate from the protocol in an arbitrary manner. Furthermore, we assume that the local clocks of the nodes may progress at different speeds (clock drift) and communication has bounded delay. In this model, we study the pulse synchronisation problem, where the task is to guarantee that eventually all correct nodes generate well-separated local pulse events (i.e., unlabelled logical clock ticks) in a synchronised manner. Compared to prior work, we achieve exponential improvements in stabilisation time and the number of communicated bits, and give the first sublinear-time algorithm for the problem: - In the deterministic setting, the state-of-the-art solutions stabilise in time Theta(f) and have each node broadcast Theta(f log f) bits per time unit. We exponentially reduce the number of bits broadcasted per time unit to Theta(log f) while retaining the same stabilisation time. - In the randomised setting, the state-of-the-art solutions stabilise in time Theta(f) and have each node broadcast O(1) bits per time unit. We exponentially reduce the stabilisation time to polylog f while each node broadcasts polylog f bits per time unit. These results are obtained by means of a recursive approach reducing the above task of self-stabilising pulse synchronisation in the bounded-delay model to non-self-stabilising binary consensus in the synchronous model. In general, our approach introduces at most logarithmic overheads in terms of stabilisation time and broadcasted bits over the underlying consensus routine.

Cite as

Christoph Lenzen and Joel Rybicki. Self-Stabilising Byzantine Clock Synchronisation is Almost as Easy as Consensus. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 32:1-32:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{lenzen_et_al:LIPIcs.DISC.2017.32,
  author =	{Lenzen, Christoph and Rybicki, Joel},
  title =	{{Self-Stabilising Byzantine Clock Synchronisation is Almost as Easy as Consensus}},
  booktitle =	{31st International Symposium on Distributed Computing (DISC 2017)},
  pages =	{32:1--32:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-053-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{91},
  editor =	{Richa, Andr\'{e}a},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.32},
  URN =		{urn:nbn:de:0030-drops-79914},
  doi =		{10.4230/LIPIcs.DISC.2017.32},
  annote =	{Keywords: Byzantine faults, self-stabilisation, clock synchronisation, consensus}
}
Document
The Container Selection Problem

Authors: Viswanath Nagarajan, Kanthi K. Sarpatwar, Baruch Schieber, Hadas Shachnai, and Joel L. Wolf

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
We introduce and study a network resource management problem that is a special case of non-metric k-median, naturally arising in cross platform scheduling and cloud computing. In the continuous d-dimensional container selection problem, we are given a set C of input points in d-dimensional Euclidean space, for some d >= 2, and a budget k. An input point p can be assigned to a "container point" c only if c dominates p in every dimension. The assignment cost is then equal to the L1-norm of the container point. The goal is to find k container points in the d-dimensional space, such that the total assignment cost for all input points is minimized. The discrete variant of the problem has one key distinction, namely, the container points must be chosen from a given set F of points. For the continuous version, we obtain a polynomial time approximation scheme for any fixed dimension d>= 2. On the negative side, we show that the problem is NP-hard for any d>=3. We further show that the discrete version is significantly harder, as it is NP-hard to approximate without violating the budget k in any dimension d>=3. Thus, we focus on obtaining bi-approximation algorithms. For d=2, the bi-approximation guarantee is (1+epsilon,3), i.e., for any epsilon>0, our scheme outputs a solution of size 3k and cost at most (1+epsilon) times the optimum. For fixed d>2, we present a (1+epsilon,O((1/epsilon)log k)) bi-approximation algorithm.

Cite as

Viswanath Nagarajan, Kanthi K. Sarpatwar, Baruch Schieber, Hadas Shachnai, and Joel L. Wolf. The Container Selection Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 416-434, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{nagarajan_et_al:LIPIcs.APPROX-RANDOM.2015.416,
  author =	{Nagarajan, Viswanath and Sarpatwar, Kanthi K. and Schieber, Baruch and Shachnai, Hadas and Wolf, Joel L.},
  title =	{{The Container Selection Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{416--434},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.416},
  URN =		{urn:nbn:de:0030-drops-53153},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.416},
  annote =	{Keywords: non-metric k-median, geometric hitting set, approximation algorithms, cloud computing, cross platform scheduling.}
}
Document
The Hardness of Approximation of Euclidean k-Means

Authors: Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d. In this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.

Cite as

Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The Hardness of Approximation of Euclidean k-Means. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 754-767, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{awasthi_et_al:LIPIcs.SOCG.2015.754,
  author =	{Awasthi, Pranjal and Charikar, Moses and Krishnaswamy, Ravishankar and Sinop, Ali Kemal},
  title =	{{The Hardness of Approximation of Euclidean k-Means}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{754--767},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.754},
  URN =		{urn:nbn:de:0030-drops-51178},
  doi =		{10.4230/LIPIcs.SOCG.2015.754},
  annote =	{Keywords: Euclidean k-means, Hardness of Approximation, Vertex Cover}
}
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