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Documents authored by Mömke, Tobias

Document
DOI: 10.4230/LIPIcs.ESA.2021.49
Faster (1+ε)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back

Authors: Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
Unsplittable flow on a path (UFP) is an important and well-studied problem. We are given a path with capacities on its edges, and a set of tasks where for each task we are given a demand, a subpath, and a weight. The goal is to select the set of tasks of maximum total weight whose total demands do not exceed the capacity on any edge. UFP admits an (1+ε)-approximation with a running time of n^{O_{ε}(poly(log n))}, i.e., a QPTAS {[}Bansal et al., STOC 2006; Batra et al., SODA 2015{]} and it is considered an important open problem to construct a PTAS. To this end, in a series of papers polynomial time approximation algorithms have been developed, which culminated in a (5/3+ε)-approximation {[}Grandoni et al., STOC 2018{]} and very recently an approximation ratio of (1+1/(e+1)+ε) < 1.269 {[}Grandoni et al., 2020{]}. In this paper, we address the search for a PTAS from a different angle: we present a faster (1+ε)-approximation with a running time of only n^{O_{ε}(log log n)}. We first give such a result in the relaxed setting of resource augmentation and then transform it to an algorithm without resource augmentation. For this, we present a framework which transforms algorithms for (a slight generalization of) UFP under resource augmentation in a black-box manner into algorithms for UFP without resource augmentation, with only negligible loss.
Cite as

Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese. Faster (1+ε)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grandoni_et_al:LIPIcs.ESA.2021.49,
  author =	{Grandoni, Fabrizio and M\"{o}mke, Tobias and Wiese, Andreas},
  title =	{{Faster (1+\epsilon)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus 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.2021.49},
  URN =		{urn:nbn:de:0030-drops-146301},
  doi =		{10.4230/LIPIcs.ESA.2021.49},
  annote =	{Keywords: Approximation Algorithms, Unsplittable Flow, Dynamic Programming}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.86
Breaking the Barrier of 2 for the Storage Allocation Problem

Authors: Tobias Mömke and Andreas Wiese

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
Packing problems are an important class of optimization problems. The probably most well-known problem if this type is knapsack and many generalizations of it have been studied in the literature like Two-dimensional Geometric Knapsack (2DKP) and Unsplittable Flow on a Path (UFP). For the latter two problems, recently the first polynomial time approximation algorithms with better approximation ratios than 2 were presented [Gálvez et al., FOCS 2017][Grandoni et al., STOC 2018]. In this paper we break the barrier of 2 for the Storage Allocation Problem (SAP), a problem which combines properties of 2DKP and UFP. In SAP, we are given a path with capacitated edges and a set of tasks where each task has a start vertex, an end vertex, a size, and a profit. We seek to select the most profitable set of tasks that we can draw as non-overlapping rectangles underneath the capacity profile of the edges where the height of each rectangle equals the size of the corresponding task. The problem SAP appears naturally in settings of allocating resources like memory, bandwidth, etc. where each request needs a contiguous portion of the resource. The best known polynomial time approximation algorithm for SAP has an approximation ratio of 2+ε [Mömke and Wiese, ICALP 2015] and no better quasi-polynomial time algorithm is known. We present a polynomial time (63/32+ε) < 1.969-approximation algorithm for the important case of uniform edge capacities and a quasi-polynomial time (1.997+ε)-approximation algorithm for non-uniform quasi-polynomially bounded edge capacities. Key to our results are building blocks consisting of stair-blocks, jammed tasks, and boxes that we use to construct profitable solutions and which allow us to compute solutions of these types efficiently. Finally, using our techniques we show that under slight resource augmentation we can obtain even approximation ratios of 3/2+ε in polynomial time and 1+ε in quasi-polynomial time, both for arbitrary edge capacities.
Cite as

Tobias Mömke and Andreas Wiese. Breaking the Barrier of 2 for the Storage Allocation Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 86:1-86:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{momke_et_al:LIPIcs.ICALP.2020.86,
  author =	{M\"{o}mke, Tobias and Wiese, Andreas},
  title =	{{Breaking the Barrier of 2 for the Storage Allocation Problem}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{86:1--86:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.86},
  URN =		{urn:nbn:de:0030-drops-124931},
  doi =		{10.4230/LIPIcs.ICALP.2020.86},
  annote =	{Keywords: Approximation Algorithms, Resource Allocation, Dynamic Programming}
}
Document
DOI: 10.4230/LIPIcs.ESA.2018.34
A QPTAS for Gapless MEC

Authors: Shilpa Garg and Tobias Mömke

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Abstract
We consider the problem Minimum Error Correction (MEC). A MEC instance is an n x m matrix M with entries from {0,1,-}. Feasible solutions are composed of two binary m-bit strings, together with an assignment of each row of M to one of the two strings. The objective is to minimize the number of mismatches (errors) where the row has a value that differs from the assigned solution string. The symbol "-" is a wildcard that matches both 0 and 1. A MEC instance is gapless, if in each row of M all binary entries are consecutive. Gapless-MEC is a relevant problem in computational biology, and it is closely related to segmentation problems that were introduced by {[}Kleinberg-Papadimitriou-Raghavan STOC'98{]} in the context of data mining. Without restrictions, it is known to be UG-hard to compute an O(1)-approximate solution to MEC. For both MEC and Gapless-MEC, the best polynomial time approximation algorithm has a logarithmic performance guarantee. We partially settle the approximation status of Gapless-MEC by providing a quasi-polynomial time approximation scheme (QPTAS). Additionally, for the relevant case where the binary part of a row is not contained in the binary part of another row, we provide a polynomial time approximation scheme (PTAS).
Cite as

Shilpa Garg and Tobias Mömke. A QPTAS for Gapless MEC. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{garg_et_al:LIPIcs.ESA.2018.34,
  author =	{Garg, Shilpa and M\"{o}mke, Tobias},
  title =	{{A QPTAS for Gapless MEC}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{34:1--34:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah 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.2018.34},
  URN =		{urn:nbn:de:0030-drops-94978},
  doi =		{10.4230/LIPIcs.ESA.2018.34},
  annote =	{Keywords: approximation algorithms, QPTAS, minimum error correction, segmentation, computational biology}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.5
Approximating Airports and Railways

Authors: Anna Adamaszek, Antonios Antoniadis, Amit Kumar, and Tobias Mömke

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
In this paper we consider the airport and railway problem (AR), which combines capacitated facility location with network design, both in the general metric and the two-dimensional Euclidean space. An instance of the airport and railway problem consists of a set of points in the corresponding metric, together with a non-negative weight for each point, and a parameter k. The points represent cities, the weights denote costs of opening an airport in the corresponding city, and the parameter k is a maximum capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting all the cities, where railways correspond to edges connecting pairs of points, and the cost of a railway is equal to the distance between the corresponding points. The network is partitioned into components, where each component contains an open airport, and spans at most k cities. For the Euclidean case, any points in the plane can be used as Steiner vertices of the network. We obtain the first bicriteria approximation algorithm for AR for the general metric case, which yields a 4-approximate solution with a resource augmentation of the airport capacity k by a factor of 2. More generally, for any parameter 0 < p <= 1 where pk is an integer we develop a (4/3)(2 + 1/p)-approximation algorithm for metric AR with a resource augmentation by a factor of 1 + p. Furthermore, we obtain the first constant factor approximation algorithm that does not resort to resource augmentation for AR in the Euclidean plane. Additionally, for the Euclidean setting we provide a quasi-polynomial time approximation scheme for the same problem with a resource augmentation by a factor of 1 + mu on the airport capacity, for any fixed mu > 0.
Cite as

Anna Adamaszek, Antonios Antoniadis, Amit Kumar, and Tobias Mömke. Approximating Airports and Railways. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{adamaszek_et_al:LIPIcs.STACS.2018.5,
  author =	{Adamaszek, Anna and Antoniadis, Antonios and Kumar, Amit and M\"{o}mke, Tobias},
  title =	{{Approximating Airports and Railways}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{5:1--5:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.5},
  URN =		{urn:nbn:de:0030-drops-85183},
  doi =		{10.4230/LIPIcs.STACS.2018.5},
  annote =	{Keywords: Network Design, Facility Location, Approximation Algorithms, PTAS, Metric, Euclidean}
}
Document
DOI: 10.4230/LIPIcs.STACS.2016.6
Airports and Railways: Facility Location Meets Network Design

Authors: Anna Adamaszek, Antonios Antoniadis, and Tobias Mömke

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Abstract
We introduce a new framework of Airport and Railway Problems, which combines capacitated facility location with network design. In this framework we are given a graph with weights on the vertices and on the edges, together with a parameter k. The vertices of the graph represent cities, and weights denote respectively the costs of opening airports in the cities and building railways that connect pairs of cities. The parameter $k$ can be thought of as the capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting the cities, where each connected component in the network spans at most k vertices, contains an open airport, and the network satisfies some additional requirements specific to the problem in the framework. We consider two problems in this framework. In the AR_F problem there are no additional requirements for the network. This problem is related to capacitated facility location. In the AR_P problem, we require each component to be a path with airports at both endpoints. AR_P is a relaxation of the capacitated vehicle routing problem (CVRP). We consider the problems in the two-dimensional Euclidean setting. We show that both AR_F and AR_P are NP-hard, even for uniform vertex weights (i.e., when the cost of building an airport is the same for all cities). On the positive side, we provide polynomial time approximation schemes for AR_F and AR_P when vertex weights are uniform. We also investigate AR_F and AR_P for k = infinity. In this setting we present an exact polynomial time algorithm for AR_F with general vertex costs, which also works for general edge costs. In contrast to AR_F, AR_P remains NP-hard when k = infinity, and we present a polynomial time approximation scheme for general vertex weights. We believe that our PTAS for AR_P with uniform vertex weights and arbitrary k brings us closer towards a PTAS for Euclidean CVRP, for which the main difficulty is to deal with paths of length at most k.
Cite as

Anna Adamaszek, Antonios Antoniadis, and Tobias Mömke. Airports and Railways: Facility Location Meets Network Design. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{adamaszek_et_al:LIPIcs.STACS.2016.6,
  author =	{Adamaszek, Anna and Antoniadis, Antonios and M\"{o}mke, Tobias},
  title =	{{Airports and Railways: Facility Location Meets Network Design}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.6},
  URN =		{urn:nbn:de:0030-drops-57074},
  doi =		{10.4230/LIPIcs.STACS.2016.6},
  annote =	{Keywords: approximation algorithms, geometric approximation, facility location, network design, PTAS}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2015.10
Robust Reoptimization of Steiner Trees

Authors: Keshav Goyal and Tobias Mömke

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Abstract
In reoptimization problems, one is given an optimal solution to a problem instance and a local modification of the instance. The goal is to obtain a solution for the modified instance. The additional information about the instance provided by the given solution plays a central role: we aim to use that information in order to obtain better solutions than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed epsilon > 0, approximating the reoptimization problem with respect to a given (1+epsilon)-approximation is as hard as approximating the Steiner tree problem itself (whereas with a given optimal solution to the original problem it is known that one can obtain considerably improved results). Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.
Cite as

Keshav Goyal and Tobias Mömke. Robust Reoptimization of Steiner Trees. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 10-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goyal_et_al:LIPIcs.FSTTCS.2015.10,
  author =	{Goyal, Keshav and M\"{o}mke, Tobias},
  title =	{{Robust Reoptimization of Steiner Trees}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{10--24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.10},
  URN =		{urn:nbn:de:0030-drops-56251},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.10},
  annote =	{Keywords: reoptimization, approximation algorithms, Steiner tree problem, robustness}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2015.294
Complexity and Approximability of Parameterized MAX-CSPs

Authors: Holger Dell, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Tobias Mömke

Published in: LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Abstract
We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k. We attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT-time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPT-AS), which computes a (1-epsilon)-approximation in time f(k,epsilon) * poly(n). 3. There is no FPT-AS unless FPT=W[1]. For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.
Cite as

Holger Dell, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Tobias Mömke. Complexity and Approximability of Parameterized MAX-CSPs. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 294-306, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dell_et_al:LIPIcs.IPEC.2015.294,
  author =	{Dell, Holger and Kim, Eun Jung and Lampis, Michael and Mitsou, Valia and M\"{o}mke, Tobias},
  title =	{{Complexity and Approximability of Parameterized MAX-CSPs}},
  booktitle =	{10th International Symposium on Parameterized and Exact Computation (IPEC 2015)},
  pages =	{294--306},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-92-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{43},
  editor =	{Husfeldt, Thore and Kanj, Iyad},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.294},
  URN =		{urn:nbn:de:0030-drops-55910},
  doi =		{10.4230/LIPIcs.IPEC.2015.294},
  annote =	{Keywords: Approximation, Structural Parameters, Constraint Satisfaction}
}
Document
DOI: 10.4230/LIPIcs.STACS.2014.470
Randomized Online Algorithms with High Probability Guarantees

Authors: Dennis Komm, Rastislav Královic, Richard Královic, and Tobias Mömke

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Abstract
We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error with high probability; the "high probability" statement is in terms of the optimal cost. Furthermore, we show that our assumptions are tight in the sense that removing any of them allows for a counterexample to the theorem.
Cite as

Dennis Komm, Rastislav Královic, Richard Královic, and Tobias Mömke. Randomized Online Algorithms with High Probability Guarantees. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 470-481, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{komm_et_al:LIPIcs.STACS.2014.470,
  author =	{Komm, Dennis and Kr\'{a}lovic, Rastislav and Kr\'{a}lovic, Richard and M\"{o}mke, Tobias},
  title =	{{Randomized Online Algorithms with High Probability Guarantees}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{470--481},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.470},
  URN =		{urn:nbn:de:0030-drops-44803},
  doi =		{10.4230/LIPIcs.STACS.2014.470},
  annote =	{Keywords: Online Algorithms, Randomization, High Probability}
}
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