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Documents authored by Jacob, Riko

Document
DOI: 10.4230/LIPIcs.ESA.2024.44
An Optimal Randomized Algorithm for Finding the Saddlepoint

Authors: Justin Dallant, Frederik Haagensen, Riko Jacob, László Kozma, and Sebastian Wild

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
A saddlepoint of an n × n matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the value of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task. For finding a strict saddlepoint (an entry that is the strict maximum of its row and the strict minimum of its column) an O(n log* n)-time algorithm was recently obtained by Dallant, Haagensen, Jacob, Kozma, and Wild, improving the O(n log n) bounds from 1991 of Bienstock, Chung, Fredman, Schäffer, Shor, Suri and of Byrne and Vaserstein. In this paper we present an optimal O(n)-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial O(n²) runtime cannot be improved even with the use of randomness.
Cite as

Justin Dallant, Frederik Haagensen, Riko Jacob, László Kozma, and Sebastian Wild. An Optimal Randomized Algorithm for Finding the Saddlepoint. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 44:1-44:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dallant_et_al:LIPIcs.ESA.2024.44,
  author =	{Dallant, Justin and Haagensen, Frederik and Jacob, Riko and Kozma, L\'{a}szl\'{o} and Wild, Sebastian},
  title =	{{An Optimal Randomized Algorithm for Finding the Saddlepoint}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{44:1--44:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.44},
  URN =		{urn:nbn:de:0030-drops-211154},
  doi =		{10.4230/LIPIcs.ESA.2024.44},
  annote =	{Keywords: saddlepoint, matrix, comparison, search, randomized algorithms}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.23
On Instance-Optimal Algorithms for a Generalization of Nuts and Bolts and Generalized Sorting

Authors: Mayank Goswami and Riko Jacob

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We generalize the classical nuts and bolts problem to a setting where the input is a collection of n nuts and m bolts, and there is no promise of any matching pairs. It is not allowed to compare a nut directly with a nut or a bolt directly with a bolt, and the goal is to perform the fewest nut-bolt comparisons to discover the partial order between the nuts and bolts. We term this problem bipartite sorting. We show that instances of bipartite sorting of the same size exhibit a wide range of complexity, and propose to perform a fine-grained analysis for this problem. We rule out straightforward notions of instance-optimality as being too stringent, and adopt a neighborhood-based definition. Our definition may be of independent interest as a unifying lens for instance-optimal algorithms for other static problems existing in literature. This includes problems like sorting (Estivill-Castro and Woods, ACM Comput. Surv. 1992), convex hull (Afshani, Barbay and Chan, JACM 2017), adaptive joins (Demaine, López-Ortiz and Munro, SODA 2000), and the recent concept of universal optimality for graphs (Haeupler, Hladík, Rozhoň, Tarjan and Tětek, 2023). As our main result on bipartite sorting, we give a randomized algorithm that is within a factor of O(log³(n+m)) of being instance-optimal w.h.p., with respect to the neighborhood-based definition. As our second contribution, we generalize bipartite sorting to DAG sorting, when the underlying DAG is not necessarily bipartite. As an unexpected consequence of a simple algorithm for DAG sorting, we rule out a potential lower bound on the widely-studied problem of sorting with priced information, posed by (Charikar, Fagin, Guruswami, Kleinberg, Raghavan and Sahai, STOC 2000). In this problem, comparing keys i and j has a known cost c_{ij} ∈ ℝ^+ ∪ {∞}, and the goal is to sort the keys in an instance-optimal way, by keeping the total cost of an algorithm as close as possible to ∑_{i=1}^{n-1} c_{x(i)x(i+1)}. Here x(1) < ⋯ < x(n) is the sorted order. While several special cases of cost functions have received a lot of attention in the community, no progress on the general version with arbitrary costs has been reported so far. One reason for this lack of progress seems to be a widely-cited Ω(n) lower bound on the competitive ratio for finding the maximum. This Ω(n) lower bound by (Gupta and Kumar, FOCS 2000) uses costs in {0,1,n, ∞}, and although not extended to sorting, this barrier seems to have stalled any progress on the general cost case. We rule out such a potential lower bound by showing the existence of an algorithm with a Õ(n^{3/4}) competitive ratio for the {0,1,n,∞} cost version. This generalizes the setting of generalized sorting proposed by (Huang, Kannan and Khanna, FOCS 2011), where the costs are either 1 or infinity, and the cost of the cheapest proof is always n-1.
Cite as

Mayank Goswami and Riko Jacob. On Instance-Optimal Algorithms for a Generalization of Nuts and Bolts and Generalized Sorting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 23:1-23:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{goswami_et_al:LIPIcs.APPROX/RANDOM.2024.23,
  author =	{Goswami, Mayank and Jacob, Riko},
  title =	{{On Instance-Optimal Algorithms for a Generalization of Nuts and Bolts and Generalized Sorting}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{23:1--23:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.23},
  URN =		{urn:nbn:de:0030-drops-210168},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.23},
  annote =	{Keywords: Sorting, Priced Information, Instance Optimality, Nuts and Bolts}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.56
An Algorithm for Bichromatic Sorting with Polylog Competitive Ratio

Authors: Mayank Goswami and Riko Jacob

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
The problem of sorting with priced information was introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons have different (potentially infinite) costs. The goal is to find a sorting algorithm with small competitive ratio, defined as the (worst-case) ratio of the algorithm’s cost to the cost of the cheapest proof of the sorted order. The simple case of bichromatic sorting posed by [CFGKRS] remains open: We are given two sets A and B of total size N, and the cost of an A-A comparison or a B-B comparison is higher than an A-B comparison. The goal is to sort A ∪ B. An Ω(log N) lower bound on competitive ratio follows from unit-cost sorting. Note that this is a generalization of the famous nuts and bolts problem, where A-A and B-B comparisons have infinite cost, and elements of A and B are guaranteed to alternate in the final sorted order. In this paper we give a randomized algorithm InversionSort with an almost-optimal w.h.p. competitive ratio of O(log³ N). This is the first algorithm for bichromatic sorting with a o(N) competitive ratio.
Cite as

Mayank Goswami and Riko Jacob. An Algorithm for Bichromatic Sorting with Polylog Competitive Ratio. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 56:1-56:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{goswami_et_al:LIPIcs.ITCS.2024.56,
  author =	{Goswami, Mayank and Jacob, Riko},
  title =	{{An Algorithm for Bichromatic Sorting with Polylog Competitive Ratio}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{56:1--56:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.56},
  URN =		{urn:nbn:de:0030-drops-195843},
  doi =		{10.4230/LIPIcs.ITCS.2024.56},
  annote =	{Keywords: Sorting, Priced Information, Nuts and Bolts}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.2
Fragile Complexity of Comparison-Based Algorithms

Authors: Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, and Nodari Sitchinava

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

Abstract
We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We give a number of upper and lower bounds on the fragile complexity for fundamental problems, including Minimum, Selection, Sorting and Heap Construction. The results include both deterministic and randomized upper and lower bounds, and demonstrate a separation between the two settings for a number of problems. The depth of a comparator network is a straight-forward upper bound on the worst case fragile complexity of the corresponding fragile algorithm. We prove that fragile complexity is a different and strictly easier property than the depth of comparator networks, in the sense that for some problems a fragile complexity equal to the best network depth can be achieved with less total work and that with randomization, even a lower fragile complexity is possible.
Cite as

Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, and Nodari Sitchinava. Fragile Complexity of Comparison-Based Algorithms. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{afshani_et_al:LIPIcs.ESA.2019.2,
  author =	{Afshani, Peyman and Fagerberg, Rolf and Hammer, David and Jacob, Riko and Kostitsyna, Irina and Meyer, Ulrich and Penschuck, Manuel and Sitchinava, Nodari},
  title =	{{Fragile Complexity of Comparison-Based Algorithms}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{2:1--2:19},
  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.2019.2},
  URN =		{urn:nbn:de:0030-drops-111235},
  doi =		{10.4230/LIPIcs.ESA.2019.2},
  annote =	{Keywords: Algorithms, comparison based algorithms, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.60
External Memory Priority Queues with Decrease-Key and Applications to Graph Algorithms

Authors: John Iacono, Riko Jacob, and Konstantinos Tsakalidis

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

Abstract
We present priority queues in the external memory model with block size B and main memory size M that support on N elements, operation Update (a combination of operations Insert and DecreaseKey) in O(1/Blog_{M/B} N/B) amortized I/Os and operations ExtractMin and Delete in O(ceil[(M^epsilon)/B log_{M/B} N/B] log_{M/B} N/B) amortized I/Os, for any real epsilon in (0,1), using O(N/Blog_{M/B} N/B) blocks. Previous I/O-efficient priority queues either support these operations in O(1/Blog_2 N/B) amortized I/Os [Kumar and Schwabe, SPDP '96] or support only operations Insert, Delete and ExtractMin in optimal O(1/Blog_{M/B} N/B) amortized I/Os, however without supporting DecreaseKey [Fadel et al., TCS '99]. We also present buffered repository trees that support on a multi-set of N elements, operation Insert in O(1/Blog_M/B N/B) I/Os and operation Extract on K extracted elements in O(M^{epsilon} log_M/B N/B + K/B) amortized I/Os, using O(N/B) blocks. Previous results achieve O(1/Blog_2 N/B) I/Os and O(log_2 N/B + K/B) I/Os, respectively [Buchsbaum et al., SODA '00]. Our results imply improved O(E/Blog_{M/B} E/B) I/Os for single-source shortest paths, depth-first search and breadth-first search algorithms on massive directed dense graphs (V,E) with E = Omega (V^(1+epsilon)), epsilon > 0 and V = Omega (M), which is equal to the I/O-optimal bound for sorting E values in external memory.
Cite as

John Iacono, Riko Jacob, and Konstantinos Tsakalidis. External Memory Priority Queues with Decrease-Key and Applications to Graph Algorithms. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{iacono_et_al:LIPIcs.ESA.2019.60,
  author =	{Iacono, John and Jacob, Riko and Tsakalidis, Konstantinos},
  title =	{{External Memory Priority Queues with Decrease-Key and Applications to Graph Algorithms}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{60:1--60:14},
  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.2019.60},
  URN =		{urn:nbn:de:0030-drops-111817},
  doi =		{10.4230/LIPIcs.ESA.2019.60},
  annote =	{Keywords: priority queues, external memory, graph algorithms, shortest paths, depth-first search, breadth-first search}
}
Document
Complete Volume
DOI: 10.4230/OASIcs.ATMOS.2006
OASIcs, Volume 5, ATMOS'06, Complete Volume

Authors: Riko Jacob and Matthias Müller-Hannemann

Published in: OASIcs, Volume 5, 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06) (2006)

Abstract
OASIcs, Volume 5, ATMOS'06, Complete Volume
Cite as

6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06). Open Access Series in Informatics (OASIcs), Volume 5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@Proceedings{jacob_et_al:OASIcs.ATMOS.2006,
  title =	{{OASIcs, Volume 5, ATMOS'06, Complete Volume}},
  booktitle =	{6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06)},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-01-9},
  ISSN =	{2190-6807},
  year =	{2012},
  volume =	{5},
  editor =	{Jacob, Riko and M\"{u}ller-Hannemann, Matthias},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2006},
  URN =		{urn:nbn:de:0030-drops-35675},
  doi =		{10.4230/OASIcs.ATMOS.2006},
  annote =	{Keywords: Analysis of Algorithms and Problem Complexity, Optimization, Graph Theory, Applications}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2007.1179
11. Multistage Methods for Freight Train Classification

Authors: Riko Jacob, Peter Marton, Jens Maue, and Marc Nunkesser

Published in: OASIcs, Volume 7, 7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'07) (2007)

Abstract
In this paper we establish a consistent encoding of freight train classification methods. This encoding scheme presents a powerful tool for efficient presentation and analysis of classification methods, which we successfully apply to illustrate the most relevant historic results from a more theoretical point of view. We analyze their performance precisely and develop new classification methods making use of the inherent optimality condition of the encoding. We conclude with deriving optimal algorithms and complexity results for restricted real-world settings.
Cite as

Riko Jacob, Peter Marton, Jens Maue, and Marc Nunkesser. 11. Multistage Methods for Freight Train Classification. In 7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'07). Open Access Series in Informatics (OASIcs), Volume 7, pp. 158-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{jacob_et_al:OASIcs.ATMOS.2007.1179,
  author =	{Jacob, Riko and Marton, Peter and Maue, Jens and Nunkesser, Marc},
  title =	{{11. Multistage Methods for Freight Train Classification}},
  booktitle =	{7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'07)},
  pages =	{158--174},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-04-0},
  ISSN =	{2190-6807},
  year =	{2007},
  volume =	{7},
  editor =	{Ahuja, Ravindra K. and Liebchen, Christian and Mesa, Juan A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2007.1179},
  URN =		{urn:nbn:de:0030-drops-11798},
  doi =		{10.4230/OASIcs.ATMOS.2007.1179},
  annote =	{Keywords: Freight trains, sorting algorithms, train classification, shunting, cargo}
}
Document
Front Matter
DOI: 10.4230/OASIcs.ATMOS.2006.690
ATMOS 2006 Preface -- Algorithmic Methods and Models for Optimization of Railways

Authors: Riko Jacob and Matthias Müller-Hannemann

Published in: OASIcs, Volume 5, 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06) (2006)

Abstract
The 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS 06) is held on September 14, 2006 in Z{\"u}rich, Switzerland (http://algo06.inf.ethz.ch/atmos), as part of the ALGO meeting.
Cite as

6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06). Open Access Series in Informatics (OASIcs), Volume 5, pp. i-v, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{jacob_et_al:OASIcs.ATMOS.2006.690,
  author =	{Jacob, Riko and M\"{u}ller-Hannemann, Matthias},
  title =	{{ATMOS 2006 Preface -- Algorithmic Methods and Models for Optimization of Railways}},
  booktitle =	{6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06)},
  pages =	{i--v},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-01-9},
  ISSN =	{2190-6807},
  year =	{2006},
  volume =	{5},
  editor =	{Jacob, Riko and M\"{u}ller-Hannemann, Matthias},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2006.690},
  URN =		{urn:nbn:de:0030-drops-6909},
  doi =		{10.4230/OASIcs.ATMOS.2006.690},
  annote =	{Keywords: Railway Optimization, Algorithmic Methods, Models}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2006.689
ATMOS 2006 Abstracts Collection -- Presentations at the 6th Workshop on Algorithmic Methods and Models for Optimization of Railways

Authors: Riko Jacob and Matthias Müller-Hannemann

Published in: OASIcs, Volume 5, 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06) (2006)

Abstract
The 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS 06) is held on September 14, 2006 in Zürich, Switzerland (http://algo06.inf.ethz.ch/atmos), as part of the ALGO meeting.
Cite as

Riko Jacob and Matthias Müller-Hannemann. ATMOS 2006 Abstracts Collection -- Presentations at the 6th Workshop on Algorithmic Methods and Models for Optimization of Railways. In 6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06). Open Access Series in Informatics (OASIcs), Volume 5, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{jacob_et_al:OASIcs.ATMOS.2006.689,
  author =	{Jacob, Riko and M\"{u}ller-Hannemann, Matthias},
  title =	{{ATMOS 2006 Abstracts Collection -- Presentations at the 6th Workshop on Algorithmic Methods and Models for Optimization of Railways}},
  booktitle =	{6th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS'06)},
  pages =	{1--4},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-01-9},
  ISSN =	{2190-6807},
  year =	{2006},
  volume =	{5},
  editor =	{Jacob, Riko and M\"{u}ller-Hannemann, Matthias},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2006.689},
  URN =		{urn:nbn:de:0030-drops-6898},
  doi =		{10.4230/OASIcs.ATMOS.2006.689},
  annote =	{Keywords: Algorithmic Methods and Models for Optimization of Railways, ATMOS}
}
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