8 Search Results for "Vondrak, Jan"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.74
Faster Submodular Maximization for Several Classes of Matroids

Authors: Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng

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

Abstract
The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Though tight approximation algorithms for general matroid constraints exist in theory, the running times of such algorithms typically scale quadratically, and are not practical for truly large scale settings. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [Alina Ene and Huy L. Nguyen, 2019]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of 1-1/e-ε and both generalize and accelerate the results of Ene and Nguyen [Alina Ene and Huy L. Nguyen, 2019]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondrák [Ashwinkumar Badanidiyuru and Jan Vondrák, 2014]. To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel Freeze operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [Stephen Alstrup et al., 2005] that maintains the maximum weight basis under insertions and deletions of elements in O(log n) time. This data structure needs to support certain subtree query and path update operations that are performed every insertion and deletion that are non-trivial to handle in conjunction. For the transversal matroid the Freeze operation corresponds to requiring the data structure to keep a certain set S of vertices matched, a property that we call S-stability. While there is a large body of work on dynamic matching algorithms, none are S-stable and maintain an approximate maximum weight matching under vertex updates. We give the first such algorithm for bipartite graphs with total running time linear (up to log factors) in the number of edges.
Cite as

Monika Henzinger, Paul Liu, Jan Vondrák, and Da Wei Zheng. Faster Submodular Maximization for Several Classes of Matroids. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 74:1-74:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{henzinger_et_al:LIPIcs.ICALP.2023.74,
  author =	{Henzinger, Monika and Liu, Paul and Vondr\'{a}k, Jan and Zheng, Da Wei},
  title =	{{Faster Submodular Maximization for Several Classes of Matroids}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{74:1--74:18},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.74},
  URN =		{urn:nbn:de:0030-drops-181267},
  doi =		{10.4230/LIPIcs.ICALP.2023.74},
  annote =	{Keywords: submodular optimization, dynamic data structures, matching algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.59
Streaming Submodular Maximization Under Matroid Constraints

Authors: Moran Feldman, Paul Liu, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank k (i.e., any solution has cardinality at most k), our main results are: - A single-pass streaming algorithm that uses Õ(k) memory and achieves an approximation guarantee of 0.3178. - A multi-pass streaming algorithm that uses Õ(k) memory and achieves an approximation guarantee of (1-1/e - ε) by taking a constant (depending on ε) number of passes over the stream. This improves on the previously best approximation guarantees of 1/4 and 1/2 for single-pass and multi-pass streaming algorithms, respectively. In fact, our multi-pass streaming algorithm is tight in that any algorithm with a better guarantee than 1/2 must make several passes through the stream and any algorithm that beats our guarantee of 1-1/e must make linearly many passes (as well as an exponential number of value oracle queries). Moreover, we show how the approach we use for multi-pass streaming can be further strengthened if the elements of the stream arrive in uniformly random order, implying an improved result for p-matchoid constraints.
Cite as

Moran Feldman, Paul Liu, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. Streaming Submodular Maximization Under Matroid Constraints. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 59:1-59:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{feldman_et_al:LIPIcs.ICALP.2022.59,
  author =	{Feldman, Moran and Liu, Paul and Norouzi-Fard, Ashkan and Svensson, Ola and Zenklusen, Rico},
  title =	{{Streaming Submodular Maximization Under Matroid Constraints}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{59:1--59:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.59},
  URN =		{urn:nbn:de:0030-drops-164007},
  doi =		{10.4230/LIPIcs.ICALP.2022.59},
  annote =	{Keywords: Submodular maximization, streaming, matroid, random order}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.61
Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard

Authors: Linda Cai, Clay Thomas, and S. Matthew Weinberg

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
State-of-the-art posted-price mechanisms for submodular bidders with m items achieve approximation guarantees of O((log log m)^3) [Sepehr Assadi and Sahil Singla, 2019]. Their truthfulness, however, requires bidders to compute an NP-hard demand-query. Some computational complexity of this form is unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an m^(1/2-ε)-approximation for any ε > 0 [Shahar Dobzinski and Jan Vondrák, 2016]. Together, these establish a stark distinction between computationally-efficient and communication-efficient truthful mechanisms. We show that this distinction disappears with a mild relaxation of truthfulness, which we term implementation in advised strategies. Specifically, advice maps a tentative strategy either to that same strategy itself, or one that dominates it. We say that a player follows advice as long as they never play actions which are dominated by advice. A poly-time mechanism guarantees an α-approximation in implementation in advised strategies if there exists advice (which runs in poly-time) for each player such that an α-approximation is achieved whenever all players follow advice. Using an appropriate bicriterion notion of approximate demand queries (which can be computed in poly-time), we establish that (a slight modification of) the [Sepehr Assadi and Sahil Singla, 2019] mechanism achieves the same O((log log m)^3)-approximation in implementation in advised strategies.
Cite as

Linda Cai, Clay Thomas, and S. Matthew Weinberg. Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 61:1-61:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cai_et_al:LIPIcs.ITCS.2020.61,
  author =	{Cai, Linda and Thomas, Clay and Weinberg, S. Matthew},
  title =	{{Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{61:1--61:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.61},
  URN =		{urn:nbn:de:0030-drops-117464},
  doi =		{10.4230/LIPIcs.ITCS.2020.61},
  annote =	{Keywords: Combinatorial auctions, Posted-Price mechanisms, Submodular valuations, Incentive compatible}
}
Document
DOI: 10.4230/OASIcs.SOSA.2019.18
Submodular Optimization in the MapReduce Model

Authors: Paul Liu and Jan Vondrak

Published in: OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

Abstract
Submodular optimization has received significant attention in both practice and theory, as a wide array of problems in machine learning, auction theory, and combinatorial optimization have submodular structure. In practice, these problems often involve large amounts of data, and must be solved in a distributed way. One popular framework for running such distributed algorithms is MapReduce. In this paper, we present two simple algorithms for cardinality constrained submodular optimization in the MapReduce model: the first is a (1/2-o(1))-approximation in 2 MapReduce rounds, and the second is a (1-1/e-epsilon)-approximation in (1+o(1))/epsilon MapReduce rounds.
Cite as

Paul Liu and Jan Vondrak. Submodular Optimization in the MapReduce Model. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 18:1-18:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{liu_et_al:OASIcs.SOSA.2019.18,
  author =	{Liu, Paul and Vondrak, Jan},
  title =	{{Submodular Optimization in the MapReduce Model}},
  booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
  pages =	{18:1--18:10},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-099-6},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{69},
  editor =	{Fineman, Jeremy T. and Mitzenmacher, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.18},
  URN =		{urn:nbn:de:0030-drops-100447},
  doi =		{10.4230/OASIcs.SOSA.2019.18},
  annote =	{Keywords: mapreduce, submodular, optimization, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2017.26
Stability and Recovery for Independence Systems

Authors: Vaggos Chatziafratis, Tim Roughgarden, and Jan Vondrak

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Abstract
Two genres of heuristics that are frequently reported to perform much better on "real-world" instances than in the worst case are greedy algorithms and local search algorithms. In this paper, we systematically study these two types of algorithms for the problem of maximizing a monotone submodular set function subject to downward-closed feasibility constraints. We consider perturbation-stable instances, in the sense of Bilu and Linial [11], and precisely identify the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution. Byproducts of our work include the first definition of perturbation-stability for non-additive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems.
Cite as

Vaggos Chatziafratis, Tim Roughgarden, and Jan Vondrak. Stability and Recovery for Independence Systems. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chatziafratis_et_al:LIPIcs.ESA.2017.26,
  author =	{Chatziafratis, Vaggos and Roughgarden, Tim and Vondrak, Jan},
  title =	{{Stability and Recovery for Independence Systems}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.26},
  URN =		{urn:nbn:de:0030-drops-78423},
  doi =		{10.4230/LIPIcs.ESA.2017.26},
  annote =	{Keywords: Submodular, approximation, stability, Local Search, Greedy, p-systems}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.22
When Are Welfare Guarantees Robust?

Authors: Tim Roughgarden, Inbal Talgam-Cohen, and Jan Vondrák

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders' valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations.
Cite as

Tim Roughgarden, Inbal Talgam-Cohen, and Jan Vondrák. When Are Welfare Guarantees Robust?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{roughgarden_et_al:LIPIcs.APPROX-RANDOM.2017.22,
  author =	{Roughgarden, Tim and Talgam-Cohen, Inbal and Vondr\'{a}k, Jan},
  title =	{{When Are Welfare Guarantees Robust?}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{22:1--22:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.22},
  URN =		{urn:nbn:de:0030-drops-75714},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.22},
  annote =	{Keywords: Valuation (set) functions, gross substitutes, linearity, approximation}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.144
Hardness of Submodular Cost Allocation: Lattice Matching and a Simplex Coloring Conjecture

Authors: Alina Ene and Jan Vondrák

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
We consider the Minimum Submodular Cost Allocation (MSCA) problem. In this problem, we are given k submodular cost functions f_1, ... , f_k: 2^V -> R_+ and the goal is to partition V into k sets A_1, ..., A_k so as to minimize the total cost sum_{i = 1}^k f_i(A_i). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function f_i satisfies f_i = g_i + h, where each g_i is monotone submodular and h is (possibly non-monotone) submodular. We give an O(k log |V|) approximation for this problem. We provide some evidence that a factor of k may be necessary, even in the special case of HyperLabel. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon) for k-uniform HyperLabel and label set [k]. We provide a proof of the simplex-coloring conjecture for k=3.
Cite as

Alina Ene and Jan Vondrák. Hardness of Submodular Cost Allocation: Lattice Matching and a Simplex Coloring Conjecture. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 144-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{ene_et_al:LIPIcs.APPROX-RANDOM.2014.144,
  author =	{Ene, Alina and Vondr\'{a}k, Jan},
  title =	{{Hardness of Submodular Cost Allocation: Lattice Matching and a Simplex Coloring Conjecture}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{144--159},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.144},
  URN =		{urn:nbn:de:0030-drops-46943},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.144},
  annote =	{Keywords: Minimum Cost Submodular Allocation, Submodular Optimization, Hypergraph Labeling}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.762
Exchangeability and Realizability: De Finetti Theorems on Graphs

Authors: T. S. Jayram and Jan Vondrák

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
A classic result in probability theory known as de Finetti's theorem states that exchangeable random variables are equivalent to a mixture of distributions where each distribution is determined by an i.i.d. sequence of random variables (an "i.i.d. mix"). Motivated by a recent application and more generally by the relationship of local vs. global correlation in randomized rounding, we study weaker notions of exchangeability that still imply the conclusion of de Finetti's theorem. We say that a bivariate distribution rho is G-realizable for a graph G if there exists a joint distribution of random variables on the vertices such that the marginal distribution on each edge equals rho. We first characterize completely the G-realizable distributions for all symmetric/arc-transitive graphs G. Our main results are forms of de Finetti's theorem for general graphs, based on spectral properties. Let lambda_1(G) >= ... >= lambda_n(G) denote the eigenvalues of the adjacency matrix of G. 1. We prove that if rho is G_n-realizable for a sequence of graphs such that lambda_n(G_n) / lambda_1(G_n) tends to 0, then rho is described by a probability matrix that is positive-semidefinite. For random variables on domains of size |D| <= 4, this implies that rho must be an i.i.d. mix. 2. If rho is G_n-realizable for a sequence of (n,d,lambda)-graphs G_n (d-regular with all eigenvalues except for one bounded by lambda in absolute value) such that lambda(G_n) / d(G_n) tends to 0, then rho is an i.i.d. mix. 3. If rho is G_n-realizable for a sequence of directed graphs such that each of them is an arbitrary orientation of an (n,d,lambda)-graph G_n, and lambda(G_n) / d(G_n) tends to 0, then rho is an i.i.d. mix.
Cite as

T. S. Jayram and Jan Vondrák. Exchangeability and Realizability: De Finetti Theorems on Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 762-778, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{jayram_et_al:LIPIcs.APPROX-RANDOM.2014.762,
  author =	{Jayram, T. S. and Vondr\'{a}k, Jan},
  title =	{{Exchangeability and Realizability: De Finetti Theorems on Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{762--778},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.762},
  URN =		{urn:nbn:de:0030-drops-47375},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.762},
  annote =	{Keywords: exchangeability, de Finetti’s Theorem, spectral graph theory, regularity lemma}
}
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