Document

RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter λ > 0; the special case λ = 1 corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete Δ-regular tree for all λ. However, Restrepo et al. (2014) showed that for sufficiently large λ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width.
We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of O(n) for the Glauber dynamics for unweighted independent sets on arbitrary trees. Moreover, for λ ≤ .44 we prove an optimal mixing time bound of O(n log n). We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree Δ. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order λ = O(1/Δ). Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance/entropy via a non-trivial inductive proof.

Charilaos Efthymiou, Thomas P. Hayes, Daniel Štefankovič, and Eric Vigoda. Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{efthymiou_et_al:LIPIcs.APPROX/RANDOM.2023.33, author = {Efthymiou, Charilaos and Hayes, Thomas P. and \v{S}tefankovi\v{c}, Daniel and Vigoda, Eric}, title = {{Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {33:1--33:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.33}, URN = {urn:nbn:de:0030-drops-188589}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.33}, annote = {Keywords: MCMC, Mixing Time, Independent Sets, Hard-Core Model, Approximate Counting Algorithms, Sampling Algorithms} }

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Track A: Algorithms, Complexity and Games

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

We study the single-site Glauber dynamics for the fugacity λ, Hard-Core model on the random graph G(n, d/n). We show that for the typical instances of the random graph G(n,d/n) and for fugacity λ < {d^d} / {(d-1)^(d+1)}, the mixing time of Glauber dynamics is n^{1 + O(1/log log n)}.
Our result improves on the recent elegant algorithm in [Bezáková, Galanis, Goldberg and Štefankovič; ICALP'22]. The algorithm there is an MCMC-based sampling algorithm, but it is not the Glauber dynamics. Our algorithm here is simpler, as we use the classic Glauber dynamics. Furthermore, the bounds on mixing time we prove are smaller than those in Bezáková et al. paper, hence our algorithm is also faster.
The main challenge in our proof is handling vertices with unbounded degrees. We provide stronger results with regard the spectral independence via branching values and show that the our Gibbs distributions satisfy the approximate tensorisation of the entropy. We conjecture that the bounds we have here are optimal for G(n,d/n).
As corollary of our analysis for the Hard-Core model, we also get bounds on the mixing time of the Glauber dynamics for the Monomer-Dimer model on G(n,d/n). The bounds we get for this model are slightly better than those we have for the Hard-Core model

Charilaos Efthymiou and Weiming Feng. On the Mixing Time of Glauber Dynamics for the Hard-Core and Related Models on G(n,d/n). In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{efthymiou_et_al:LIPIcs.ICALP.2023.54, author = {Efthymiou, Charilaos and Feng, Weiming}, title = {{On the Mixing Time of Glauber Dynamics for the Hard-Core and Related Models on G(n,d/n)}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {54:1--54:17}, 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.2023.54}, URN = {urn:nbn:de:0030-drops-181064}, doi = {10.4230/LIPIcs.ICALP.2023.54}, annote = {Keywords: spin-system, spin-glass, sparse random (hyper)graph, approximate sampling, efficient algorithm} }

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Track A: Algorithms, Complexity and Games

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

Motivated by the theory of spin-glasses in physics, we study the so-called reconstruction problem on the tree, and on the sparse random graph G(n,d/n). Both cases reduce naturally to analysing broadcasting models, where each edge has its own broadcasting matrix, and this matrix is drawn independently from a predefined distribution.
We establish the reconstruction threshold for the cases where the broadcasting matrices give rise to symmetric, 2-spin Gibbs distributions. This threshold seems to be a natural extension of the well-known Kesten-Stigum bound that manifests in the classic version of the reconstruction problem. Our results determine, as a special case, the reconstruction threshold for the prominent Edwards–Anderson model of spin-glasses, on the tree.
Also, we extend our analysis to the setting of the Galton-Watson random tree, and the (sparse) random graph G(n,d/n), where we establish the corresponding thresholds. Interestingly, for the Edwards–Anderson model on the random graph, we show that the replica symmetry breaking phase transition, established by Guerra and and Toninelli in [Guerra and Toninelli, 2004], coincides with the reconstruction threshold.
Compared to classical Gibbs distributions, spin-glasses have several unique features. In that respect, their study calls for new ideas, e.g. we introduce novel estimators for the reconstruction problem. The main technical challenge in the analysis of such systems, is the presence of (too) many levels of randomness, which we manage to circumvent by utilising recently proposed tools coming from the analysis of Markov chains.

Charilaos Efthymiou and Kostas Zampetakis. Broadcasting with Random Matrices. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{efthymiou_et_al:LIPIcs.ICALP.2023.55, author = {Efthymiou, Charilaos and Zampetakis, Kostas}, title = {{Broadcasting with Random Matrices}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {55:1--55:14}, 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.2023.55}, URN = {urn:nbn:de:0030-drops-181070}, doi = {10.4230/LIPIcs.ICALP.2023.55}, annote = {Keywords: spin-system, spin-glass, sparse random graph, reconstruction, phase transitions} }

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Track A: Algorithms, Complexity and Games

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

In this paper, we present a novel, polynomial time, algorithm for approximate sampling from symmetric Gibbs distributions on the sparse random graph and hypergraph. The examples of symmetric distributions we consider here include some important distributions on spin-systems and spin-glasses. These are: the q-state antiferromagnetic Potts model for q ≥ 2, including the (hyper)graph Ising model and random colourings. The uniform distribution over the Not-All-Equal solutions of a random k-SAT formula. Finally, we consider sampling from the spin-glass distribution called the k-spin model, i.e., this is the "diluted" version of the well-known Sherrington-Kirkpatrick model. Spin-glasses give rise to very intricate distributions which are also studied in mathematics, in neural computation, computational biology and many other areas. To our knowledge, this is the first rigorously analysed efficient sampling algorithm for spin-glasses which operates in a non trivial range of the parameters of the distribution.
We present, what we believe to be, an elegant sampling algorithm. Our algorithm is unique in its approach and does not belong to any of the well-known families of sampling algorithms. We derive it by investigating the power and the limits of the approach that was introduced in [Efthymiou: SODA 2012] and combine it, in a novel way, with powerful notions from the Cavity method.
Specifically, for a symmetric Gibbs distribution μ on the random (hyper)graph whose parameters are within an appropriate range, our sampling algorithm has the following properties: with probability 1-o(1) over the instances of the input (hyper)graph, it generates a configuration which is distributed within total variation distance n^{-Ω(1)} from μ. The time complexity is O((nlog n)²), where n is the size of the input (hyper)graph.
We make a notable progress regarding impressive predictions of physicists relating phase-transitions of Gibbs distributions with the efficiency of the corresponding sampling algorithms. For most (if not all) the cases we consider here, our algorithm outperforms by far any other sampling algorithms in terms of the permitted range of the parameters of the Gibbs distributions.
The use of notions and ideas from the Cavity method provides a new insight to the sampling problem. Our results imply that there is a lot of potential for further exploiting the Cavity method for algorithmic design.

Charilaos Efthymiou. On Sampling Symmetric Gibbs Distributions on Sparse Random Graphs and Hypergraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{efthymiou:LIPIcs.ICALP.2022.57, author = {Efthymiou, Charilaos}, title = {{On Sampling Symmetric Gibbs Distributions on Sparse Random Graphs and Hypergraphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {57:1--57:16}, 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.2022.57}, URN = {urn:nbn:de:0030-drops-163982}, doi = {10.4230/LIPIcs.ICALP.2022.57}, annote = {Keywords: spin-system, spin-glass, sparse random (hyper)graph, approximate sampling, efficient algorithm} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Strong spatial mixing (SSM) is a form of correlation decay that has played an essential role in the design of approximate counting algorithms for spin systems. A notable example is the algorithm of Weitz (2006) for the hard-core model on weighted independent sets. We study SSM for the q-colorings problem on the infinite (d+1)-regular tree. Weak spatial mixing (WSM) captures whether the influence of the leaves on the root vanishes as the height of the tree grows. Jonasson (2002) established WSM when q>d+1. In contrast, in SSM, we first fix a coloring on a subset of internal vertices, and we again ask if the influence of the leaves on the root is vanishing. It was known that SSM holds on the (d+1)-regular tree when q>alpha d where alpha ~~ 1.763... is a constant that has arisen in a variety of results concerning random colorings. Here we improve on this bound by showing SSM for q>1.59d. Our proof establishes an L^2 contraction for the BP operator. For the contraction we bound the norm of the BP Jacobian by exploiting combinatorial properties of the coloring of the tree.

Charilaos Efthymiou, Andreas Galanis, Thomas P. Hayes, Daniel Štefankovič, and Eric Vigoda. Improved Strong Spatial Mixing for Colorings on Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{efthymiou_et_al:LIPIcs.APPROX-RANDOM.2019.48, author = {Efthymiou, Charilaos and Galanis, Andreas and Hayes, Thomas P. and \v{S}tefankovi\v{c}, Daniel and Vigoda, Eric}, title = {{Improved Strong Spatial Mixing for Colorings on Trees}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {48:1--48:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.48}, URN = {urn:nbn:de:0030-drops-112630}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.48}, annote = {Keywords: colorings, regular tree, spatial mixing, phase transitions} }

Document

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

Random graph models and associated inference problems such as the stochastic block model play an eminent role in computer science, discrete mathematics and statistics. Based on non-rigorous arguments physicists predicted the existence of a generic phase transition that separates a "replica symmetric phase" where statistical inference is impossible from a phase where the detection of the "ground truth" is information-theoretically possible.
In this paper we prove a contiguity result that shows that detectability is indeed impossible within the replica-symmetric phase for a broad class of models. In particular, this implies the detectability conjecture for the disassortative stochastic block model from [Decelle et al.: Phys Rev E 2011]. Additionally, we investigate key features of the replica symmetric phase such as the nature of point-to-set correlations (`reconstruction').

Amin Coja-Oghlan, Charilaos Efthymiou, Nor Jaafari, Mihyun Kang, and Tobias Kapetanopoulos. Charting the Replica Symmetric Phase. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cojaoghlan_et_al:LIPIcs.APPROX-RANDOM.2017.40, author = {Coja-Oghlan, Amin and Efthymiou, Charilaos and Jaafari, Nor and Kang, Mihyun and Kapetanopoulos, Tobias}, title = {{Charting the Replica Symmetric Phase}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {40:1--40:17}, 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.2017.40}, URN = {urn:nbn:de:0030-drops-75895}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.40}, annote = {Keywords: Random factor graph, bounds for condensation phase transition, Potts antiferromagnet, diluted k-spin model, stochastic block model} }

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

Let G=G(n,m) be a random graph whose average degree d=2m/n is below the k-colorability threshold. If we sample a k-coloring Sigma of G uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart? According to a prediction from statistical physics, for average degrees below the so-called condensation threshold d_c, the colors assigned to far away vertices are asymptotically independent [Krzakala et al: PNAS 2007]. We prove this conjecture for k exceeding a certain constant k_0. More generally, we determine the joint distribution of the k-colorings that Sigma induces locally on the bounded-depth neighborhoods of a fixed number of vertices.

Amin Coja-Oghlan, Charilaos Efthymiou, and Nor Jaafari. Local Convergence of Random Graph Colorings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 726-737, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cojaoghlan_et_al:LIPIcs.APPROX-RANDOM.2015.726, author = {Coja-Oghlan, Amin and Efthymiou, Charilaos and Jaafari, Nor}, title = {{Local Convergence of Random Graph Colorings}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {726--737}, 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.2015.726}, URN = {urn:nbn:de:0030-drops-53321}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.726}, annote = {Keywords: Random graph, Galton-Watson tree, phase transitions, graph coloring, Gibbs distribution, convergence} }

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

The broadcasting models on trees arise in many contexts such as discrete mathematics, biology, information theory, statistical physics and computer science. In this work, we consider the k-colouring model. A basic question here is whether the assignment at the root affects the distribution of the colourings at the vertices at distance h from the root. This is the so-called reconstruction problem. For the case where the underlying tree is d -ary it is well known that d/ln(d) is the reconstruction threshold. That is, for k=(1+epsilon)*d/ln(d) we have non-reconstruction while for k=(1-epsilon)*d/ln(d) we have reconstruction.
Here, we consider the largely unstudied case where the underlying tree is chosen according to a predefined distribution. In particular, we consider the well-known Galton-Watson trees. The corresponding model arises naturally in many contexts such as
the theory of spin-glasses and its applications on random Constraint Satisfaction Problems (rCSP). The study on rCSP focuses on Galton-Watson trees with offspring distribution B(n,d/n), i.e. the binomial with parameters n and d/n, where d is fixed. Here we consider a broader version of the problem, as we assume general offspring distribution which includes B(n,d/n) as a special case.
Our approach relates the corresponding bounds for (non)reconstruction to certain concentration properties of the offspring distribution. This allows to derive reconstruction thresholds for a very wide family of offspring distributions, which includes B(n,d/n). A very interesting corollary is that for distributions with expected offspring d, we get reconstruction threshold d/ln(d) under weaker concentration conditions than what we have in B(n,d/n).
Furthermore, our reconstruction threshold for the random colorings of Galton-Watson with offspring B(n,d/n), implies the reconstruction threshold for the random colourings of G(n,d/n).

Charilaos Efthymiou. Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 756-774, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{efthymiou:LIPIcs.APPROX-RANDOM.2015.756, author = {Efthymiou, Charilaos}, title = {{Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {756--774}, 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.2015.756}, URN = {urn:nbn:de:0030-drops-53346}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.756}, annote = {Keywords: Random Colouring, Reconstruction Problem, Galton-Watson Tree} }

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