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RANDOM

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

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments.

Amin Coja-Oghlan, Jane Gao, Max Hahn-Klimroth, Joon Lee, Noela Müller, and Maurice Rolvien. The Full Rank Condition for Sparse Random Matrices. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cojaoghlan_et_al:LIPIcs.APPROX/RANDOM.2023.54, author = {Coja-Oghlan, Amin and Gao, Jane and Hahn-Klimroth, Max and Lee, Joon and M\"{u}ller, Noela and Rolvien, Maurice}, title = {{The Full Rank Condition for Sparse Random Matrices}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {54:1--54:14}, 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.2023.54}, URN = {urn:nbn:de:0030-drops-188792}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.54}, annote = {Keywords: random matrices, rank, finite fields, rationals} }

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**Published in:** LIPIcs, Volume 253, 26th International Conference on Principles of Distributed Systems (OPODIS 2022)

We study the consensus problem among n agents, defined as follows. Initially, each agent holds one of two possible opinions. The goal is to reach a consensus configuration in which every agent shares the same opinion. To this end, agents randomly sample other agents and update their opinion according to a simple update function depending on the sampled opinions.
We consider two communication models: the gossip model and a variant of the population model. In the gossip model, agents are activated in parallel, synchronous rounds. In the population model, one agent is activated after the other in a sequence of discrete time steps. For both models we analyze the following natural family of majority processes called j-Majority: when activated, every agent samples j other agents uniformly at random (with replacement) and adopts the majority opinion among the sample (breaking ties uniformly at random). As our main result we show a hierarchy among majority protocols: (j+1)-Majority (for j > 1) converges stochastically faster than j-Majority for any initial opinion configuration. In our analysis we use Strassen’s Theorem to prove the existence of a coupling. This gives an affirmative answer for the case of two opinions to an open question asked by Berenbrink et al. [PODC 2017].

Petra Berenbrink, Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, Dominik Kaaser, and Malin Rau. On the Hierarchy of Distributed Majority Protocols. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.OPODIS.2022.23, author = {Berenbrink, Petra and Coja-Oghlan, Amin and Gebhard, Oliver and Hahn-Klimroth, Max and Kaaser, Dominik and Rau, Malin}, title = {{On the Hierarchy of Distributed Majority Protocols}}, booktitle = {26th International Conference on Principles of Distributed Systems (OPODIS 2022)}, pages = {23:1--23:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-265-5}, ISSN = {1868-8969}, year = {2023}, volume = {253}, editor = {Hillel, Eshcar and Palmieri, Roberto and Rivi\`{e}re, Etienne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2022.23}, URN = {urn:nbn:de:0030-drops-176434}, doi = {10.4230/LIPIcs.OPODIS.2022.23}, annote = {Keywords: Consensus, Majority, Hierarchy, Stochastic Dominance, Population Protocols, Gossip Model, Strassen’s Theorem} }

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**Published in:** LIPIcs, Volume 233, 20th International Symposium on Experimental Algorithms (SEA 2022)

The group testing problem asks for efficient pooling schemes and inference algorithms that allow to screen moderately large numbers of samples for rare infections. The goal is to accurately identify the infected individuals while minimizing the number of tests.
We propose the novel adaptive pooling scheme adaptive Belief Propagation (ABP) that acknowledges practical limitations such as limited pooling sizes and noisy tests that may give imperfect answers. We demonstrate that the accuracy of ABP surpasses that of individual testing despite using few overall tests. The new design comes with Belief Propagation as an efficient inference algorithm. While the development of ABP is guided by mathematical analyses and asymptotic insights, we conduct an experimental study to obtain results on practical population sizes.

Amin Coja-Oghlan, Max Hahn-Klimroth, Philipp Loick, and Manuel Penschuck. Efficient and Accurate Group Testing via Belief Propagation: An Empirical Study. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cojaoghlan_et_al:LIPIcs.SEA.2022.8, author = {Coja-Oghlan, Amin and Hahn-Klimroth, Max and Loick, Philipp and Penschuck, Manuel}, title = {{Efficient and Accurate Group Testing via Belief Propagation: An Empirical Study}}, booktitle = {20th International Symposium on Experimental Algorithms (SEA 2022)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-251-8}, ISSN = {1868-8969}, year = {2022}, volume = {233}, editor = {Schulz, Christian and U\c{c}ar, Bora}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2022.8}, URN = {urn:nbn:de:0030-drops-165422}, doi = {10.4230/LIPIcs.SEA.2022.8}, annote = {Keywords: Group testing, Probabilistic Construction, Belief Propagation, Simulation} }

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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)

We study the performance of Markov chains for the q-state ferromagnetic Potts model on random regular graphs. While the cases of the grid and the complete graph are by now well-understood, the case of random regular graphs has resisted a detailed analysis and, in fact, even analysing the properties of the Potts distribution has remained elusive. It is conjectured that the performance of Markov chains is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape, and therefore cause slow mixing. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these phases based on optimisation arguments fall short of the task.
Our first contribution is to detail the emergence of the metastable phases for the q-state Potts model on the d-regular random graph for all integers q,d ≥ 3, and establish that for an interval of temperatures, delineated by the uniqueness and a broadcasting threshold on the d-regular tree, the two phases coexist. The proofs are based on a conceptual connection between spatial properties and the structure of the Potts distribution on the random regular graph, rather than complicated moment calculations. This significantly refines earlier results by Helmuth, Jenssen, and Perkins who had established phase coexistence for a small interval around the so-called ordered-disordered threshold (via different arguments) that applied for large q and d ≥ 5.
Based on our new structural understanding of the model, we obtain various algorithmic consequences. We first complement recent fast mixing results for Glauber dynamics by Blanca and Gheissari below the uniqueness threshold, showing an exponential lower bound on the mixing time above the uniqueness threshold. Then, we obtain tight results even for the non-local and more elaborate Swendsen-Wang chain, where we establish slow mixing/metastability for the whole interval of temperatures where the chain is conjectured to mix slowly on the random regular graph. The key is to bound the conductance of the chains using a random graph "planting" argument combined with delicate bounds on random-graph percolation.

Amin Coja-Oghlan, Andreas Galanis, Leslie Ann Goldberg, Jean Bernoulli Ravelomanana, Daniel Štefankovič, and Eric Vigoda. Metastability of the Potts Ferromagnet on Random Regular Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 45:1-45:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cojaoghlan_et_al:LIPIcs.ICALP.2022.45, author = {Coja-Oghlan, Amin and Galanis, Andreas and Goldberg, Leslie Ann and Ravelomanana, Jean Bernoulli and \v{S}tefankovi\v{c}, Daniel and Vigoda, Eric}, title = {{Metastability of the Potts Ferromagnet on Random Regular Graphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {45:1--45:20}, 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.2022.45}, URN = {urn:nbn:de:0030-drops-163865}, doi = {10.4230/LIPIcs.ICALP.2022.45}, annote = {Keywords: Markov chains, sampling, random regular graph, Potts model} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Random factor graphs provide a powerful framework for the study of inference problems such as decoding problems or the stochastic block model. Information-theoretically the key quantity of interest is the mutual information between the observed factor graph and the underlying ground truth around which the factor graph was created; in the stochastic block model, this would be the planted partition. The mutual information gauges whether and how well the ground truth can be inferred from the observable data. For a very general model of random factor graphs we verify a formula for the mutual information predicted by physics techniques. As an application we prove a conjecture about low-density generator matrix codes from [Montanari: IEEE Transactions on Information Theory 2005]. Further applications include phase transitions of the stochastic block model and the mixed k-spin model from physics.

Amin Coja-Oghlan, Max Hahn-Klimroth, Philipp Loick, Noela Müller, Konstantinos Panagiotou, and Matija Pasch. Inference and Mutual Information on Random Factor Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cojaoghlan_et_al:LIPIcs.STACS.2021.24, author = {Coja-Oghlan, Amin and Hahn-Klimroth, Max and Loick, Philipp and M\"{u}ller, Noela and Panagiotou, Konstantinos and Pasch, Matija}, title = {{Inference and Mutual Information on Random Factor Graphs}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.24}, URN = {urn:nbn:de:0030-drops-136692}, doi = {10.4230/LIPIcs.STACS.2021.24}, annote = {Keywords: Information theory, random factor graphs, inference problems, phase transitions} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al. 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al. 2014, Johnson et al. 2019].

Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, and Philipp Loick. Information-Theoretic and Algorithmic Thresholds for Group Testing. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cojaoghlan_et_al:LIPIcs.ICALP.2019.43, author = {Coja-Oghlan, Amin and Gebhard, Oliver and Hahn-Klimroth, Max and Loick, Philipp}, title = {{Information-Theoretic and Algorithmic Thresholds for Group Testing}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {43:1--43:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.43}, URN = {urn:nbn:de:0030-drops-106196}, doi = {10.4230/LIPIcs.ICALP.2019.43}, annote = {Keywords: Group testing problem, phase transitions, information theory, efficient algorithms, sharp threshold, Bayesian inference} }

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 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Much of the recent work on phase transitions in discrete structures has been inspired by ingenious but non-rigorous approaches from physics. The physics predictions typically come in the form of distributional fixed point problems that mimic Belief Propagation, a message passing algorithm. In this paper we show how the Belief Propagation calculation can be turned into a rigorous proof of such a prediction, namely the existence and location of a condensation phase transition in the regular k-SAT model.

Victor Bapst and Amin Coja-Oghlan. The Condensation Phase Transition in the Regular k-SAT Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bapst_et_al:LIPIcs.APPROX-RANDOM.2016.22, author = {Bapst, Victor and Coja-Oghlan, Amin}, title = {{The Condensation Phase Transition in the Regular k-SAT Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.22}, URN = {urn:nbn:de:0030-drops-66452}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.22}, annote = {Keywords: random k-SAT, phase transitions, Belief Propagation, condensation} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

According to physics predictions, the free energy of random factor graph models that satisfy a certain "static replica symmetry" condition can be calculated via the Belief Propagation message passing scheme [Krzakala et al. PNAS, 2007]. Here we prove this conjecture for a wide class of random factor graph models. Specifically, we show that the messages constructed just as in the case of acyclic factor graphs asymptotically satisfy the Belief Propagation equations and that the free energy density is given by the Bethe free energy formula.

Amin Coja-Oghlan and Will Perkins. Belief Propagation on Replica Symmetric Random Factor Graph Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cojaoghlan_et_al:LIPIcs.APPROX-RANDOM.2016.27, author = {Coja-Oghlan, Amin and Perkins, Will}, title = {{Belief Propagation on Replica Symmetric Random Factor Graph Models}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {27:1--27:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.27}, URN = {urn:nbn:de:0030-drops-66500}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.27}, annote = {Keywords: Gibbs distributions, Belief Propagation, Bethe Free Energy, Random k-SAT} }

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

Gibbs measures induced by random factor graphs play a prominent role in computer science, combinatorics and physics. A key problem is to calculate the typical value of the partition function. According to the "replica symmetric cavity method", a heuristic that rests on non-rigorous considerations from statistical mechanics, in many cases this problem can be tackled by way of maximising a functional called the "Bethe free energy". In this paper we prove that the Bethe free energy upper-bounds the partition function in a broad class of models. Additionally, we provide a sufficient condition for this upper bound to be tight.

Victor Bapst and Amin Coja-Oghlan. Harnessing the Bethe Free Energy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 467-480, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bapst_et_al:LIPIcs.APPROX-RANDOM.2015.467, author = {Bapst, Victor and Coja-Oghlan, Amin}, title = {{Harnessing the Bethe Free Energy}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {467--480}, 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.2015.467}, URN = {urn:nbn:de:0030-drops-53180}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.467}, annote = {Keywords: Belief Propagation, free energy, Gibbs measure, partition function} }

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

In the planted bisection model a random graph G(n,p_+,p_-) with n vertices is created by partitioning the vertices randomly into two classes of equal size (up to plus or minus 1). Any two vertices that belong to the same class are linked by an edge with probability p_+ and any two that belong to different classes with probability (p_-) <(p_+) independently. The planted bisection model has been used extensively to benchmark graph partitioning algorithms. If (p_+)=2(d_+)/n and (p_-)=2(d_-)/n for numbers 0 <= (d_-) <(d_+) that remain fixed as n tends to infinity, then with high probability the "planted" bisection (the one used to construct the graph) will not be a minimum bisection. In this paper we derive an asymptotic formula for the minimum bisection width under the assumption that (d_+)-(d_-) > c * sqrt((d_+)ln(d_+)) for a certain constant c>0.

Amin Coja-Oghlan, Oliver Cooley, Mihyun Kang, and Kathrin Skubch. The Minimum Bisection in the Planted Bisection Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 710-725, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cojaoghlan_et_al:LIPIcs.APPROX-RANDOM.2015.710, author = {Coja-Oghlan, Amin and Cooley, Oliver and Kang, Mihyun and Skubch, Kathrin}, title = {{The Minimum Bisection in the Planted Bisection Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {710--725}, 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.2015.710}, URN = {urn:nbn:de:0030-drops-53315}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.710}, annote = {Keywords: Random graphs, minimum bisection, planted bisection, belief propagation.} }

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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 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random k-SAT or random graph k-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called condensation [Krzakala et al., PNAS 2007]. The existence of this phase transition appears to be intimately related to the difficulty of proving precise results on, e.g., the k-colorability threshold as well as to the performance of message passing algorithms. In random graph k-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k0.

Victor Bapst, Amin Coja-Oghlan, Samuel Hetterich, Felicia Raßmann, and Dan Vilenchik. The Condensation Phase Transition in Random Graph Coloring. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 449-464, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bapst_et_al:LIPIcs.APPROX-RANDOM.2014.449, author = {Bapst, Victor and Coja-Oghlan, Amin and Hetterich, Samuel and Ra{\ss}mann, Felicia and Vilenchik, Dan}, title = {{The Condensation Phase Transition in Random Graph Coloring}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {449--464}, 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.2014.449}, URN = {urn:nbn:de:0030-drops-47168}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.449}, annote = {Keywords: random graphs, graph coloring, phase transitions, message-passing algorithm} }

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