19 Search Results for "Vigoda, Eric"


Document
Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm

Authors: Sam Olesker-Taylor

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)


Abstract
In the hardcore model, certain vertices in a graph are active: the active vertices must form an independent set. We extend this to a multicoloured version: instead of simply being active or not, the active vertices are assigned a colour; active vertices of the same colour must not be adjacent. This models a scenario in which two neighbouring resources may interfere when active - eg, short-range radio communication. However, there are multiple channels (colours) available; they only interfere if both use the same channel. Other applications include routing in fibreoptic networks. We analyse Glauber dynamics. Vertices update their status at random times, at which a uniform colour is proposed: the vertex is assigned that colour if it is available; otherwise, it is set inactive. We find conditions for fast mixing of these dynamics. We also use them to model a queueing system: vertices only serve customers in their queue whilst active. The mixing estimates are applied to establish positive recurrence of the queue lengths, and bound their expectation in equilibrium.

Cite as

Sam Olesker-Taylor. Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{oleskertaylor:LIPIcs.AofA.2024.20,
  author =	{Olesker-Taylor, Sam},
  title =	{{Multicoloured Hardcore Model: Fast Mixing and Its Applications as a Scheduling Algorithm}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.20},
  URN =		{urn:nbn:de:0030-drops-204558},
  doi =		{10.4230/LIPIcs.AofA.2024.20},
  annote =	{Keywords: mixing time, queueing theory, hardcore model, proper colourings, independent set, data transmission, randomised algorithms, routing, scheduling, multihop wireless networks}
}
Document
Track A: Algorithms, Complexity and Games
Approximate Counting for Spin Systems in Sub-Quadratic Time

Authors: Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, and Jiaheng Wang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
We present two randomised approximate counting algorithms with Õ(n^{2-c}/ε²) running time for some constant c > 0 and accuracy ε: 1) for the hard-core model with fugacity λ on graphs with maximum degree Δ when λ = O(Δ^{-1.5-c₁}) where c₁ = c/(2-2c); 2) for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as ℤ². For the hard-core model, Weitz’s algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when λ = o(Δ^{-2}). Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist. Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as ℤ^d, but with a running time of the form Õ(n²ε^{-2}/2^{c(log n)^{1/d}}) where d is the exponent of the polynomial growth and c > 0 is some constant.

Cite as

Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, and Jiaheng Wang. Approximate Counting for Spin Systems in Sub-Quadratic Time. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{anand_et_al:LIPIcs.ICALP.2024.11,
  author =	{Anand, Konrad and Feng, Weiming and Freifeld, Graham and Guo, Heng and Wang, Jiaheng},
  title =	{{Approximate Counting for Spin Systems in Sub-Quadratic Time}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.11},
  URN =		{urn:nbn:de:0030-drops-201543},
  doi =		{10.4230/LIPIcs.ICALP.2024.11},
  annote =	{Keywords: Randomised algorithm, Approximate counting, Spin system, Sub-quadratic algorithm}
}
Document
Track A: Algorithms, Complexity and Games
A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs

Authors: Charlie Carlson, Ewan Davies, Alexandra Kolla, and Aditya Potukuchi

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to "high-temperature" problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e., bounded threshold rank), which, via subspace enumeration, lets us enumerate small cuts efficiently.

Cite as

Charlie Carlson, Ewan Davies, Alexandra Kolla, and Aditya Potukuchi. A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{carlson_et_al:LIPIcs.ICALP.2024.35,
  author =	{Carlson, Charlie and Davies, Ewan and Kolla, Alexandra and Potukuchi, Aditya},
  title =	{{A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{35:1--35:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.35},
  URN =		{urn:nbn:de:0030-drops-201782},
  doi =		{10.4230/LIPIcs.ICALP.2024.35},
  annote =	{Keywords: approximate counting, independent sets, bipartite graphs, graph containers}
}
Document
Track A: Algorithms, Complexity and Games
Problems on Group-Labeled Matroid Bases

Authors: Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, and Tamás Schwarcz

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
Consider a matroid equipped with a labeling of its ground set to an abelian group. We define the label of a subset of the ground set as the sum of the labels of its elements. We study a collection of problems on finding bases and common bases of matroids with restrictions on their labels. For zero bases and zero common bases, the results are mostly negative. While finding a non-zero basis of a matroid is not difficult, it turns out that the complexity of finding a non-zero common basis depends on the group. Namely, we show that the problem is hard for a fixed group if it contains an element of order two, otherwise it is polynomially solvable. As a generalization of both zero and non-zero constraints, we further study F-avoiding constraints where we seek a basis or common basis whose label is not in a given set F of forbidden labels. Using algebraic techniques, we give a randomized algorithm for finding an F-avoiding common basis of two matroids represented over the same field for finite groups given as operation tables. The study of F-avoiding bases with groups given as oracles leads to a conjecture stating that whenever an F-avoiding basis exists, an F-avoiding basis can be obtained from an arbitrary basis by exchanging at most |F| elements. We prove the conjecture for the special cases when |F| ≤ 2 or the group is ordered. By relying on structural observations on matroids representable over fixed, finite fields, we verify a relaxed version of the conjecture for these matroids. As a consequence, we obtain a polynomial-time algorithm in these special cases for finding an F-avoiding basis when |F| is fixed.

Cite as

Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, and Tamás Schwarcz. Problems on Group-Labeled Matroid Bases. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 86:1-86:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{horsch_et_al:LIPIcs.ICALP.2024.86,
  author =	{H\"{o}rsch, Florian and Imolay, Andr\'{a}s and Mizutani, Ryuhei and Oki, Taihei and Schwarcz, Tam\'{a}s},
  title =	{{Problems on Group-Labeled Matroid Bases}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{86:1--86:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.86},
  URN =		{urn:nbn:de:0030-drops-202299},
  doi =		{10.4230/LIPIcs.ICALP.2024.86},
  annote =	{Keywords: matroids, matroid intersection, congruency constraint, exact-weight constraint, additive combinatorics, algebraic algorithm, strongly base orderability}
}
Document
Improved Distributed Algorithms for Random Colorings

Authors: Charlie Carlson, Daniel Frishberg, and Eric Vigoda

Published in: LIPIcs, Volume 286, 27th International Conference on Principles of Distributed Systems (OPODIS 2023)


Abstract
Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph G of maximum degree Δ and an integer k ≥ Δ+1, the goal is to generate a random proper vertex k-coloring of G. Jerrum (1995) proved that the Glauber dynamics has O(nlog{n}) mixing time when k > 2Δ. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in O(log{n}) rounds for k > (2+ε)Δ for any ε > 0. We improve this result to k > (11/6-δ)Δ for a fixed δ > 0. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.

Cite as

Charlie Carlson, Daniel Frishberg, and Eric Vigoda. Improved Distributed Algorithms for Random Colorings. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{carlson_et_al:LIPIcs.OPODIS.2023.13,
  author =	{Carlson, Charlie and Frishberg, Daniel and Vigoda, Eric},
  title =	{{Improved Distributed Algorithms for Random Colorings}},
  booktitle =	{27th International Conference on Principles of Distributed Systems (OPODIS 2023)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-308-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{286},
  editor =	{Bessani, Alysson and D\'{e}fago, Xavier and Nakamura, Junya and Wada, Koichi and Yamauchi, Yukiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2023.13},
  URN =		{urn:nbn:de:0030-drops-195030},
  doi =		{10.4230/LIPIcs.OPODIS.2023.13},
  annote =	{Keywords: Distributed Graph Algorithms, Local Algorithms, Coloring, Glauber Dynamics, Sampling, Markov Chains}
}
Document
Rapid Mixing for the Hardcore Glauber Dynamics and Other Markov Chains in Bounded-Treewidth Graphs

Authors: David Eppstein and Daniel Frishberg

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)


Abstract
We give a new rapid mixing result for a natural random walk on the independent sets of a graph G. We show that when G has bounded treewidth, this random walk - known as the Glauber dynamics for the hardcore model - mixes rapidly for all fixed values of the standard parameter λ > 0, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for analogous Markov chains on dominating sets, b-edge covers, b-matchings, maximal independent sets, and maximal b-matchings. (For b-matchings, maximal independent sets, and maximal b-matchings we also require bounded degree.) Our results imply simpler alternatives to known algorithms for the sampling and approximate counting problems in these graphs. We prove our results by applying a divide-and-conquer framework we developed in a previous paper, as an alternative to the projection-restriction technique introduced by Jerrum, Son, Tetali, and Vigoda. We extend this prior framework to handle chains for which the application of that framework is not straightforward, strengthening existing results by Dyer, Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on q-colorings of graphs of bounded treewidth and bounded degree.

Cite as

David Eppstein and Daniel Frishberg. Rapid Mixing for the Hardcore Glauber Dynamics and Other Markov Chains in Bounded-Treewidth Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{eppstein_et_al:LIPIcs.ISAAC.2023.30,
  author =	{Eppstein, David and Frishberg, Daniel},
  title =	{{Rapid Mixing for the Hardcore Glauber Dynamics and Other Markov Chains in Bounded-Treewidth Graphs}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{30:1--30:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.30},
  URN =		{urn:nbn:de:0030-drops-193324},
  doi =		{10.4230/LIPIcs.ISAAC.2023.30},
  annote =	{Keywords: Glauber dynamics, mixing time, projection-restriction, multicommodity flow}
}
Document
RANDOM
Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees

Authors: Charilaos Efthymiou, Thomas P. Hayes, Daniel Štefankovič, and Eric Vigoda

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


Abstract
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.

Cite as

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}
}
Document
Counting and Sampling Labeled Chordal Graphs in Polynomial Time

Authors: Úrsula Hébert-Johnson, Daniel Lokshtanov, and Eric Vigoda

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
We present the first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on n vertices. Our algorithm solves a more general problem: given n and ω as input, it computes the number of ω-colorable labeled chordal graphs on n vertices, using O(n⁷) arithmetic operations. A standard sampling-to-counting reduction then yields a polynomial-time exact sampler that generates an ω-colorable labeled chordal graph on n vertices uniformly at random. Our counting algorithm improves upon the previous best result by Wormald (1985), which computes the number of labeled chordal graphs on n vertices in time exponential in n. An implementation of the polynomial-time counting algorithm gives the number of labeled chordal graphs on up to 30 vertices in less than three minutes on a standard desktop computer. Previously, the number of labeled chordal graphs was only known for graphs on up to 15 vertices.

Cite as

Úrsula Hébert-Johnson, Daniel Lokshtanov, and Eric Vigoda. Counting and Sampling Labeled Chordal Graphs in Polynomial Time. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{hebertjohnson_et_al:LIPIcs.ESA.2023.58,
  author =	{H\'{e}bert-Johnson, \'{U}rsula and Lokshtanov, Daniel and Vigoda, Eric},
  title =	{{Counting and Sampling Labeled Chordal Graphs in Polynomial Time}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{58:1--58:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.58},
  URN =		{urn:nbn:de:0030-drops-187119},
  doi =		{10.4230/LIPIcs.ESA.2023.58},
  annote =	{Keywords: Counting algorithms, graph sampling, chordal graphs}
}
Document
Track A: Algorithms, Complexity and Games
Metastability of the Potts Ferromagnet on Random Regular Graphs

Authors: Amin Coja-Oghlan, Andreas Galanis, Leslie Ann Goldberg, Jean Bernoulli Ravelomanana, Daniel Štefankovič, and Eric Vigoda

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


Abstract
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.

Cite as

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}
}
Document
Track A: Algorithms, Complexity and Games
Approximating Observables Is as Hard as Counting

Authors: Andreas Galanis, Daniel Štefankovič, and Eric Vigoda

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


Abstract
We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. A recent work of Galanis et al (2021) established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. We instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results.

Cite as

Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Approximating Observables Is as Hard as Counting. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 63:1-63:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{galanis_et_al:LIPIcs.ICALP.2022.63,
  author =	{Galanis, Andreas and \v{S}tefankovi\v{c}, Daniel and Vigoda, Eric},
  title =	{{Approximating Observables Is as Hard as Counting}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{63:1--63:18},
  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.2022.63},
  URN =		{urn:nbn:de:0030-drops-164047},
  doi =		{10.4230/LIPIcs.ICALP.2022.63},
  annote =	{Keywords: Approximate Counting, Averages, Phase Transitions, Random Structures}
}
Document
RANDOM
The Swendsen-Wang Dynamics on Trees

Authors: Antonio Blanca, Zongchen Chen, Daniel Štefankovič, and Eric Vigoda

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)


Abstract
The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree. Our bounds extend to the non-uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli et al. (2003), imply Ω(1) spectral gap and O(log n) mixing time, respectively, for the Swendsen-Wang dynamics on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish Θ(log n) mixing for the Swendsen-Wang dynamics for all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model, and for the q-state Potts model when q is small with respect to d. Our proofs feature a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders and utilize recent work on block factorization of entropy under spatial mixing conditions.

Cite as

Antonio Blanca, Zongchen Chen, Daniel Štefankovič, and Eric Vigoda. The Swendsen-Wang Dynamics on Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2021.43,
  author =	{Blanca, Antonio and Chen, Zongchen and \v{S}tefankovi\v{c}, Daniel and Vigoda, Eric},
  title =	{{The Swendsen-Wang Dynamics on Trees}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{43:1--43:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.43},
  URN =		{urn:nbn:de:0030-drops-147366},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.43},
  annote =	{Keywords: Markov Chains, mixing times, Ising and Potts models, Swendsen-Wang dynamics, trees}
}
Document
RANDOM
Fast Algorithms at Low Temperatures via Markov Chains

Authors: Zongchen Chen, Andreas Galanis, Leslie Ann Goldberg, Will Perkins, James Stewart, and Eric Vigoda

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


Abstract
For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time O(n log n) for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^{O(log Delta)} for algorithms based on Barvinok’s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces.

Cite as

Zongchen Chen, Andreas Galanis, Leslie Ann Goldberg, Will Perkins, James Stewart, and Eric Vigoda. Fast Algorithms at Low Temperatures via Markov Chains. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2019.41,
  author =	{Chen, Zongchen and Galanis, Andreas and Goldberg, Leslie Ann and Perkins, Will and Stewart, James and Vigoda, Eric},
  title =	{{Fast Algorithms at Low Temperatures via Markov Chains}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{41:1--41:14},
  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.2019.41},
  URN =		{urn:nbn:de:0030-drops-112560},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.41},
  annote =	{Keywords: Markov chains, approximate counting, Potts model, hard-core model, expander graphs}
}
Document
RANDOM
Improved Strong Spatial Mixing for Colorings on Trees

Authors: Charilaos Efthymiou, Andreas Galanis, Thomas P. Hayes, Daniel Štefankovič, and Eric Vigoda

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


Abstract
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.

Cite as

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
RANDOM
Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions

Authors: Antonio Blanca, Reza Gheissari, and Eric Vigoda

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


Abstract
The random-cluster (FK) model is a key tool for the study of phase transitions and for the design of efficient Markov chain Monte Carlo (MCMC) sampling algorithms for the Ising/Potts model. It is well-known that in the high-temperature region beta<beta_c(q) of the q-state Ising/Potts model on an n x n box Lambda_n of the integer lattice Z^2, spin correlations decay exponentially fast; this property holds even arbitrarily close to the boundary of Lambda_n and uniformly over all boundary conditions. A direct consequence of this property is that the corresponding single-site update Markov chain, known as the Glauber dynamics, mixes in optimal O(n^2 log{n}) steps on Lambda_{n} for all choices of boundary conditions. We study the effect of boundary conditions on the FK-dynamics, the analogous Glauber dynamics for the random-cluster model. On Lambda_n the random-cluster model with parameters (p,q) has a sharp phase transition at p = p_c(q). Unlike the Ising/Potts model, the random-cluster model has non-local interactions which can be forced by boundary conditions: external wirings of boundary vertices of Lambda_n. We consider the broad and natural class of boundary conditions that are realizable as a configuration on Z^2 \ Lambda_n. Such boundary conditions can have many macroscopic wirings and impose long-range correlations even at very high temperatures (p << p_c(q)). In this paper, we prove that when q>1 and p != p_c(q) the mixing time of the FK-dynamics is polynomial in n for every realizable boundary condition. Previously, for boundary conditions that do not carry long-range information (namely wired and free), Blanca and Sinclair (2017) had proved that the FK-dynamics in the same setting mixes in optimal O(n^2 log n) time. To illustrate the difficulties introduced by general boundary conditions, we also construct a class of non-realizable boundary conditions that induce slow (stretched-exponential) convergence at high temperatures.

Cite as

Antonio Blanca, Reza Gheissari, and Eric Vigoda. Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2019.67,
  author =	{Blanca, Antonio and Gheissari, Reza and Vigoda, Eric},
  title =	{{Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{67:1--67:19},
  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.2019.67},
  URN =		{urn:nbn:de:0030-drops-112827},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.67},
  annote =	{Keywords: Markov chain, mixing time, random-cluster model, Glauber dynamics, spatial mixing}
}
Document
Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region

Authors: Antonio Blanca, Zongchen Chen, and Eric Vigoda

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)


Abstract
The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G=(V,E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V|^{1/4}) steps for any graph G at any (inverse) temperature beta. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V|) upper bounds on its convergence time. We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when beta < beta_c(d) where beta_c(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is Theta(1) on any graph of maximum degree d >= 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log{|V|}) and relaxation time Theta(1) on any graph of maximum degree d for all beta < beta_c(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.

Cite as

Antonio Blanca, Zongchen Chen, and Eric Vigoda. Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2018.32,
  author =	{Blanca, Antonio and Chen, Zongchen and Vigoda, Eric},
  title =	{{Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.32},
  URN =		{urn:nbn:de:0030-drops-94365},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.32},
  annote =	{Keywords: Swendsen-Wang dynamics, mixing time, relaxation time, spatial mixing, censoring}
}
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