15 Search Results for "Chen, Zongchen"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.48
Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries

Authors: Tomer Adar, Eldar Fischer, and Amit Levi

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

Abstract
We study property testing in the subcube conditional model introduced by Bhattacharyya and Chakraborty (2017). We obtain the first equivalence test for n-dimensional distributions that is quasi-linear in n, improving the previously known Õ(n²/ε²) query complexity bound to Õ(n/ε²). We extend this result to general finite alphabets with logarithmic cost in the alphabet size. By exploiting the specific structure of the queries that we use (which are more restrictive than general subcube queries), we obtain a cubic improvement over the best known test for distributions over {1,…,N} under the interval querying model of Canonne, Ron and Servedio (2015), attaining a query complexity of Õ((log N)/ε²), which for fixed ε almost matches the known lower bound of Ω((log N)/log log N). We also derive a product test for n-dimensional distributions with Õ(n/ε²) queries, and provide an Ω(√n/ε²) lower bound for this property.
Cite as

Tomer Adar, Eldar Fischer, and Amit Levi. Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 48:1-48:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{adar_et_al:LIPIcs.APPROX/RANDOM.2024.48,
  author =	{Adar, Tomer and Fischer, Eldar and Levi, Amit},
  title =	{{Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{48:1--48:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.48},
  URN =		{urn:nbn:de:0030-drops-210418},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.48},
  annote =	{Keywords: Distribution testing, conditional sampling, sub-cube sampling}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.49
Parallelising Glauber Dynamics

Authors: Holden Lee

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

Abstract
For distributions over discrete product spaces ∏_{i=1}^n Ω_i', Glauber dynamics is a Markov chain that at each step, resamples a random coordinate conditioned on the other coordinates. We show that k-Glauber dynamics, which resamples a random subset of k coordinates, mixes k times faster in χ²-divergence, and assuming approximate tensorization of entropy, mixes k times faster in KL-divergence. We apply this to obtain parallel algorithms in two settings: (1) For the Ising model μ_{J,h}(x) ∝ exp(1/2 ⟨x,Jx⟩ + ⟨h,x⟩) with ‖J‖ < 1-c (the regime where fast mixing is known), we show that we can implement each step of Θ(n/‖J‖_F)-Glauber dynamics efficiently with a parallel algorithm, resulting in a parallel algorithm with running time Õ(‖J‖_F) = Õ(√n). (2) For the mixed p-spin model at high enough temperature, we show that with high probability we can implement each step of Θ(√n)-Glauber dynamics efficiently and obtain running time Õ(√n).
Cite as

Holden Lee. Parallelising Glauber Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lee:LIPIcs.APPROX/RANDOM.2024.49,
  author =	{Lee, Holden},
  title =	{{Parallelising Glauber Dynamics}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{49:1--49:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.49},
  URN =		{urn:nbn:de:0030-drops-210424},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.49},
  annote =	{Keywords: sampling, Ising model, parallel algorithm, Markov chain, Glauber dynamics}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.56
Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs

Authors: Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap

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

Abstract
We study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree Δ. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree Δ graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs.
Cite as

Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap. Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 56:1-56:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kuchukova_et_al:LIPIcs.APPROX/RANDOM.2024.56,
  author =	{Kuchukova, Aiya and Pappik, Marcus and Perkins, Will and Yap, Corrine},
  title =	{{Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{56:1--56:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.56},
  URN =		{urn:nbn:de:0030-drops-210493},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.56},
  annote =	{Keywords: ferromagnetic Ising model, fixed-magnetization Ising model, Kawasaki dynamics, Glauber dynamics, mixing time}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.58
Expanderizing Higher Order Random Walks

Authors: Vedat Levi Alev and Shravas Rao

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

Abstract
We study a variant of the down-up (also known as the Glauber dynamics) and up-down walks over an n-partite simplicial complex, which we call expanderized higher order random walks - where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph H. When H is the clique with self loops on [n], this random walk reduces to the usual down-up walk and when H is the directed cycle on [n], this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph H. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan (RANDOM 2005). We study the mixing times of our expanderized walks in two example cases: We show that when initiated with an expander graph our expanderized random walks have mixing time (i) O(n log n) for sampling a uniformly random list colorings of a graph G of maximum degree Δ = O(1) where each vertex has at least (11/6 - ε) Δ and at most O(Δ) colors, (ii) O_h((n log n)/(1 - ‖J‖_op)²) for sampling the Ising model with a PSD interaction matrix J ∈ ℝ^{n×n} satisfying ‖J‖_op ≤ 1 and the external field h ∈ ℝⁿ- here the O(•) notation hides a constant that depends linearly on the largest entry of h. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor. We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global Φ-entropy contraction in simplicial complexes - giving simpler proofs for many pre-existing results.
Cite as

Vedat Levi Alev and Shravas Rao. Expanderizing Higher Order Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 58:1-58:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alev_et_al:LIPIcs.APPROX/RANDOM.2024.58,
  author =	{Alev, Vedat Levi and Rao, Shravas},
  title =	{{Expanderizing Higher Order Random Walks}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{58:1--58:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.58},
  URN =		{urn:nbn:de:0030-drops-210510},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.58},
  annote =	{Keywords: Higher Order Random Walks, Expander Graphs, Glauber Dynamics, Derandomized Squaring, High Dimensional Expansion, Spectral Independence, Entropic Independence}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.63
Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size

Authors: Vishesh Jain and Clayton Mizgerd

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

Abstract
Let G = (V,E) be a graph on n vertices and let m^*(G) denote the size of a maximum matching in G. We show that for any δ > 0 and for any 1 ≤ k ≤ (1-δ)m^*(G), the down-up walk on matchings of size k in G mixes in time polynomial in n. Previously, polynomial mixing was not known even for graphs with maximum degree Δ, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time O_{Δ,δ}(nlog{n}). In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting.
Cite as

Vishesh Jain and Clayton Mizgerd. Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jain_et_al:LIPIcs.APPROX/RANDOM.2024.63,
  author =	{Jain, Vishesh and Mizgerd, Clayton},
  title =	{{Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{63:1--63:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.63},
  URN =		{urn:nbn:de:0030-drops-210563},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.63},
  annote =	{Keywords: Down-up walk, Matchings, MCMC}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.70
On Sampling from Ising Models with Spectral Constraints

Authors: Andreas Galanis, Alkis Kalavasis, and Anthimos Vardis Kandiros

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

Abstract
We consider the problem of sampling from the Ising model when the underlying interaction matrix has eigenvalues lying within an interval of length γ. Recent work in this setting has shown various algorithmic results that apply roughly when γ < 1, notably with nearly-linear running times based on the classical Glauber dynamics. However, the optimality of the range of γ was not clear since previous inapproximability results developed for the antiferromagnetic case (where the matrix has entries ≤ 0) apply only for γ > 2. To this end, Kunisky (SODA'24) recently provided evidence that the problem becomes hard already when γ > 1 based on the low-degree hardness for an inference problem on random matrices. Based on this, he conjectured that sampling from the Ising model in the same range of γ is NP-hard. Here we confirm this conjecture, complementing in particular the known algorithmic results by showing NP-hardness results for approximately counting and sampling when γ > 1, with strong inapproximability guarantees; we also obtain a more refined hardness result for matrices where only a constant number of entries per row are allowed to be non-zero. The main observation in our reductions is that, for γ > 1, Glauber dynamics mixes slowly when the interactions are all positive (ferromagnetic) for the complete and random regular graphs, due to a bimodality in the underlying distribution. While ferromagnetic interactions typically preclude NP-hardness results, here we work around this by introducing in an appropriate way mild antiferromagnetism, keeping the spectrum roughly within the same range. This allows us to exploit the bimodality of the aforementioned graphs and show the target NP-hardness by adapting suitably previous inapproximability techniques developed for antiferromagnetic systems.
Cite as

Andreas Galanis, Alkis Kalavasis, and Anthimos Vardis Kandiros. On Sampling from Ising Models with Spectral Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{galanis_et_al:LIPIcs.APPROX/RANDOM.2024.70,
  author =	{Galanis, Andreas and Kalavasis, Alkis and Kandiros, Anthimos Vardis},
  title =	{{On Sampling from Ising Models with Spectral Constraints}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{70:1--70:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.70},
  URN =		{urn:nbn:de:0030-drops-210638},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.70},
  annote =	{Keywords: Ising model, spectral constraints, Glauber dynamics, mean-field Ising, random regular graphs}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.35
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
DOI: 10.4230/LIPIcs.ICALP.2024.62
An FPRAS for Two Terminal Reliability in Directed Acyclic Graphs

Authors: Weiming Feng and Heng Guo

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

Abstract
We give a fully polynomial-time randomized approximation scheme (FPRAS) for two terminal reliability in directed acyclic graphs (DAGs). In contrast, we also show the complementing problem of approximating two terminal unreliability in DAGs is #BIS-hard.
Cite as

Weiming Feng and Heng Guo. An FPRAS for Two Terminal Reliability in Directed Acyclic Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{feng_et_al:LIPIcs.ICALP.2024.62,
  author =	{Feng, Weiming and Guo, Heng},
  title =	{{An FPRAS for Two Terminal Reliability in Directed Acyclic Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{62:1--62:19},
  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.2024.62},
  URN =		{urn:nbn:de:0030-drops-202057},
  doi =		{10.4230/LIPIcs.ICALP.2024.62},
  annote =	{Keywords: Approximate counting, Network reliability, Sampling algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.11
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
DOI: 10.4230/LIPIcs.ITCS.2024.30
Influence Maximization in Ising Models

Authors: Zongchen Chen and Elchanan Mossel

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

Abstract
Given a complex high-dimensional distribution over {± 1}ⁿ, what is the best way to increase the expected number of +1’s by controlling the values of only a small number of variables? Such a problem is known as influence maximization and has been widely studied in social networks, biology, and computer science. In this paper, we consider influence maximization on the Ising model which is a prototypical example of undirected graphical models and has wide applications in many real-world problems. We establish a sharp computational phase transition for influence maximization on sparse Ising models under a bounded budget: In the high-temperature regime, we give a linear-time algorithm for finding a small subset of variables and their values which achieve nearly optimal influence; In the low-temperature regime, we show that the influence maximization problem cannot be solved in polynomial time under commonly-believed complexity assumption. The critical temperature coincides with the tree uniqueness/non-uniqueness threshold for Ising models which is also a critical point for other computational problems including approximate sampling and counting.
Cite as

Zongchen Chen and Elchanan Mossel. Influence Maximization in Ising Models. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2024.30,
  author =	{Chen, Zongchen and Mossel, Elchanan},
  title =	{{Influence Maximization in Ising Models}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{30:1--30:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.30},
  URN =		{urn:nbn:de:0030-drops-195588},
  doi =		{10.4230/LIPIcs.ITCS.2024.30},
  annote =	{Keywords: Influence maximization, Ising model, phase transition, correlation decay}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.30
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
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.43
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
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.41
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
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.64
Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions

Authors: Zongchen Chen and Santosh S. Vempala

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

Abstract
We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to e^{-f} where f:R^d -> R is mu-strongly convex and L-smooth (the condition number is kappa = L/mu). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is O(kappa), improving on the previous best bound of O(kappa^{1.5}); we complement this with an example where the relaxation time is Omega(kappa). When implemented using a nearly optimal ODE solver, HMC returns an epsilon-approximate point in 2-Wasserstein distance using O~((kappa d)^{0.5} epsilon^{-1}) gradient evaluations per step and O~((kappa d)^{1.5}epsilon^{-1}) total time.
Cite as

Zongchen Chen and Santosh S. Vempala. Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 64:1-64:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2019.64,
  author =	{Chen, Zongchen and Vempala, Santosh S.},
  title =	{{Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{64:1--64:12},
  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.2019.64},
  URN =		{urn:nbn:de:0030-drops-112790},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.64},
  annote =	{Keywords: logconcave distribution, sampling, Hamiltonian Monte Carlo, spectral gap, strong convexity}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.32
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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