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Documents authored by Wein, Alexander S.

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RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.47
Time Lower Bounds for the Metropolis Process and Simulated Annealing

Authors: Zongchen Chen, Dan Mikulincer, Daniel Reichman, and Alexander S. Wein

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
The Metropolis process (MP) and Simulated Annealing (SA) are stochastic local search heuristics that are often used in solving combinatorial optimization problems. Despite significant interest, there are very few theoretical results regarding the quality of approximation obtained by MP and SA (with polynomially many iterations) for NP-hard optimization problems. We provide rigorous lower bounds for MP and SA with respect to the classical maximum independent set problem when the algorithms are initialized from the empty set. We establish the existence of a family of graphs for which both MP and SA fail to find approximate solutions in polynomial time. More specifically, we show that for any ε ∈ (0,1) there are n-vertex graphs for which the probability SA (when limited to polynomially many iterations) will approximate the optimal solution within ratio Ω(1/n^{1-ε}) is exponentially small. Our lower bounds extend to graphs of constant average degree d, illustrating the failure of MP to achieve an approximation ratio of Ω(log(d)/d) in polynomial time. In some cases, our lower bounds apply even when the temperature is chosen adaptively. Finally, we prove exponential-time lower bounds when the inputs to these algorithms are bipartite graphs, and even trees, which are known to admit polynomial-time algorithms for the independent set problem.
Cite as

Zongchen Chen, Dan Mikulincer, Daniel Reichman, and Alexander S. Wein. Time Lower Bounds for the Metropolis Process and Simulated Annealing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 47:1-47:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.47,
  author =	{Chen, Zongchen and Mikulincer, Dan and Reichman, Daniel and Wein, Alexander S.},
  title =	{{Time Lower Bounds for the Metropolis Process and Simulated Annealing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{47:1--47:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.47},
  URN =		{urn:nbn:de:0030-drops-244138},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.47},
  annote =	{Keywords: Metropolis Process, Simulated Annealing, Independent Set}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.94
Is It Easier to Count Communities Than Find Them?

Authors: Cynthia Rush, Fiona Skerman, Alexander S. Wein, and Dana Yang

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
Random graph models with community structure have been studied extensively in the literature. For both the problems of detecting and recovering community structure, an interesting landscape of statistical and computational phase transitions has emerged. A natural unanswered question is: might it be possible to infer properties of the community structure (for instance, the number and sizes of communities) even in situations where actually finding those communities is believed to be computationally hard? We show the answer is no. In particular, we consider certain hypothesis testing problems between models with different community structures, and we show (in the low-degree polynomial framework) that testing between two options is as hard as finding the communities. In addition, our methods give the first computational lower bounds for testing between two different "planted" distributions, whereas previous results have considered testing between a planted distribution and an i.i.d. "null" distribution.
Cite as

Cynthia Rush, Fiona Skerman, Alexander S. Wein, and Dana Yang. Is It Easier to Count Communities Than Find Them?. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 94:1-94:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{rush_et_al:LIPIcs.ITCS.2023.94,
  author =	{Rush, Cynthia and Skerman, Fiona and Wein, Alexander S. and Yang, Dana},
  title =	{{Is It Easier to Count Communities Than Find Them?}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{94:1--94:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.94},
  URN =		{urn:nbn:de:0030-drops-175970},
  doi =		{10.4230/LIPIcs.ITCS.2023.94},
  annote =	{Keywords: Community detection, Hypothesis testing, Low-degree polynomials}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.75
Counterexamples to the Low-Degree Conjecture

Authors: Justin Holmgren and Alexander S. Wein

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
A conjecture of Hopkins (2018) posits that for certain high-dimensional hypothesis testing problems, no polynomial-time algorithm can outperform so-called "simple statistics", which are low-degree polynomials in the data. This conjecture formalizes the beliefs surrounding a line of recent work that seeks to understand statistical-versus-computational tradeoffs via the low-degree likelihood ratio. In this work, we refute the conjecture of Hopkins. However, our counterexample crucially exploits the specifics of the noise operator used in the conjecture, and we point out a simple way to modify the conjecture to rule out our counterexample. We also give an example illustrating that (even after the above modification), the symmetry assumption in the conjecture is necessary. These results do not undermine the low-degree framework for computational lower bounds, but rather aim to better understand what class of problems it is applicable to.
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Justin Holmgren and Alexander S. Wein. Counterexamples to the Low-Degree Conjecture. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 75:1-75:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{holmgren_et_al:LIPIcs.ITCS.2021.75,
  author =	{Holmgren, Justin and Wein, Alexander S.},
  title =	{{Counterexamples to the Low-Degree Conjecture}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{75:1--75:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.75},
  URN =		{urn:nbn:de:0030-drops-136148},
  doi =		{10.4230/LIPIcs.ITCS.2021.75},
  annote =	{Keywords: Low-degree likelihood ratio, error-correcting codes}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.78
Computational Hardness of Certifying Bounds on Constrained PCA Problems

Authors: Afonso S. Bandeira, Dmitriy Kunisky, and Alexander S. Wein

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

Abstract
Given a random n × n symmetric matrix ? drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form ?^⊤ ? ? over all vectors ? in a constraint set ? ⊂ ℝⁿ. For a certain class of normalized constraint sets we show that, conditional on a certain complexity-theoretic conjecture, no polynomial-time algorithm can certify a better upper bound than the largest eigenvalue of ?. A notable special case included in our results is the hypercube ? = {±1/√n}ⁿ, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. Our results suggest a striking gap between optimization and certification for this problem. Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for predicting computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is conjectured to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over ? ∈ ? is much larger than that of a GOE matrix.
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Afonso S. Bandeira, Dmitriy Kunisky, and Alexander S. Wein. Computational Hardness of Certifying Bounds on Constrained PCA Problems. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 78:1-78:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bandeira_et_al:LIPIcs.ITCS.2020.78,
  author =	{Bandeira, Afonso S. and Kunisky, Dmitriy and Wein, Alexander S.},
  title =	{{Computational Hardness of Certifying Bounds on Constrained PCA Problems}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{78:1--78:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.78},
  URN =		{urn:nbn:de:0030-drops-117633},
  doi =		{10.4230/LIPIcs.ITCS.2020.78},
  annote =	{Keywords: Certification, Sherrington-Kirkpatrick model, spiked Wishart model, low-degree likelihood ratio}
}
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