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Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs

Author Zhangsong Li

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ACM Subject Classification
  • Theory of computation → Random network models
  • Mathematics of computing → Random graphs
Keywords
  • Algorithmic Contiguity
  • Low-degree Conjecture
  • Correlated Random Graphs

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Abstract

In this paper, assuming a natural strengthening of the low-degree conjecture, we provide evidence of computational hardness for two problems: (1) the (partial) matching recovery problem in the sparse correlated Erdős-Rényi graphs G(n,q;ρ) when the edge-density q = n^{-1+o(1)} and the correlation ρ < √{α} lies below the Otter’s threshold, this resolves a remaining problem in [Jian Ding et al., 2023]; (2) the detection problem between a pair of correlated sparse stochastic block model S(n,λ/n;k,ε;s) and a pair of independent stochastic block models S(n,λs/n;k,ε) when ε² λ s < 1 lies below the Kesten-Stigum (KS) threshold and s < √α lies below the Otter’s threshold, this resolves a remaining problem in [Guanyi Chen et al., 2024].
One of the main ingredient in our proof is to derive certain forms of algorithmic contiguity between two probability measures based on bounds on their low-degree advantage. To be more precise, consider the high-dimensional hypothesis testing problem between two probability measures ℙ and ℚ based on the sample Y. We show that if the low-degree advantage Adv_{≤D}(dℙ/dℚ) = O(1), then (assuming the low-degree conjecture) there is no efficient algorithm A such that ℚ(A(Y) = 0) = 1-o(1) and ℙ(A(Y) = 1) = Ω(1). This framework provides a useful tool for performing reductions between different inference tasks.

Cite As Get BibTex

Zhangsong Li. Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.30

Author Details

Zhangsong Li
  • School of Mathematical Sciences, Peking University, Beijing, China

Funding

  • Li, Zhangsong: Partially supported by National Key R&D program of China (Project No. 2023YFA1010103) and NSFC Key Program (Project No. 12231002)

Acknowledgements

Z.L. thanks Jian Ding and Jingqiu Ding for helpful discussions, Alexander S. Wein for helpful comments on the revised low-degree conjecture, and Hang Du for pointing him to the reference [Youngtak Sohn and Alexander S. Wein, 2025].

References

  1. László Babai. Graph isomorphism in quasipolynomial time. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC), pages 684-697. ACM, 2016. Google Scholar
  2. Afonso S. Bandeira, Ahmed El Alaoui, Samuel B. Hopkins, Tselil Schramm, Alexander S. Wein, and Ilias Zadik. The Franz-Parisi criterion and computational trade-offs in high dimensional statistics. In Advances in Neural Information Processing Systems (NIPS), volume 35, pages 33831-33844. Curran Associates, Inc., 2022. Google Scholar
  3. Afonso S. Bandeira, Dmitriy Kunisky, and Alexander S. Wein. Computational hardness of certifying bounds on constrained PCA problems. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.78.
  4. Boaz Barak, Chi-Ning Chou, Zhixian Lei, Tselil Schramm, and Yueqi Sheng. (Nearly) efficient algorithms for the graph matching problem on correlated random graphs. In Advances in Neural Information Processing Systems (NIPS), volume 32, pages 9190-9198. Curran Associates, Inc., 2019. Google Scholar
  5. Boaz Barak, Samuel B. Hopkins, Jonathan Kelner, Pravesh K. Kothari, Ankur Moitra, and Aaron Potechin. A nearly tight sum-of-squares lower bound for the planted clique problem. SIAM Journal on Computing, 48(2):687-735, 2019. URL: https://doi.org/10.1137/17M1138236.
  6. Guy Bresler and Brice Huang. The algorithmic phase transition of random k-SAT for low degree polynomials. In Proceedings of the IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 298-309. IEEE, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00038.
  7. Rainer E. Burkard, Eranda Cela an Panos M. Pardalos, and Leonidas S. Pitsoulis. The quadratic assignment problem. Handbook of combinatorial optimization, pages 1713-1809, 1998. Google Scholar
  8. Shuwen Chai and Miklós Z. Rácz. Efficient graph matching for correlated stochastic block models. In Advances in Neural Information Processing Systems (NIPS), volume 38, pages 116388-116461. Curran Associates, Inc., 2024. Google Scholar
  9. Guanyi Chen, Jian Ding, Shuyang Gong, and Zhangsong Li. A computational transition for detecting correlated stochastic block models by low-degree polynomials. arXiv preprint, 2024. URL: https://arxiv.org/abs/2409.00966.
  10. Daniel Cullina and Negar Kiyavash. Exact alignment recovery for correlated Erdős-Rényi graphs. arXiv preprint, 2024. URL: https://arxiv.org/abs/1711.06783.
  11. Osman E. Dai, Daniel Cullina, Negar Kiyavash, and Matthias Grossglauser. Analysis of a canonical labeling algorithm for the alignment of correlated Erdős-Rényi graphs. Proceedings of the ACM on Measurement and Analysis of Computing Systems, 3(2):1-25, 2019. URL: https://doi.org/10.1145/3341617.3326151.
  12. Abhishek Dhawan, Cheng Mao, and Alexander S. Wein. Detection of dense subhypergraphs by low-degree polynomials. arXiv preprint, 2024. to appear in Random Structures and Algorithms. URL: https://arxiv.org/abs/2304.08135.
  13. Jian Ding and Hang Du. Detection threshold for correlated Erdős-Rényi graphs via densest subgraph. IEEE Transactions on Information Theory, 69(8):5289-5298, 2023. URL: https://doi.org/10.1109/TIT.2023.3265009.
  14. Jian Ding and Hang Du. Matching recovery threshold for correlated random graphs. Annals of Statistics, 51(4):1718-1743, 2023. Google Scholar
  15. Jian Ding, Hang Du, and Zhangsong Li. Low-degree hardness of detection for correlated Erdős-Rényi graphs. arXiv preprint, 2023. to appear in Annals of Statistics. URL: https://arxiv.org/abs/2311.15931.
  16. Jian Ding and Zhangsong Li. A polynomial time iterative algorithm for matching Gaussian matrices with non-vanishing correlation. arXiv preprint, 2023. to appear in Foundations of Computational Mathematics. URL: https://arxiv.org/abs/2212.13677.
  17. Jian Ding and Zhangsong Li. A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation. arXiv preprint, 2023. URL: https://arxiv.org/abs/2306.00266.
  18. Jian Ding, Zongming Ma, Yihong Wu, and Jiaming Xu. Efficient random graph matching via degree profiles. Probability Theory and Related Fields, 179(1-2):29-115, 2021. Google Scholar
  19. Jingqiu Ding, Yiding Hua, Lucas Slot, and David Steurer. Low degree conjecture implies sharp computational thresholds in stochastic block model. arXiv preprint, 2025. URL: https://arxiv.org/abs/arXiv:2502.15024.
  20. Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein, and Afonso S. Bandeira. Subexponential-time algorithms for sparse PCA. Foundations of Computational Mathematics, 22(1):1-50, 2022. Google Scholar
  21. Hang Du. Optimal recovery of correlated Erdős-Rényi graphs. arXiv preprint, 2025. URL: https://arxiv.org/abs/2502.12077.
  22. Zhou Fan, Cheng Mao, Yihong Wu, and Jiaming Xu. Spectral graph matching and regularized quadratic relaxations I: Algorithm and Gaussian analysis. Foundations of Computational Mathematics, 23(5):1511-1565, 2023. URL: https://doi.org/10.1007/S10208-022-09570-Y.
  23. Zhou Fan, Cheng Mao, Yihong Wu, and Jiaming Xu. Spectral graph matching and regularized quadratic relaxations II: Erdős-Rényi graphs and universality. Foundations of Computational Mathematics, 23(5):1567-1617, 2023. Google Scholar
  24. David Gamarnik. The overlap gap property: A topological barrier to optimizing over random structures. Proceedings of the National Academy of Sciences of the United States of America, 118(41):e2108492118, 2021. Google Scholar
  25. David Gamarnik, Aukosh Jagannath, and Alexander S. Wein. Low-degree hardness of random optimization problems. In Proceedings of the IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 131-140. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00021.
  26. Luca Ganassali and Laurent Massoulié. From tree matching to sparse graph alignment. In Proceedings of the 33rd Conference on Learning Theory (COLT), pages 1633-1665. PMLR, 2020. URL: http://proceedings.mlr.press/v125/ganassali20a.html.
  27. Luca Ganassali, Laurent Massoulié, and Marc Lelarge. Impossibility of partial recovery in the graph alignment problem. In Proceedings of the 34th Conference on Learning Theory (COLT), pages 2080-2102. PMLR, 2021. URL: http://proceedings.mlr.press/v134/ganassali21a.html.
  28. Luca Ganassali, Laurent Massoulié, and Marc Lelarge. Correlation detection in trees for planted graph alignment. Annals of Applied Probability, 34(3):2799-2843, 2024. Google Scholar
  29. Luca Ganassali, Laurent Massoulié, and Guilhem Semerjian. Statistical limits of correlation detection in trees. Annals of Applied Probability, 34(4):3701-3734, 2024. Google Scholar
  30. Julia Gaudio, Miklós Z. Rácz, and Anirudh Sridhar. Exact community recovery in correlated stochastic block models. In Proceedings of the 35th Conference on Learning Theory (COLT), pages 2183-2241. PMLR, 2022. URL: https://proceedings.mlr.press/v178/gaudio22a.html.
  31. Georgina Hall and Laurent Massoulié. Partial recovery in the graph alignment problem. Operations Research, 71(1):259-272, 2023. URL: https://doi.org/10.1287/OPRE.2022.2355.
  32. Samuel B. Hopkins. Statistical inference and the sum of squares method. Cornell University, 2018. PHD thesis. Google Scholar
  33. Samuel B. Hopkins, Pravesh K. Kothari, Aaron Potechin, Prasad Raghavendra, Tselil Schramm, and David Steurer. The power of sum-of-squares for detecting hidden structures. In Proceedings of the IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 720-731. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.72.
  34. Samuel B. Hopkins and David Steurer. Efficient Bayesian estimation from few samples: community detection and related problems. In Proceedings of the IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 379-390. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.42.
  35. Harry Kesten and Bernt P. Stigum. Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Annals of Mathematical Statistics, 37:1463-1481, 1966. Google Scholar
  36. Pravesh K. Kothari, Santosh S. Vempala, Alexander S. Wein, and Jeff Xu. Is planted coloring easier than planted clique? In Proceedings of the 36th Conference on Learning Theory (COLT), pages 5343-5372. PMLR, 2023. URL: https://proceedings.mlr.press/v195/kothari23a.html.
  37. Dmitriy Kunisky. Hypothesis testing with low-degree polynomials in the Morris class of exponential families. In Proceedings of the 34th Conference on Learning Theory (COLT), pages 2822-2848. PMLR, 2021. URL: http://proceedings.mlr.press/v134/kunisky21a.html.
  38. Dmitriy Kunisky, Cristopher Moore, and Alexander S. Wein. Tensor cumulants for statistical inference on invariant distributions. In Proceedings of the IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 1007-1026. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00067.
  39. Dmitriy Kunisky, Alexander S. Wein, and Afonso S. Bandeira. Notes on computational hardness of hypothesis testing: Predictions using the low-degree likelihood ratio. In Mathematical Analysis, its Applications and Computation: ISAAC, pages 1-50. Springer, 2022. Google Scholar
  40. Zhangsong Li. Robust random graph matching in dense graphs via vector approximate message passing. arXiv preprint, 2024. URL: https://arxiv.org/abs/2412.16457.
  41. Zhangsong Li. Algorithmic contiguity from low-degree conjecture and applications in correlated random graphs. arXiv preprint, 2025. URL: https://arxiv.org/abs/arXiv:2502.09832.
  42. Konstantin Makarychev, Rajsekar Manokaran, and Maxim Sviridenko. Maximum quadratic assignment problem: Reduction from maximum label cover and lp-based approximation algorithm. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), pages 594-604. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-14165-2_50.
  43. Cheng Mao, Mark Rudelson, and Konstantin Tikhomirov. Random graph matching with improved noise robustness. In Proceedings of the 34th Conference on Learning Theory (COLT), pages 3296-3329. PMLR, 2021. URL: http://proceedings.mlr.press/v134/mao21a.html.
  44. Cheng Mao, Mark Rudelson, and Konstantin Tikhomirov. Exact matching of random graphs with constant correlation. Probability Theory and Related Fields, 186(2):327-389, 2023. Google Scholar
  45. Cheng Mao, Yihong Wu, Jiaming Xu, and Sophie H. Yu. Random graph matching at Otter’s threshold via counting chandeliers. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC), pages 1345-1356. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585156.
  46. Cheng Mao, Yihong Wu, Jiaming Xu, and Sophie H. Yu. Testing network correlation efficiently via counting trees. Annals of Statistics, 52(6):2483-2505, 2024. Google Scholar
  47. Ankur Moitra and Alexander S. Wein. Precise error rates for computationally efficient testing. Annals of Statistics, 53(2):854-878, 2023. Google Scholar
  48. Andrea Montanari and Alexander S. Wein. Equivalence of approximate message passing and low-degree polynomials in rank-one matrix estimation. Probability Theory and Related Fields, 191(1-2):181-233, 2025. Google Scholar
  49. Arvind Narayanan and Vitaly Shmatikov. Robust de-anonymization of large sparse datasets. In IEEE Symposium on Security and Privacy, pages 111-125. IEEE, 2008. URL: https://doi.org/10.1109/SP.2008.33.
  50. Richard Otter. The number of trees. Annals of Mathematics, 49(3):583-599, 1948. Google Scholar
  51. Miklós Z. Rácz and Anirudh Sridhar. Correlated stochastic block models: exact graph matching with applications to recovering communities. In Advances in Neural Information Processing Systems (NIPS), volume 34, pages 22259-22273. Curran Associates, Inc., 2021. URL: https://proceedings.neurips.cc/paper/2021/hash/baf4f1a5938b8d520b328c13b51ccf11-Abstract.html.
  52. Prasad Raghavendra, Tselil Schramm, and David Steurer. High-dimensional estimation via sum-of-squares proofs. In Proceedings of the International Congress of Mathematicians (ICM 2018), pages 3389-3423, 2019. Google Scholar
  53. Cynthia Rush, Fiona Skerman, Alexander S. Wein, and Dana Yang. Is it easier to count communities than find them? In Proceedings of the 14th Innovations in Theoretical Computer Science Conference (ITCS). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.94.
  54. Tselil Schramm and Alexander S. Wein. Computational barriers to estimation from low-degree polynomials. Annals of Statistics, 50(3):1833-1858, 2022. Google Scholar
  55. Rohit Singh, Jinbo Xu, and Bonnie Berger. Global alignment of multiple protein interaction networks with application to functional orthology detection. Proceedings of the National Academy of Sciences of the United State of America, 105(35):12763-12768, 2008. URL: https://doi.org/10.1073/PNAS.0806627105.
  56. Youngtak Sohn and Alexander S. Wein. Sharp phase transitions in estimation with low-degree polynomials. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC), pages 891-902. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718294.
  57. Alexander S. Wein. Optimal low-degree hardness of maximum independent set. Mathematical Statistics and Learning, 4(3/4):221-251, 2022. Google Scholar
  58. Yihong Wu, Jiaming Xu, and Sophie H. Yu. Settling the sharp reconstruction thresholds of random graph matching. IEEE Transactions on Information Theory, 68(8):5391-5417, 2022. URL: https://doi.org/10.1109/TIT.2022.3169005.
  59. Yihong Wu, Jiaming Xu, and Sophie H. Yu. Testing correlation of unlabeled random graphs. Annals of Applied Probability, 33(4):2519-2558, 2023. Google Scholar
  60. Ilias Zadik, Min Jae Song, Alexander S. Wein, and Joan Bruna. Lattice-based methods surpass sum-of-squares in clustering. In Proceedings of the 35th Conference on Learning Theory (COLT), pages 1247-1248. PMLR, 2022. URL: https://proceedings.mlr.press/v178/zadik22a.html.
  61. Lenka Zdeborová and Florent Krzakala. Statistical physics of inference: thresholds and algorithms. Advances in Phyisics, 65(5):453-552, 2016. Google Scholar
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