Document Open Access Logo

Dimension-Free Correlated Sampling for the Hypersimplex

Authors Joseph (Seffi) Naor, Nitya Raju, Abhishek Shetty, Aravind Srinivasan, Renata Valieva, David Wajc

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2026.104.pdf
  • Filesize: 0.89 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
Keywords
  • Correlated Rounding
  • Dependent Rounding

Metrics

Abstract

Sampling from multiple distributions so as to maximize overlap has been studied by statisticians since the 1950s. Since the 2000s, such correlated sampling from the probability simplex has been a powerful building block in disparate areas of theoretical computer science. We study a generalization of this problem to sampling sets from given vectors in the hypersimplex, i.e., outputting sets of size (at most) k ∈ [n], while maximizing the overlap of the sampled sets. Specifically, the expected difference between two output sets should be at most α times their input vectors' 𝓁₁ distance. A value of α = O(log n) is known to be achievable, due to Chen et al. (ICALP'17). We improve this factor to O(log k), independent of the ambient dimension n. Our algorithm satisfies other desirable properties, including (up to a log^* n factor) input-sparsity sampling time, logarithmic parallel depth and dynamic update time, as well as preservation of submodular objectives. Anticipating broader use of correlated sampling algorithms for the hypersimplex, we present applications of our algorithm to online paging, offline approximation of metric multi-labeling, and swift multi-scenario submodular welfare approximating reallocation.

Cite As Get BibTex

Joseph (Seffi) Naor, Nitya Raju, Abhishek Shetty, Aravind Srinivasan, Renata Valieva, and David Wajc. Dimension-Free Correlated Sampling for the Hypersimplex. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 104:1-104:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.104

Author Details

Joseph (Seffi) Naor
  • Technion - Israel Institute of Technology, Haifa, Israel
Nitya Raju
  • University of Maryland, College Park, MD, USA
Abhishek Shetty
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Aravind Srinivasan
  • University of Maryland, College Park, MD, USA
Renata Valieva
  • University of Maryland, College Park, MD, USA
David Wajc
  • Technion - Israel Institute of Technology, Haifa, Israel

Funding

  • Naor, Joseph (Seffi): Supported in part by ISF grant 3001/24 and United States - Israel BSF grant 2022418.
  • Shetty, Abhishek: Supported in part by ARO award W911NF-21-1-0328, the Simons Foundation, NSF award DMS-2031883, a DARPA AIQ award, an NSF FODSI Postdoctoral Fellowship and an Apple AI/ML Fellowship.
  • Srinivasan, Aravind: Supported in part by NSF award number CCF-1918749.
  • Valieva, Renata: Supported in part by NSF award numbers CCF-1918749 and CNS-2317194.
  • Wajc, David: Supported in part by the Taub Family Foundation "Leader in Science and Technology" fellowship, Grand Technion Energy Program (GTEP) and ISF grant 3200/24.

References

  1. Nima Anari, Ruiquan Gao, and Aviad Rubinstein. Parallel sampling via counting. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 537-548, 2024. URL: https://doi.org/10.1145/3618260.3649744.
  2. Omer Angel and Yinon Spinka. Pairwise optimal coupling of multiple random variables. arXiv preprint arXiv:1903.00632, 2019. Google Scholar
  3. Ghazi Badih, Kumar Ravi, and Manurangsi Pasin. User-level private learning via correlated sampling. Advances in Neural Information Processing Systems, 2021. Google Scholar
  4. Boaz Barak, Moritz Hardt, Ishay Haviv, Anup Rao, Oded Regev, and David Steurer. Rounding parallel repetitions of unique games. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 374-383, 2008. URL: https://doi.org/10.1109/FOCS.2008.55.
  5. Mohammad Bavarian, Badih Ghazi, Elad Haramaty, Pritish Kamath, Ronald L Rivest, and Madhu Sudan. Optimality of correlated sampling strategies. Theory of Computing, 16(1), 2020. URL: https://doi.org/10.4086/TOC.2020.V016A012.
  6. Petter Brändén and Johan Jonasson. Negative dependence in sampling. Scandinavian Journal of Statistics, 39(4):830-838, 2012. Google Scholar
  7. Andrei Z Broder. On the resemblance and containment of documents. In Proceedings. Compression and Complexity of SEQUENCES 1997 (Cat. No. 97TB100171), pages 21-29, 1997. URL: https://doi.org/10.1109/SEQUEN.1997.666900.
  8. Niv Buchbinder, Kamal Jain, and Joseph (Seffi) Naor. Online primal-dual algorithms for maximizing ad-auctions revenue. In Proceedings of the 15th Annual European Symposium on Algorithms (ESA), pages 253-264. 2007. Google Scholar
  9. Niv Buchbinder, Joseph (Seffi) Naor, and Roy Schwartz. Simplex partitioning via exponential clocks and the multiway-cut problem. SIAM Journal on Computing (SICOMP), 47(4):1463-1482, 2018. URL: https://doi.org/10.1137/15M1045521.
  10. Niv Buchbinder, Joseph (Seffi) Naor, David Wajc, et al. Chasing submodular objectives, and submodular maximization via cutting planes. arXiv preprint arXiv:2511.13605, 2025. Google Scholar
  11. Mark Bun, Marco Gaboardi, Max Hopkins, Russell Impagliazzo, Rex Lei, Toniann Pitassi, Satchit Sivakumar, and Jessica Sorrell. Stability is stable: Connections between replicability, privacy, and adaptive generalization. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 520-527, 2023. URL: https://doi.org/10.1145/3564246.3585246.
  12. Jarosław Byrka, Piotr Skowron, and Krzysztof Sornat. Proportional approval voting, harmonic k-median, and negative association. In Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP), page 26, 2018. Google Scholar
  13. Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing (SICOMP), 40(6):1740-1766, 2011. URL: https://doi.org/10.1137/080733991.
  14. Moses S Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 380-388, 2002. URL: https://doi.org/10.1145/509907.509965.
  15. Chandra Chekuri, Jan Vondrak, and Rico Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In Proceedings of the 51st Symposium on Foundations of Computer Science (FOCS), pages 575-584, 2010. Google Scholar
  16. Shahar Chen, Dotan Di Castro, Zohar Karnin, Liane Lewin-Eytan, Joseph Seffi Naor, and Roy Schwartz. Correlated rounding of multiple uniform matroids and multi-label classification. In Proceedings of the 44th International Colloquium on Automata, Languages and Programming (ICALP), 2017. Google Scholar
  17. Tasos C Christofides and Eutichia Vaggelatou. A connection between supermodular ordering and positive/negative association. Journal of Multivariate analysis, 88(1):138-151, 2004. Google Scholar
  18. Jean-Claude Deville and Yves Tille. Unequal probability sampling without replacement through a splitting method. Biometrika, 85(1):89-101, 1998. Google Scholar
  19. Devdatt Dubhashi, Johan Jonasson, and Desh Ranjan. Positive influence and negative dependence. Combinatorics, Probability and Computing, 16(01):29-41, 2007. URL: https://doi.org/10.1017/S0963548306007772.
  20. Devdatt Dubhashi and Desh Ranjan. Balls and bins: A study in negative dependence. Random Struct. Algorithms, 13(2):99-124, 1998. URL: https://doi.org/10.1002/(SICI)1098-2418(199809)13:2%3C99::AID-RSA1%3E3.0.CO;2-M.
  21. Devdatt P Dubhashi, Volker Priebe, and Desh Ranjan. Negative dependence through the FKG inequality. BRICS Report Series, 3(27), 1996. Google Scholar
  22. Lawrence R Ernst. The maximization and minimization of sample overlap problems: a half century of results. In Bulletin of the International Statistical Institute, Proceedings, volume 58, pages 293-296, 1999. Google Scholar
  23. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  24. Amos Fiat, Richard M Karp, Michael Luby, Lyle A McGeoch, Daniel D Sleator, and Neal E Young. Competitive paging algorithms. Journal of Algorithms, 12(4):685-699, 1991. URL: https://doi.org/10.1016/0196-6774(91)90041-V.
  25. Dongdong Ge, Simai He, Yinyu Ye, and Jiawei Zhang. Geometric rounding: a dependent randomized rounding scheme. Journal of combinatorial optimization, 22:699-725, 2011. URL: https://doi.org/10.1007/S10878-010-9320-Z.
  26. Thomas Holenstein. Parallel repetition: simplifications and the no-signaling case. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), pages 411-419, 2007. URL: https://doi.org/10.1145/1250790.1250852.
  27. Kumar Joag-Dev and Frank Proschan. Negative association of random variables with applications. The Annals of Statistics, pages 286-295, 1983. Google Scholar
  28. Nathan Keyfitz. Sampling with probabilities proportional to size: adjustment for changes in the probabilities. Journal of the American Statistical Association, 46(253):105-109, 1951. Google Scholar
  29. Subhash Khot, Richard J Lipton, Evangelos Markakis, and Aranyak Mehta. Inapproximability results for combinatorial auctions with submodular utility functions. In Proceedings of the 1st Conference on Web and Internet Economics (WINE), pages 92-101, 2005. URL: https://doi.org/10.1007/11600930_10.
  30. Jon Kleinberg and Éva Tardos. Approximation algorithms for classification problems with pairwise relationships: Metric labeling and Markov random fields. Journal of the ACM (JACM), 49(5):616-639, 2002. URL: https://doi.org/10.1145/585265.585268.
  31. Josh Brown Kramer, Jonathan Cutler, and AJ Radcliffe. Negative dependence and srinivasan’s sampling process. Combinatorics, Probability and Computing, 20(3):347-361, 2011. URL: https://doi.org/10.1017/S0963548311000095.
  32. Hongyang Liu and Yitong Yin. Simple parallel algorithms for single-site dynamics. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1431-1444, 2022. URL: https://doi.org/10.1145/3519935.3519999.
  33. Vahab Mirrokni, Michael Schapira, and Jan Vondrák. Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions. In Proceedings of the 9th ACM conference on Electronic commerce, pages 70-77, 2008. URL: https://doi.org/10.1145/1386790.1386805.
  34. George L Nemhauser and Laurence A Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of Operations Research (Math of OR), 3(3):177-188, 1978. URL: https://doi.org/10.1287/MOOR.3.3.177.
  35. Frederick Qiu and Sahil Singla. Submodular dominance and applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), 2022. Google Scholar
  36. Ronald L Rivest. Symmetric encryption via keyrings and ecc. In Northernmost Crypto Workshop, Longyearbyen, Norway, Midnight Lect, 2016. Google Scholar
  37. Ankit Sharma and Jan Vondrák. Multiway cut, pairwise realizable distributions, and descending thresholds. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 724-733, 2014. URL: https://doi.org/10.1145/2591796.2591866.
  38. Aravind Srinivasan. Distributions on level-sets with applications to approximation algorithms. In Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS), pages 588-597, 2001. URL: https://doi.org/10.1109/SFCS.2001.959935.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact