Sampling and certifying symmetric functions

Yuval Filmus, Itai Leigh, Artur Riazanov, and Dmitry Sokolov

Viola considered the complexity of (approximately) sampling from a given distribution. We consider the uniform distribution over vectors in $\{0,1\}^n$ of weight $k$. We show that when $k$ is constant, any approximate sampler must have locality $\tilde\Omega(\log n)$, almost matching the upper bound $O(\log n)$.

Beyersdorff et al. considered the complexity of generating from a given set. One natural example is the set of vectors in $\{0,1\}^n$ with a majority of ones. They gave a “proof system” with locality $O(\log^2 n)$, and proved a lower bound of $\Omega(\log^* n)$. We improve the lower bound to $\Omega(\sqrt{\log n})$.


 author = {Filmus, Yuval and Leigh, Itai and Riazanov, Artur and Sokolov, Dmitry},
 title = {Sampling and Certifying Symmetric Functions},
 booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
 pages = {36:1--36:21},
 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 = {},
 URN = {urn:nbn:de:0030-drops-188611},
 doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.36},
 annote = {Keywords: sampling, lower bounds, robust sunflowers, decision trees, switching networks}
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