Sampling and Certifying Symmetric Functions

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

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Author Details

Yuval Filmus
  • Technion - Israel Institute of Technology, Haifa, Israel
Itai Leigh
  • Tel-Aviv University, Israel
Artur Riazanov
  • École Polytechnique Fédérale de Lausanne, Switzerland
Dmitry Sokolov
  • École Polytechnique Fédérale de Lausanne, Switzerland


We thank Mika Göös, Aleksandr Smal, and Anastasia Sofronova for fruitful discussions and suggestions. We thank the RANDOM 2023 anonymous referees for their helpful comments.

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Yuval Filmus, Itai Leigh, Artur Riazanov, and Dmitry Sokolov. Sampling and Certifying Symmetric Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A circuit 𝒞 samples a distribution X with an error ε if the statistical distance between the output of 𝒞 on the uniform input and X is ε. We study the hardness of sampling a uniform distribution over the set of n-bit strings of Hamming weight k denoted by Uⁿ_k for decision forests, i.e. every output bit is computed as a decision tree of the inputs. For every k there is an O(log n)-depth decision forest sampling Uⁿ_k with an inverse-polynomial error [Emanuele Viola, 2012; Czumaj, 2015]. We show that for every ε > 0 there exists τ such that for decision depth τ log (n/k) / log log (n/k), the error for sampling U_kⁿ is at least 1-ε. Our result is based on the recent robust sunflower lemma [Ryan Alweiss et al., 2021; Rao, 2019]. Our second result is about matching a set of n-bit strings with the image of a d-local circuit, i.e. such that each output bit depends on at most d input bits. We study the set of all n-bit strings whose Hamming weight is at least n/2. We improve the previously known locality lower bound from Ω(log^* n) [Beyersdorff et al., 2013] to Ω(√log n), leaving only a quartic gap from the best upper bound of O(log² n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Generating random combinatorial structures
  • sampling
  • lower bounds
  • robust sunflowers
  • decision trees
  • switching networks


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