@inproceedings{8c6e4f445ce0487a99ae940b4adf7628,
title = "Sampling and Certifying Symmetric Functions",
abstract = "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 Ukn 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 Ukn with an inverse-polynomial error [26, 11]. 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 Ukn is at least 1 − ε. Our result is based on the recent robust sunflower lemma [1, 23]. 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) [5] to Ω(√log n), leaving only a quartic gap from the best upper bound of O(log2 n).",
keywords = "decision trees, lower bounds, robust sunflowers, sampling, switching networks",
author = "Yuval Filmus and Itai Leigh and Artur Riazanov and Dmitry Sokolov",
note = "Publisher Copyright: {\textcopyright} 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.; 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 ; Conference date: 11-09-2023 Through 13-09-2023",
year = "2023",
month = sep,
doi = "10.4230/LIPIcs.APPROX/RANDOM.2023.36",
language = "אנגלית",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
editor = "Nicole Megow and Adam Smith",
booktitle = "Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023",
}