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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

A polynomial threshold function (PTF) f: ℝⁿ → ℝ is a function of the form f(x) = sign(p(x)) where p is a polynomial of degree at most d. PTFs are a classical and well-studied complexity class with applications across complexity theory, learning theory, approximation theory, quantum complexity and more. We address the question of designing pseudorandom generators (PRGs) for polynomial threshold functions (PTFs) in the gaussian space: design a PRG that takes a seed of few bits of randomness and outputs a n-dimensional vector whose distribution is indistinguishable from a standard multivariate gaussian by a degree d PTF.
Our main result is a PRG that takes a seed of d^O(1) log(n/ε) log(1/ε)/ε² random bits with output that cannot be distinguished from an n-dimensional gaussian distribution with advantage better than ε by degree d PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seed length of d^O(log d)) in the degree d. Along the way we prove a few nearly-tight structural properties of restrictions of PTFs that may be of independent interest.
Similar results were obtained in [Ryan O'Donnell et al., 2021] (independently and concurrently).

Zander Kelley and Raghu Meka. Random Restrictions and PRGs for PTFs in Gaussian Space. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kelley_et_al:LIPIcs.CCC.2022.21, author = {Kelley, Zander and Meka, Raghu}, title = {{Random Restrictions and PRGs for PTFs in Gaussian Space}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {21:1--21:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.21}, URN = {urn:nbn:de:0030-drops-165836}, doi = {10.4230/LIPIcs.CCC.2022.21}, annote = {Keywords: polynomial threshold function, pseudorandom generator, multivariate gaussian} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

The well-known Komlós conjecture states that given n vectors in ℝ^d with Euclidean norm at most one, there always exists a ± 1 coloring such that the 𝓁_∞ norm of the signed-sum vector is a constant independent of n and d. We prove this conjecture in a smoothed analysis setting where the vectors are perturbed by adding a small Gaussian noise and when the number of vectors n = ω(d log d). The dependence of n on d is the best possible even in a completely random setting.
Our proof relies on a weighted second moment method, where instead of considering uniformly randomly colorings we apply the second moment method on an implicit distribution on colorings obtained by applying the Gram-Schmidt walk algorithm to a suitable set of vectors. The main technical idea is to use various properties of these colorings, including subgaussianity, to control the second moment.

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Smoothed Analysis of the Komlós Conjecture. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ICALP.2022.14, author = {Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand}, title = {{Smoothed Analysis of the Koml\'{o}s Conjecture}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {14:1--14:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.14}, URN = {urn:nbn:de:0030-drops-163556}, doi = {10.4230/LIPIcs.ICALP.2022.14}, annote = {Keywords: Koml\'{o}s conjecture, smoothed analysis, weighted second moment method, subgaussian coloring} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of T unit vectors in ℝ^d, find ± signs for each of them such that the signed sum vector along any prefix has a small 𝓁_∞-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Komlós problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes.
Banaszczyk gave an O(√{log d+ log T}) bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk’s bound and consider natural generalizations of prefix discrepancy:
- We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in T compared to Banaszczyk’s bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of O(√{log d+ log log T}) with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk’s bound in the worst case.
- We also introduce a generalization of the prefix discrepancy problem to arbitrary DAGs. Here, vertices correspond to unit vectors, and the discrepancy constraints correspond to paths on a DAG on T vertices - prefix discrepancy is precisely captured when the DAG is a simple path. We show that an analog of Banaszczyk’s O(√{log d+ log T}) bound continues to hold in this setting for adversarially given unit vectors and that the √{log T} factor is unavoidable for DAGs. We also show that unlike for prefix discrepancy, the dependence on T cannot be improved significantly in the smoothed case for DAGs.
- We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. In this problem, the discrepancy constraints are generalized from paths of a DAG to an arbitrary set system. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ITCS.2022.13, author = {Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand}, title = {{Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {13:1--13:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.13}, URN = {urn:nbn:de:0030-drops-156092}, doi = {10.4230/LIPIcs.ITCS.2022.13}, annote = {Keywords: Prefix discrepancy, smoothed analysis, combinatorial vector balancing} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Our proof uses elementary counting together with a novel connection to the sunflower lemma.
In addition to a simplified proof, our approach opens up a new avenue of attack towards proving lifting theorems with improved gadget size - one of the main challenges in the area. Focusing on one of the most widely used gadgets - the index gadget - existing lifting techniques are known to require at least a quadratic gadget size. Our new approach combined with robust sunflower lemmas allows us to reduce the gadget size to near linear. We conjecture that it can be further improved to polylogarithmic, similar to the known bounds for the corresponding robust sunflower lemmas.

Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with Sunflowers. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 104:1-104:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lovett_et_al:LIPIcs.ITCS.2022.104, author = {Lovett, Shachar and Meka, Raghu and Mertz, Ian and Pitassi, Toniann and Zhang, Jiapeng}, title = {{Lifting with Sunflowers}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {104:1--104:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.104}, URN = {urn:nbn:de:0030-drops-157004}, doi = {10.4230/LIPIcs.ITCS.2022.104}, annote = {Keywords: Lifting theorems, communication complexity, combinatorics, sunflowers} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Motivated by the derandomization of space-bounded computation, there has been a long line of work on constructing pseudorandom generators (PRGs) against various forms of read-once branching programs (ROBPs), with a goal of improving the O(log² n) seed length of Nisan’s classic construction [Noam Nisan, 1992] to the optimal O(log n).
In this work, we construct an explicit PRG with seed length Õ(log n) for constant-width ROBPs that are monotone, meaning that the states at each time step can be ordered so that edges with the same labels never cross each other. Equivalently, for each fixed input, the transition functions are a monotone function of the state. This result is complementary to a line of work that gave PRGs with seed length O(log n) for (ordered) permutation ROBPs of constant width [Braverman et al., 2014; Koucký et al., 2011; De, 2011; Thomas Steinke, 2012], since the monotonicity constraint can be seen as the "opposite" of the permutation constraint.
Our PRG also works for monotone ROBPs that can read the input bits in any order, which are strictly more powerful than read-once AC⁰. Our PRG achieves better parameters (in terms of the dependence on the depth of the circuit) than the best previous pseudorandom generator for read-once AC⁰, due to Doron, Hatami, and Hoza [Doron et al., 2019].
Our pseudorandom generator construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989; Gopalan et al., 2012]. We give a randomness-efficient width-reduction process which proves that the branching program simplifies to an O(log n)-junta after only O(log log n) independent applications of the Forbes-Kelley pseudorandom restrictions [Michael A. Forbes and Zander Kelley, 2018].

Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan. Pseudorandom Generators for Read-Once Monotone Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 58:1-58:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2021.58, author = {Doron, Dean and Meka, Raghu and Reingold, Omer and Tal, Avishay and Vadhan, Salil}, title = {{Pseudorandom Generators for Read-Once Monotone Branching Programs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {58:1--58:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.58}, URN = {urn:nbn:de:0030-drops-147513}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.58}, annote = {Keywords: Branching programs, pseudorandom generators, constant depth circuits} }

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Complete Volume

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

LIPIcs, Volume 176, APPROX/RANDOM 2020, Complete Volume

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 1-1228, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{byrka_et_al:LIPIcs.APPROX/RANDOM.2020, title = {{LIPIcs, Volume 176, APPROX/RANDOM 2020, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {1--1228}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020}, URN = {urn:nbn:de:0030-drops-126021}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020}, annote = {Keywords: LIPIcs, Volume 176, APPROX/RANDOM 2020, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Front Matter, Table of Contents, Preface, Conference Organization

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{byrka_et_al:LIPIcs.APPROX/RANDOM.2020.0, author = {Byrka, Jaros{\l}aw and Meka, Raghu}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {0:i--0:xx}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.0}, URN = {urn:nbn:de:0030-drops-126037}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Hashing is one of the main techniques in data processing and algorithm design for very large data sets. While random hash functions satisfy most desirable properties, it is often too expensive to store a fully random hash function. Motivated by this, much attention has been given to designing small families of hash functions suitable for various applications. In this work, we study the question of designing space-efficient hash families H = {h:[U] -> [N]} with the natural property of 'covering': H is said to be covering if any set of Omega(N log N) distinct items from the universe (the "coupon-collector limit") are hashed to cover all N bins by most hash functions in H. We give an explicit covering family H of size poly(N) (which is optimal), so that hash functions in H can be specified efficiently by O(log N) bits.
We build covering hash functions by drawing a connection to "dispersers", which are quite well-studied and have a variety of applications themselves. We in fact need strong dispersers and we give new constructions of strong dispersers which may be of independent interest. Specifically, we construct strong dispersers with optimal entropy loss in the high min-entropy, but very small error (poly(n)/2^n for n bit sources) regimes. We also provide a strong disperser construction with constant error but for any min-entropy. Our constructions achieve these by using part of the source to replace seed from previous non-strong constructions in surprising ways. In doing so, we take two of the few constructions of dispersers with parameters better than known extractors and make them strong.

Raghu Meka, Omer Reingold, and Yuan Zhou. Deterministic Coupon Collection and Better Strong Dispersers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 872-884, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{meka_et_al:LIPIcs.APPROX-RANDOM.2014.872, author = {Meka, Raghu and Reingold, Omer and Zhou, Yuan}, title = {{Deterministic Coupon Collection and Better Strong Dispersers}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {872--884}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.872}, URN = {urn:nbn:de:0030-drops-47440}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.872}, annote = {Keywords: Coupon collection; dispersers, strong dispersers, hashing, pseudorandomness} }

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