Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients.
In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang, Wong, Xie, and Zhang (FOCS'13), and the second is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni. Refuting Approaches to the Log-Rank Conjecture for XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 82:1-82:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hatami_et_al:LIPIcs.ICALP.2024.82, author = {Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar and Ostuni, Anthony}, title = {{Refuting Approaches to the Log-Rank Conjecture for XOR Functions}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {82:1--82:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.82}, URN = {urn:nbn:de:0030-drops-202252}, doi = {10.4230/LIPIcs.ICALP.2024.82}, annote = {Keywords: Communication complexity, log-rank conjecture, XOR functions, additive structure} }

Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

A folklore conjecture in quantum computing is that the acceptance probability of a quantum query algorithm can be approximated by a classical decision tree, with only a polynomial increase in the number of queries. Motivated by this conjecture, Aaronson and Ambainis (Theory of Computing, 2014) conjectured that this should hold more generally for any bounded function computed by a low degree polynomial.
In this work we prove two new results towards establishing this conjecture: first, that any such polynomial has a small fractional certificate complexity; and second, that many inputs have a small sensitive block. We show that these would imply the Aaronson and Ambainis conjecture, assuming a conjectured extension of Talagrand’s concentration inequality.
On the technical side, many classical techniques used in the analysis of Boolean functions seem to fail when applied to bounded functions. Here, we develop a new technique, based on a mix of combinatorics, analysis and geometry, and which in part extends a recent technique of Knop et al. (STOC 2021) to bounded functions.

Shachar Lovett and Jiapeng Zhang. Fractional Certificates for Bounded Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 84:1-84:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lovett_et_al:LIPIcs.ITCS.2023.84, author = {Lovett, Shachar and Zhang, Jiapeng}, title = {{Fractional Certificates for Bounded Functions}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {84:1--84:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.84}, URN = {urn:nbn:de:0030-drops-175871}, doi = {10.4230/LIPIcs.ITCS.2023.84}, annote = {Keywords: Aaronson-Ambainis conjecture, fractional block sensitivity, Talagrand inequality} }

Document

RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Fast mixing of random walks on hypergraphs (simplicial complexes) has recently led to myriad breakthroughs throughout theoretical computer science. Many important applications, however, (e.g. to LTCs, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial structure. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify the advantage of different poset architectures in both a spectral and combinatorial sense, highlighting how regularity controls the spectral decay and edge-expansion of corresponding random walks.
We show that the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha APPROX-RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the regularity of the underlying poset. This gives a simple condition to identify poset architectures (e.g. the Grassmann) that exhibit strong (even exponential) decay of eigenvalues, versus architectures like hypergraphs whose eigenvalues decay linearly - a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight characterization of edge-expansion on expanding posets in the 𝓁₂-regime (generalizing recent work of Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization of expansion used in the proof of the 2-2 Games Conjecture which relies on 𝓁_∞ rather than 𝓁₂-structure.

Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, and Ruizhe Zhang. Eigenstripping, Spectral Decay, and Edge-Expansion on Posets. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 16:1-16:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gaitonde_et_al:LIPIcs.APPROX/RANDOM.2022.16, author = {Gaitonde, Jason and Hopkins, Max and Kaufman, Tali and Lovett, Shachar and Zhang, Ruizhe}, title = {{Eigenstripping, Spectral Decay, and Edge-Expansion on Posets}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {16:1--16:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.16}, URN = {urn:nbn:de:0030-drops-171381}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.16}, annote = {Keywords: High-dimensional expanders, posets, eposets} }

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Complete Volume

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

LIPIcs, Volume 234, CCC 2022, Complete Volume

37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1-960, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{lovett:LIPIcs.CCC.2022, title = {{LIPIcs, Volume 234, CCC 2022, Complete Volume}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {1--960}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022}, URN = {urn:nbn:de:0030-drops-165614}, doi = {10.4230/LIPIcs.CCC.2022}, annote = {Keywords: LIPIcs, Volume 234, CCC 2022, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lovett:LIPIcs.CCC.2022.0, author = {Lovett, Shachar}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {0:i--0:xvi}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.0}, URN = {urn:nbn:de:0030-drops-165621}, doi = {10.4230/LIPIcs.CCC.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**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.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)

We study the singularity probability of random integer matrices. Concretely, the probability that a random n × n matrix, with integer entries chosen uniformly from {-m,…,m}, is singular. This problem has been well studied in two regimes: large n and constant m; or large m and constant n. In this paper, we extend previous techniques to handle the regime where both n,m are large. We show that the probability that such a matrix is singular is m^{-cn} for some absolute constant c > 0. We also provide some connections of our result to coding theory.

Sankeerth Rao Karingula and Shachar Lovett. Singularity of Random Integer Matrices with Large Entries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{karingula_et_al:LIPIcs.APPROX/RANDOM.2021.33, author = {Karingula, Sankeerth Rao and Lovett, Shachar}, title = {{Singularity of Random Integer Matrices with Large Entries}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {33:1--33:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.33}, URN = {urn:nbn:de:0030-drops-147260}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.33}, annote = {Keywords: Coding Theory, Random matrix theory, Singularity probability MDS codes, Error correction codes, Littlewood Offord, Fourier Analysis} }

Document

**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al. [Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2019] that exploit L₁ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k. This interpolates previous works, which either require Fourier bounds on all levels [Chattopadhyay et al., 2019], or have polynomial dependence on the error parameter in the seed length [Eshan Chattopadhyay et al., 2019], and thus answers an open question in [Eshan Chattopadhyay et al., 2019]. As an example, we show that for polynomial error, Fourier bounds on the first O(log n) levels is sufficient to recover the seed length in [Chattopadhyay et al., 2019], which requires bounds on the entire tail.
We obtain our results by an alternate analysis of fractional PRGs using Taylor’s theorem and bounding the degree-k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the level-k unsigned Fourier sum, which is potentially a much smaller quantity than the L₁ notion in previous works. By generalizing a connection established in [Chattopadhyay et al., 2020], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for 𝔽₂ polynomials with seed length close to the state-of-the-art construction due to Viola [Emanuele Viola, 2009].

Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, and Abhishek Shetty. Fractional Pseudorandom Generators from Any Fourier Level. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2021.10, author = {Chattopadhyay, Eshan and Gaitonde, Jason and Lee, Chin Ho and Lovett, Shachar and Shetty, Abhishek}, title = {{Fractional Pseudorandom Generators from Any Fourier Level}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {10:1--10:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.10}, URN = {urn:nbn:de:0030-drops-142843}, doi = {10.4230/LIPIcs.CCC.2021.10}, annote = {Keywords: Derandomization, pseudorandomness, pseudorandom generators, Fourier analysis} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2^{-Ω(n)}. We note that every matrix of sign-rank 2 has discrepancy n^{-O(1)}.
Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.

Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Sign Rank vs Discrepancy. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hatami_et_al:LIPIcs.CCC.2020.18, author = {Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar}, title = {{Sign Rank vs Discrepancy}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.18}, URN = {urn:nbn:de:0030-drops-125700}, doi = {10.4230/LIPIcs.CCC.2020.18}, annote = {Keywords: Discrepancy, sign rank, Unbounded-error communication complexity, weakly unbounded error communication complexity} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

The sunflower conjecture is one of the most well-known open problems in combinatorics. It has several applications in theoretical computer science, one of which is DNF compression, due to Gopalan, Meka and Reingold (Computational Complexity, 2013). In this paper, we show that improved bounds for DNF compression imply improved bounds for the sunflower conjecture, which is the reverse direction of the DNF compression result. The main approach is based on regularity of set systems and a structure-vs-pseudorandomness approach to the sunflower conjecture.

Shachar Lovett, Noam Solomon, and Jiapeng Zhang. From DNF Compression to Sunflower Theorems via Regularity. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lovett_et_al:LIPIcs.CCC.2019.5, author = {Lovett, Shachar and Solomon, Noam and Zhang, Jiapeng}, title = {{From DNF Compression to Sunflower Theorems via Regularity}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.5}, URN = {urn:nbn:de:0030-drops-108277}, doi = {10.4230/LIPIcs.CCC.2019.5}, annote = {Keywords: DNF sparsification, sunflower conjecture, regular set systems} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

We study the relation between streaming algorithms and linear sketching algorithms, in the context of binary updates. We show that for inputs in n dimensions, the existence of efficient streaming algorithms which can process Omega(n^2) updates implies efficient linear sketching algorithms with comparable cost. This improves upon the previous work of Li, Nguyen and Woodruff [Yi Li et al., 2014] and Ai, Hu, Li and Woodruff [Yuqing Ai et al., 2016] which required a triple-exponential number of updates to achieve a similar result for updates over integers. We extend our results to updates modulo p for integers p >= 2, and to approximation instead of exact computation.

Kaave Hosseini, Shachar Lovett, and Grigory Yaroslavtsev. Optimality of Linear Sketching Under Modular Updates. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hosseini_et_al:LIPIcs.CCC.2019.13, author = {Hosseini, Kaave and Lovett, Shachar and Yaroslavtsev, Grigory}, title = {{Optimality of Linear Sketching Under Modular Updates}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.13}, URN = {urn:nbn:de:0030-drops-108355}, doi = {10.4230/LIPIcs.CCC.2019.13}, annote = {Keywords: communication complexity, linear sketching, streaming algorithm} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it.
Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.

Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality Alone Does not Simulate Randomness. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2019.14, author = {Chattopadhyay, Arkadev and Lovett, Shachar and Vinyals, Marc}, title = {{Equality Alone Does not Simulate Randomness}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {14:1--14:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.14}, URN = {urn:nbn:de:0030-drops-108368}, doi = {10.4230/LIPIcs.CCC.2019.14}, annote = {Keywords: Communication lower bound, derandomization} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

We propose an algebraic approach to proving circuit lower bounds for ACC^0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC^0 and ACC^0 can be reformulated in this framework, implying that ACC^0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC^0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.

Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao. Torus Polynomials: An Algebraic Approach to ACC Lower Bounds. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bhrushundi_et_al:LIPIcs.ITCS.2019.13, author = {Bhrushundi, Abhishek and Hosseini, Kaave and Lovett, Shachar and Rao, Sankeerth}, title = {{Torus Polynomials: An Algebraic Approach to ACC Lower Bounds}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.13}, URN = {urn:nbn:de:0030-drops-101066}, doi = {10.4230/LIPIcs.ITCS.2019.13}, annote = {Keywords: Circuit complexity, ACC, lower bounds, polynomials} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH.
As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.

Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2019.22, author = {Chattopadhyay, Eshan and Hatami, Pooya and Lovett, Shachar and Tal, Avishay}, title = {{Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {22:1--22:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.22}, URN = {urn:nbn:de:0030-drops-101150}, doi = {10.4230/LIPIcs.ITCS.2019.22}, annote = {Keywords: Derandomization, Pseudorandom generator, Explicit construction, Random walk, Small-depth circuits with parity gates} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

The Erdös-Rado sunflower theorem (Journal of Lond. Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding sunflower conjecture is a central open problem. Motivated by applications in complexity theory, Rossman (FOCS 2010) extended the result to quasi-sunflowers, where similar conjectures emerge about the optimal parameters for which it holds.
In this work, we exhibit a surprising connection between the existence of sunflowers and quasi-sunflowers in large enough set systems, and the problem of constructing (or existing) certain randomness extractors. This allows us to re-derive the known results in a systematic manner, and to reduce the relevant conjectures to the problem of obtaining improved constructions of the randomness extractors.

Xin Li, Shachar Lovett, and Jiapeng Zhang. Sunflowers and Quasi-Sunflowers from Randomness Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{li_et_al:LIPIcs.APPROX-RANDOM.2018.51, author = {Li, Xin and Lovett, Shachar and Zhang, Jiapeng}, title = {{Sunflowers and Quasi-Sunflowers from Randomness Extractors}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {51:1--51:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.51}, URN = {urn:nbn:de:0030-drops-94555}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.51}, annote = {Keywords: Sunflower conjecture, Quasi-sunflowers, Randomness Extractors} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Let H be an arbitrary family of hyper-planes in d-dimensions. We show that the point-location problem for H can be solved by a linear decision tree that only uses a special type of queries called generalized comparison queries. These queries correspond to hyperplanes that can be written as a linear combination of two hyperplanes from H; in particular, if all hyperplanes in H are k-sparse then generalized comparisons are 2k-sparse. The depth of the obtained linear decision tree is polynomial in d and logarithmic in |H|, which is comparable to previous results in the literature that use general linear queries.
This extends the study of comparison trees from a previous work by the authors [Kane {et al.}, FOCS 2017]. The main benefit is that using generalized comparison queries allows to overcome limitations that apply for the more restricted type of comparison queries.
Our analysis combines a seminal result of Forster regarding sets in isotropic position [Forster, JCSS 2002], the margin-based inference dimension analysis for comparison queries from [Kane {et al.}, FOCS 2017], and compactness arguments.

Daniel M. Kane, Shachar Lovett, and Shay Moran. Generalized Comparison Trees for Point-Location Problems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 82:1-82:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kane_et_al:LIPIcs.ICALP.2018.82, author = {Kane, Daniel M. and Lovett, Shachar and Moran, Shay}, title = {{Generalized Comparison Trees for Point-Location Problems}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {82:1--82:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.82}, URN = {urn:nbn:de:0030-drops-90862}, doi = {10.4230/LIPIcs.ICALP.2018.82}, annote = {Keywords: linear decision trees, comparison queries, point location problems} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [-1,1]^n. Next, we use a fractional pseudorandom generator as steps of a random walk in [-1,1]^n that converges to {-1,1}^n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.

Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom Generators from Polarizing Random Walks. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2018.1, author = {Chattopadhyay, Eshan and Hatami, Pooya and Hosseini, Kaave and Lovett, Shachar}, title = {{Pseudorandom Generators from Polarizing Random Walks}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {1:1--1:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.1}, URN = {urn:nbn:de:0030-drops-88880}, doi = {10.4230/LIPIcs.CCC.2018.1}, annote = {Keywords: AC0, branching program, polarization, pseudorandom generators, random walks, Sensitivity} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire.
This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism.

Marco L. Carmosino, Russell Impagliazzo, Shachar Lovett, and Ivan Mihajlin. Hardness Amplification for Non-Commutative Arithmetic Circuits. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carmosino_et_al:LIPIcs.CCC.2018.12, author = {Carmosino, Marco L. and Impagliazzo, Russell and Lovett, Shachar and Mihajlin, Ivan}, title = {{Hardness Amplification for Non-Commutative Arithmetic Circuits}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.12}, URN = {urn:nbn:de:0030-drops-88772}, doi = {10.4230/LIPIcs.CCC.2018.12}, annote = {Keywords: arithmetic circuits, hardness amplification, circuit lower bounds, non-commutative computation} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

An important theorem of Banaszczyk (Random Structures & Algorithms 1998) states that for any sequence of vectors of l_2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R^n, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination.
We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric.
As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/sqrt{log n}), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)-subgaussian signing distributions when the vectors have length O(1/sqrt{log n}), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies.

Daniel Dadush, Shashwat Garg, Shachar Lovett, and Aleksandar Nikolov. Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 28:1-28:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dadush_et_al:LIPIcs.APPROX-RANDOM.2016.28, author = {Dadush, Daniel and Garg, Shashwat and Lovett, Shachar and Nikolov, Aleksandar}, title = {{Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {28:1--28:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.28}, URN = {urn:nbn:de:0030-drops-66512}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.28}, annote = {Keywords: Discrepancy, Vector Balancing, Convex Geometry} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Sigma), where each element x in X lies in t randomly selected sets of Sigma, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |\Sigma| >= |X| the hereditary discrepancy of (X,Sigma) is with high probability O(sqrt{t log t}); and when |X| >> |\Sigma|^t the hereditary discrepancy of (X,Sigma) is with high probability O(1). The first bound combines the Lovasz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.

Esther Ezra and Shachar Lovett. On the Beck-Fiala Conjecture for Random Set Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 29:1-29:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ezra_et_al:LIPIcs.APPROX-RANDOM.2016.29, author = {Ezra, Esther and Lovett, Shachar}, title = {{On the Beck-Fiala Conjecture for Random Set Systems}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {29:1--29:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.29}, URN = {urn:nbn:de:0030-drops-66526}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.29}, annote = {Keywords: Discrepancy theory, Beck-Fiala conjecture, Random set systems} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We prove that for any constant k and any epsilon < 1, there exist bimatrix win-lose games for which every epsilon-WSNE requires supports of cardinality greater than k. To do this, we provide a graph-theoretic characterization of win-lose games that possess epsilon-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight to construct win-lose games that do not satisfy the requirements of the characterization. These constructions disprove graph theoretic conjectures of Daskalakis, Mehta and Papadimitriou and Myers.

Yogesh Anbalagan, Hao Huang, Shachar Lovett, Sergey Norin, Adrian Vetta, and Hehui Wu. Large Supports are Required for Well-Supported Nash Equilibria. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 78-84, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{anbalagan_et_al:LIPIcs.APPROX-RANDOM.2015.78, author = {Anbalagan, Yogesh and Huang, Hao and Lovett, Shachar and Norin, Sergey and Vetta, Adrian and Wu, Hehui}, title = {{Large Supports are Required for Well-Supported Nash Equilibria}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {78--84}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.78}, URN = {urn:nbn:de:0030-drops-52959}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.78}, annote = {Keywords: bimatrix games, well-supported Nash equilibria} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.

Abhishek Bhowmick and Shachar Lovett. Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 72-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bhowmick_et_al:LIPIcs.CCC.2015.72, author = {Bhowmick, Abhishek and Lovett, Shachar}, title = {{Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {72--87}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.72}, URN = {urn:nbn:de:0030-drops-50491}, doi = {10.4230/LIPIcs.CCC.2015.72}, annote = {Keywords: nonclassical polynomials, polynomials, lower bounds, barrier} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Linearity tests are randomized algorithms which have oracle access
to the truth table of some function f, and are supposed to
distinguish between linear functions and functions which are far
from linear. Linearity tests were first introduced by (Blum, Luby
and Rubenfeld, 1993), and were later used in the PCP theorem,
among other applications. The quality of a linearity test is
described by its correctness c - the probability it accepts linear
functions, its soundness s - the probability it accepts functions
far from linear, and its query complexity q - the number of queries
it makes.
Linearity tests were studied in order to decrease the soundness of
linearity tests, while keeping the query complexity small (for one
reason, to improve PCP constructions). Samorodnitsky and Trevisan
(Samorodnitsky and Trevisan 2000) constructed the Complete Graph
Test, and prove that no Hyper Graph Test can perform better than
the Complete Graph Test. Later in (Samorodnitsky and Trevisan
2006) they prove, among other results, that no non-adaptive
linearity test can perform better than the Complete Graph Test.
Their proof uses the algebraic machinery of the Gowers Norm. A
result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to
generalize this lower bound also to adaptive linearity tests.
We also prove the same optimal lower bound for adaptive linearity
test, but our proof technique is arguably simpler and more direct
than the one used in (Samorodnitsky and Trevisan 2006). We also
study, like (Samorodnitsky and Trevisan 2006), the behavior of
linearity tests on quadratic functions. However, instead of
analyzing the Gowers Norm of certain functions, we provide a more
direct combinatorial proof, studying the behavior of linearity
tests on random quadratic functions. This proof technique also
lets us prove directly the lower bound also for adaptive linearity
tests.

Shachar Lovett. Lower bounds for adaptive linearity tests. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 515-526, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{lovett:LIPIcs.STACS.2008.1313, author = {Lovett, Shachar}, title = {{Lower bounds for adaptive linearity tests}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {515--526}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1313}, URN = {urn:nbn:de:0030-drops-13132}, doi = {10.4230/LIPIcs.STACS.2008.1313}, annote = {Keywords: Property testing, Linearity testing, Adaptive tests, Lower Property testing, Linearity testing, Adaptive tests, Lower} }