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Electronic Colloquium on Computational Complexity

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All reports by Author Shachar Lovett:

TR19-145 | 31st October 2019
Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, David Zuckerman

XOR Lemmas for Resilient Functions Against Polynomials

A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over $F_2$. We introduce a new technique to prove such correlation bounds with $F_2$ polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In ... more >>>

TR19-110 | 23rd August 2019
Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang

Improved bounds for the sunflower lemma

A sunflower with $r$ petals is a collection of $r$ sets so that the
intersection of each pair is equal to the intersection of all. Erd\H{o}s and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must ... more >>>

TR19-067 | 6th May 2019
Hamed Hatami, Kaave Hosseini, Shachar Lovett

Sign rank vs Discrepancy

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 ... more >>>

TR19-028 | 1st March 2019
Shachar Lovett, Noam Solomon, Jiapeng Zhang

From DNF compression to sunflower theorems via regularity

Revisions: 1

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 ... more >>>

TR18-206 | 3rd December 2018
Arkadev Chattopadhyay, Shachar Lovett, Marc Vinyals

Equality Alone Does Not Simulate Randomness

Revisions: 1

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 ... more >>>

TR18-155 | 8th September 2018
Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, Avishay Tal

Pseudorandom generators from the second Fourier level and applications to AC0 with parity gates

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 ... more >>>

TR18-142 | 17th August 2018
Kaave Hosseini, Shachar Lovett

A bilinear Bogolyubov-Ruzsa lemma with poly-logarithmic bounds

The Bogolyubov-Ruzsa lemma, in particular the quantitative bounds obtained by Sanders, plays a central role
in obtaining effective bounds for the inverse $U^3$ theorem for the Gowers norms. Recently, Gowers and Mili\'cevi\'c
applied a bilinear Bogolyubov-Ruzsa lemma as part of a proof of the inverse $U^4$ theorem
with effective bounds.
more >>>

TR18-095 | 11th May 2018
Marco Carmosino, Russell Impagliazzo, Shachar Lovett, Ivan Mihajlin

Hardness Amplification for Non-Commutative Arithmetic Circuits

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 ... more >>>

TR18-076 | 22nd April 2018
Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, Sankeerth Rao

Torus polynomials: an algebraic approach to ACC lower bounds

Revisions: 2

We propose an algebraic approach to proving circuit lower bounds for ACC0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC0 and ACC0 can be reformulated in this framework, implying that ACC0 can be approximated by low-degree torus polynomials. Furthermore, ... more >>>

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