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All reports by Author Avi Wigderson:

TR16-129 | 16th August 2016
Emanuele Viola, Avi Wigderson

Local Expanders

Revisions: 1

Abstract A map $f:{0,1}^{n}\to {0,1}^{n}$ has locality t if every output bit of f depends only on t input bits. Arora, Steurer, and Wigderson (2009) ask if there exist bounded-degree expander graphs on $2^{n}$ nodes such that the neighbors of a node $x\in {0,1}^{n}$ can be computed by maps of ... more >>>

TR16-098 | 16th June 2016
Michael Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson

Proof Complexity Lower Bounds from Algebraic Circuit Complexity

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the ... more >>>

TR16-069 | 25th April 2016
Parikshit Gopalan, Rocco Servedio, Avishay Tal, Avi Wigderson

Degree and Sensitivity: tails of two distributions

The sensitivity of a Boolean function $f$ is the maximum, over all inputs $x$, of the number of sensitive coordinates of $x$ (namely the number of Hamming neighbors of $x$ with different $f$-value). The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-$s$ Boolean function ... more >>>

TR15-165 | 14th October 2015
Ran Gelles, Bernhard Haeupler, Gillat Kol, Noga Ron-Zewi, Avi Wigderson

Towards Optimal Deterministic Coding for Interactive Communication

Revisions: 1

We study \emph{efficient, deterministic} interactive coding schemes that simulate any interactive protocol both under random and adversarial errors, and can achieve a constant communication rate independent of the protocol length.

For channels that flip bits independently with probability~$\epsilon<1/2$, our coding scheme achieves a communication rate of $1 - O(\sqrt{H({\epsilon})})$ and ... more >>>

TR15-131 | 10th August 2015
Parikshit Gopalan, Noam Nisan, Rocco Servedio, Kunal Talwar, Avi Wigderson

Smooth Boolean functions are easy: efficient algorithms for low-sensitivity functions

A natural measure of smoothness of a Boolean function is its sensitivity (the largest number of Hamming neighbors of a point which differ from it in function value). The structure of smooth or equivalently low-sensitivity functions is still a mystery. A well-known conjecture states that every such Boolean function can ... more >>>

TR15-025 | 22nd February 2015
Shay Moran, Amir Shpilka, Avi Wigderson, Amir Yehudayoff

Teaching and compressing for low VC-dimension

In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. We present relatively efficient constructions of {\em sample compression schemes} and
for classes of low VC-dimension. Let $C$ be a finite boolean concept class of VC-dimension $d$. Set $k ... more >>>

TR15-003 | 3rd January 2015
Oded Goldreich, Emanuele Viola, Avi Wigderson

On Randomness Extraction in AC0

We consider randomness extraction by AC0 circuits. The main parameter, $n$, is the length of the source, and all other parameters are functions of it. The additional extraction parameters are the min-entropy bound $k=k(n)$, the seed length $r=r(n)$, the output length $m=m(n)$, and the (output) deviation bound $\epsilon=\epsilon(n)$.

For $k ... more >>>

TR13-190 | 28th December 2013
Dmitry Gavinsky, Or Meir, Omri Weinstein, Avi Wigderson

Toward Better Formula Lower Bounds: An Information Complexity Approach to the KRW Composition Conjecture

Revisions: 10

One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}_1~$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the ... more >>>

TR13-160 | 20th November 2013
Zeev Dvir, Shubhangi Saraf, Avi Wigderson

Breaking the quadratic barrier for 3-LCCs over the Reals

We prove that 3-query linear locally correctable codes over the Reals of dimension $d$ require block length $n>d^{2+\lambda}$ for some fixed, positive $\lambda >0$. Geometrically, this means that if $n$ vectors in $R^d$ are such that each vector is spanned by a linear number of disjoint triples of others, then ... more >>>

TR13-152 | 7th November 2013
Oded Goldreich, Avi Wigderson

On Derandomizing Algorithms that Err Extremely Rarely

Revisions: 2

{\em Does derandomization of probabilistic algorithms become easier when the number of ``bad'' random inputs is extremely small?}

In relation to the above question, we put forward the following {\em quantified derandomization challenge}:
For a class of circuits $\cal C$ (e.g., P/poly or $AC^0,AC^0[2]$) and a bounding function $B:\N\to\N$ (e.g., ... more >>>

TR13-105 | 29th July 2013
Raghu Meka, Avi Wigderson

Association schemes, non-commutative polynomial concentration, and sum-of-squares lower bounds for planted clique

Revisions: 1

Finding cliques in random graphs and the closely related ``planted'' clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for t = ... more >>>

TR13-043 | 25th March 2013
Oded Goldreich, Avi Wigderson

On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

Revisions: 1

We propose that multi-linear functions of relatively low degree
over GF(2) may be good candidates for obtaining exponential
lower bounds on the size of constant-depth Boolean circuits
(computing explicit functions).
Specifically, we propose to move gradually from linear functions
to multilinear ones, and conjecture that, for any $t\geq2$,
more >>>

TR12-139 | 2nd November 2012
Albert Ai, Zeev Dvir, Shubhangi Saraf, Avi Wigderson

Sylvester-Gallai type theorems for approximate collinearity

We study questions in incidence geometry where the precise position of points is `blurry' (e.g. due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of ... more >>>

TR12-138 | 2nd November 2012
Zeev Dvir, Shubhangi Saraf, Avi Wigderson

Improved rank bounds for design matrices and a new proof of Kelly's theorem

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in which they were used to answer questions regarding point configurations. In ... more >>>

TR12-118 | 18th September 2012
Avi Wigderson, Amir Yehudayoff

Population Recovery and Partial Identification

We study several problems in which an {\em unknown} distribution over an {\em unknown} population of vectors needs to be recovered from partial or noisy samples, each of which nearly completely erases or obliterates the original vector. For example, consider a distribution $p$ over a population $V \subseteq \{0,1\}^n$. A ... more >>>

TR11-160 | 1st December 2011
Zeev Dvir, Anup Rao, Avi Wigderson, Amir Yehudayoff

Restriction Access

We introduce a notion of non-black-box access to computational devices (such as circuits, formulas, decision trees, and so forth) that we call \emph{restriction access}. Restrictions are partial assignments to input variables. Each restriction simplifies the device, and yields a new device for the restricted function on the unassigned variables. On ... more >>>

TR10-149 | 22nd September 2010
Boaz Barak, Zeev Dvir, Avi Wigderson, Amir Yehudayoff

Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

Revisions: 1

A $(q,k,t)$-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most $q$ non-zeros, each column has at least $k$ non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank ... more >>>

TR10-040 | 10th March 2010
Pavel Hrubes, Avi Wigderson, Amir Yehudayoff

Relationless completeness and separations

This paper extends Valiant's work on $\vp$ and $\vnp$ to the settings in which variables are not multiplicatively commutative and/or associative. Our main result is a theory of completeness for these algebraic worlds.
We define analogs of Valiant's classes $\vp$ and $\vnp$, as well as of the polynomials permanent ... more >>>

TR10-037 | 8th March 2010
Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, Avi Wigderson

Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors

We present new explicit constructions of *deterministic* randomness extractors, dispersers and related objects. We say that a
distribution $X$ on binary strings of length $n$ is a
$\delta$-source if $X$ assigns probability at most $2^{-\delta n}$
to any string of length $n$. For every $\delta>0$ we construct the
following poly($n$)-time ... more >>>

TR10-021 | 21st February 2010
Pavel Hrubes, Avi Wigderson, Amir Yehudayoff

Non-commutative circuits and the sum-of-squares problem

We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of \emph{non-commutative} arithmetic circuits and a problem about \emph{commutative} degree four polynomials, the classical sum-of-squares problem: find the smallest $n$ such that ... more >>>

TR09-135 | 10th December 2009
Zeev Dvir, Avi Wigderson

Monotone expanders - constructions and applications

The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to:
(1) Constant degree dimension expanders in finite ... more >>>

TR09-090 | 6th October 2009
Russell Impagliazzo, Valentine Kabanets, Avi Wigderson

New Direct-Product Testers and 2-Query PCPs

The “direct product code” of a function f gives its values on all k-tuples (f(x1), . . . , f(xk)).
This basic construct underlies “hardness amplification” in cryptography, circuit complexity and
PCPs. Goldreich and Safra [GS00] pioneered its local testing and its PCP application. A recent
result by Dinur and ... more >>>

TR09-084 | 24th September 2009
Arkadev Chattopadhyay, Avi Wigderson

Linear systems over composite moduli

We study solution sets to systems of generalized linear equations of the following form:
$\ell_i (x_1, x_2, \cdots , x_n)\, \in \,A_i \,\, (\text{mod } m)$,
where $\ell_1, \ldots ,\ell_t$ are linear forms in $n$ Boolean variables, each $A_i$ is an arbitrary subset of $\mathbb{Z}_m$, and $m$ is a composite ... more >>>

TR08-079 | 31st August 2008
Russell Impagliazzo, Ragesh Jaiswal, Valentine Kabanets, Avi Wigderson

Uniform Direct-Product Theorems: Simplified, Optimized, and Derandomized

The classical Direct-Product Theorem for circuits says
that if a Boolean function $f:\{0,1\}^n\to\{0,1\}$ is somewhat hard
to compute on average by small circuits, then the corresponding
$k$-wise direct product function
$f^k(x_1,\dots,x_k)=(f(x_1),\dots,f(x_k))$ (where each
$x_i\in\{0,1\}^n$) is significantly harder to compute on average by
slightly smaller ... more >>>

TR08-058 | 1st June 2008
Zeev Dvir, Avi Wigderson

Kakeya sets, new mergers and old extractors

A merger is a probabilistic procedure which extracts the
randomness out of any (arbitrarily correlated) set of random
variables, as long as one of them is uniform. Our main result is
an efficient, simple, optimal (to constant factors) merger, which,
for $k$ random vairables on $n$ bits each, uses a ... more >>>

TR08-005 | 15th January 2008
Scott Aaronson, Avi Wigderson

Algebrization: A New Barrier in Complexity Theory

Any proof of P!=NP will have to overcome two barriers: relativization
and natural proofs. Yet over the last decade, we have seen circuit
lower bounds (for example, that PP does not have linear-size circuits)
that overcome both barriers simultaneously. So the question arises of
whether there ... more >>>

TR07-079 | 11th August 2007
Emanuele Viola, Avi Wigderson

One-way multi-party communication lower bound for pointer jumping with applications

In this paper we study the one-way multi-party communication model,
in which every party speaks exactly once in its turn. For every
fixed $k$, we prove a tight lower bound of
$\Omega{n^{1/(k-1)}}$ on the probabilistic communication
complexity of pointer jumping in a $k$-layered tree, where the
pointers of the $i$-th ... more >>>

TR07-056 | 10th July 2007
Zeev Dvir, Ariel Gabizon, Avi Wigderson

Extractors and Rank Extractors for Polynomial Sources

In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial ... more >>>

TR06-118 | 2nd September 2006
Irit Dinur, Madhu Sudan, Avi Wigderson

Robust Local Testability of Tensor Products of LDPC Codes

Given two binary linear codes R and C, their tensor product R \otimes C consists of all matrices with rows in R and columns in C. We analyze the "robustness" of the following test for this code (suggested by Ben-Sasson and Sudan~\cite{BenSasson-Sudan04}): Pick a random row (or column) and check ... more >>>

TR06-105 | 23rd August 2006
Avi Wigderson, David Xiao

Derandomizing the AW matrix-valued Chernoff bound using pessimistic estimators and applications

Ahlswede and Winter introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan). As a consequence, we derandomize a construction of Alon ... more >>>

TR05-107 | 28th September 2005
Avi Wigderson, David Xiao

A Randomness-Efficient Sampler for Matrix-valued Functions and Applications

Revisions: 1

In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter, in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is ... more >>>

TR02-039 | 30th June 2002
Oded Goldreich, Avi Wigderson

Derandomization that is rarely wrong from short advice that is typically good

Comments: 1

For every $\epsilon>0$,
we present a {\em deterministic}\/ log-space algorithm
that correctly decides undirected graph connectivity
on all but at most $2^{n^\epsilon}$ of the $n$-vertex graphs.
The same holds for every problem in Symmetric Log-space (i.e., $\SL$).

Making no assumptions (and in particular not assuming the ... more >>>

TR01-046 | 2nd July 2001
Oded Goldreich, Salil Vadhan, Avi Wigderson

On Interactive Proofs with a Laconic Prover

We continue the investigation of interactive proofs with bounded
communication, as initiated by Goldreich and Hastad (IPL 1998).
Let $L$ be a language that has an interactive proof in which the prover
sends few (say $b$) bits to the verifier.
We prove that the complement $\bar L$ has ... more >>>

TR01-018 | 23rd February 2001
Omer Reingold, Salil Vadhan, Avi Wigderson

Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors

The main contribution of this work is a new type of graph product, which we call the zig-zag
product. Taking a product of a large graph with a small graph, the resulting graph inherits
(roughly) its size from the large one, its degree from the small one, and ... more >>>

TR00-059 | 11th August 2000
Omer Reingold, Ronen Shaltiel, Avi Wigderson

Extracting Randomness via Repeated Condensing

On an input probability distribution with some (min-)entropy
an {\em extractor} outputs a distribution with a (near) maximum
entropy rate (namely the uniform distribution).
A natural weakening of this concept is a condenser, whose
output distribution has a higher entropy rate than the
input distribution (without losing
much of ... more >>>

TR00-056 | 20th July 2000
Oded Goldreich, Avi Wigderson

On Pseudorandomness with respect to Deterministic Observers.

In the theory of pseudorandomness, potential (uniform) observers
are modeled as probabilistic polynomial-time machines.
In fact many of the central results in
that theory are proven via probabilistic polynomial-time reductions.
In this paper we show that analogous deterministic reductions
are unlikely to hold. We conclude that randomness ... more >>>

TR00-023 | 11th May 2000
Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

Pseudorandom Generators in Propositional Proof Complexity

We call a pseudorandom generator $G_n:\{0,1\}^n\to \{0,1\}^m$ {\em
hard} for a propositional proof system $P$ if $P$ can not efficiently
prove the (properly encoded) statement $G_n(x_1,\ldots,x_n)\neq b$ for
{\em any} string $b\in\{0,1\}^m$. We consider a variety of
``combinatorial'' pseudorandom generators inspired by the
Nisan-Wigderson generator on the one hand, and ... more >>>

TR00-009 | 21st February 2000
Russell Impagliazzo, Ronen Shaltiel, Avi Wigderson

Extractors and pseudo-random generators with optimal seed length

We give the first construction of a pseudo-random generator with
optimal seed length that uses (essentially) arbitrary hardness.
It builds on the novel recursive use of the NW-generator in
a previous paper by the same authors, which produced many optimal
generators one of which was pseudo-random. This is achieved ... more >>>

TR00-005 | 17th January 2000
Eli Ben-Sasson, Russell Impagliazzo, Avi Wigderson

Near-Optimal Separation of Treelike and General Resolution

We present the best known separation
between tree-like and general resolution, improving
on the recent $\exp(n^\epsilon)$ separation of \cite{BEGJ98}.
This is done by constructing a natural family of contradictions, of
size $n$, that have $O(n)$-size resolution
refutations, but only $\exp (\Omega(n/\log n))$-size tree-like refutations.
This result ... more >>>

TR00-004 | 14th January 2000
Oded Goldreich, Salil Vadhan, Avi Wigderson

Simplified derandomization of BPP using a hitting set generator.

A hitting-set generator is a deterministic
algorithm which generates a set of strings that intersects
every dense set recognizable by a small circuit.
A polynomial time hitting-set generator readily implies $RP=P$.
Andreev \etal\/ (ICALP'96, and JACM 1998)
showed that if polynomial-time hitting-set
generator in fact implies ... more >>>

TR99-040 | 20th October 1999
Michael Alekhnovich, Eli Ben-Sasson, Alexander Razborov, Avi Wigderson

Space Complexity in Propositional Calculus

We study space complexity in the framework of
propositional proofs. We consider a natural model analogous to
Turing machines with a read-only input tape, and such
popular propositional proof systems as Resolution, Polynomial
Calculus and Frege systems. We propose two different space measures,
corresponding to the maximal number of bits, ... more >>>

TR99-023 | 16th June 1999
Amir Shpilka, Avi Wigderson

Depth-3 Arithmetic Formulae over Fields of Characteristic Zero

In this paper we prove near quadratic lower bounds for
depth-3 arithmetic formulae over fields of characteristic zero.
Such bounds are obtained for the elementary symmetric
functions, the (trace of) iterated matrix multiplication, and the
determinant. As corollaries we get the first nontrivial lower
bounds for ... more >>>

TR99-022 | 14th June 1999
Eli Ben-Sasson, Avi Wigderson

Short Proofs are Narrow - Resolution made Simple

The width of a Resolution proof is defined to be the maximal number of
literals in any clause of the proof. In this paper we relate proof width
to proof length (=size), in both general Resolution, and its tree-like
variant. The following consequences of these relations reveal width as ... more >>>

TR98-072 | 14th December 1998
Ziv Bar-Yossef, Oded Goldreich, Avi Wigderson

Deterministic Amplification of Space Bounded Probabilistic Algorithms.

This paper initiates the study of deterministic amplification of space
bounded probabilistic algorithms. The straightforward implementations of
known amplification methods cannot be used for such algorithms, since they
consume too much space. We present a new implementation of the
Ajtai-Koml\'{o}s-Szemer\'{e}di method, that enables to amplify an $S$ ... more >>>

TR96-041 | 24th July 1996
Oded Goldreich, Avi Wigderson

On the Circuit Complexity of Perfect Hashing

Revisions: 1 , Comments: 2

We consider the size of circuits which perfectly hash
an arbitrary subset $S\!\subset\!\bitset^n$ of cardinality $2^k$
into $\bitset^m$.
We observe that, in general, the size of such circuits is
exponential in $2k-m$,
and provide a matching upper bound.

more >>>

TR95-050 | 15th October 1995
Oded Goldreich, Noam Nisan, Avi Wigderson

On Yao's XOR-Lemma

Revisions: 2 , Comments: 1

TR95-049 | 19th October 1995
Anna Gal, Avi Wigderson

Boolean complexity classes vs. their arithmetic analogs

This paper provides logspace and small circuit depth analogs
of the result of Valiant-Vazirani, which is a randomized (or
nonuniform) reduction from NP to its arithmetic analog ParityP.
We show a similar randomized reduction between the
Boolean classes NL and semi-unbounded fan-in Boolean circuits and
their arithmetic counterparts. These ... more >>>

TR94-002 | 12th December 1994
Oded Goldreich, Avi Wigderson

Tiny Families of Functions with Random Properties: A Quality--Size Trade--off for Hashing

Revisions: 2

We present three explicit constructions of hash functions,
which exhibit a trade-off between the size of the family
(and hence the number of random bits needed to generate a member of the family),
and the quality (or error parameter) of the pseudo-random property it
achieves. Unlike previous constructions, ... more >>>

TR94-001 | 12th December 1994
Noam Nisan, Avi Wigderson

On Rank vs. Communication Complexity

This paper concerns the open problem of Lovasz and
Saks regarding the relationship between the communication complexity
of a boolean function and the rank of the associated matrix.
We first give an example exhibiting the largest gap known. We then
prove two related theorems.

more >>>

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