We study the $n$-party noisy broadcast channel with a constant fraction of malicious parties. Specifically, we assume that each non-malicious party holds an input bit, and communicates with the others in order to learn the input bits of all non-malicious parties. In each communication round, one of the parties broadcasts ... more >>>

We present new constructions of pseudorandom generators (PRGs) for two of the most widely-studied non-uniform circuit classes in complexity theory. Our main result is a construction of the first non-trivial PRG for linear threshold (LTF) circuits of arbitrary constant depth and super-linear size. This PRG fools circuits with depth $d\in\mathbb{N}$ ... more >>>

In this work we prove that there is a function $f \in \textrm{E}^\textrm{NP}$ such that, for every sufficiently large $n$ and $d = \sqrt{n}/\log n$, $f_n$ ($f$ restricted to $n$-bit inputs) cannot be $(1/2 + 2^{-d})$-approximated by $\textrm{F}_2$-polynomials of degree $d$. We also observe that a minor improvement ... more >>>

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function $f:\{\pm1\}^{n} \to \{\pm1\}$ we give a poly$(k, \frac{1}{\varepsilon})$ query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from ... more >>>

A natural problem in high-dimensional inference is to decide if a classifier $f:\mathbb{R}^n \rightarrow \{-1,1\}$ depends on a small number of linear directions of its input data. Call a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$, a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. ... more >>>

We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large ... more >>>

Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be $d$-dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. ... more >>>

Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and n be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, for a sufficiently small ? > 0, one removes
?dn ... more >>>

One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence ... more >>>

A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT ... more >>>

A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal ... more >>>

The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical ... more >>>

In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary $m$-facet polytopes in $n$ variables with seed length poly-logarithmic in $m,n$, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, ... more >>>

In this paper we study polynomials in VP$_e$ (polynomial-sized formulas) and in $\Sigma\Pi\Sigma$ (polynomial-size depth-$3$ circuits) whose orbits, under the action of the affine group GL$^{aff}_n({\mathbb F})$, are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms.

As ... more >>>

The orbit of an $n$-variate polynomial $f(\mathbf{x})$ over a field $\mathbb{F}$ is the set $\mathrm{orb}(f) := \{f(A\mathbf{x}+\mathbf{b}) : A \in \mathrm{GL}(n,\mathbb{F}) \ \mathrm{and} \ \mathbf{b} \in \mathbb{F}^n\}$. This paper studies explicit hitting sets for the orbits of polynomials computable by certain well-studied circuit classes. This version of the hitting set ... more >>>

We exhibit an unambiguous $k$-DNF formula that requires CNF width $\tilde{\Omega}(k^{1.5})$. Our construction is inspired by the board game Hex and it is vastly simpler than previous ones, which achieved at best an exponent of $1.22$. Our result is known to imply several other improved separations in query and communication ... more >>>

We initiate a systematic study of mixing in non-quasirandom groups.
Let $A$ and $B$ be two independent, high-entropy distributions over
a group $G$. We show that the product distribution $AB$ is statistically
close to the distribution $F(AB)$ for several choices of $G$ and
$F$, including:

(1) $G$ is the affine ... more >>>

We study monotone branching programs, wherein 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.

We prove that constant-width monotone branching programs of ... more >>>

A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ... more >>>

Weighted pseudorandom generators (WPRGs), introduced by Braverman, Cohen and Garg [BCG20], is a generalization of pseudorandom generators (PRGs) in which arbitrary real weights are considered rather than a probability mass. Braverman et al. constructed WPRGs against read once branching programs (ROBPs) with near-optimal dependence on the error parameter. Chattopadhyay and ... more >>>

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n/\log n)$ in the Polynomial ... more >>>

We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in ... more >>>

Proving circuit lower bounds has been an important but extremely hard problem for decades. Although one may show that almost every function $f:\mathbb{F}_2^n\to\mathbb{F}_2$ requires circuit of size $\Omega(2^n/n)$ by a simple counting argument, it remains unknown whether there is an explicit function (for example, a function in $NP$) not computable ... more >>>

We show that computing the majority of $n$ copies of a boolean function $g$ has randomised query complexity $\mathrm{R}(\mathrm{Maj} \circ g^n) = \Theta(n\cdot \bar{\mathrm{R}}_{1/n}(g))$. In fact, we show that to obtain a similar result for any composed function $f\circ g^n$, it suffices to prove a sufficiently strong form of the ... more >>>

An $(n,r,h,a,q)$-Local Reconstruction Code is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to recover from `$a$' erasures in that local group and there are further $h$ global parity ... more >>>

Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their ... more >>>

We give an almost quadratic $n^{2-o(1)}$ lower bound on the space consumption of any $o(\sqrt{\log n})$-pass streaming algorithm solving the (directed) $s$-$t$ reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including ... more >>>

Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. '95 who showed the equivalence between regular Resolution and read-once branching programs ... more >>>

Suppose that a problem $\Pi$ has a statistical zero-knowledge (SZK) proof with communication complexity $m$. The question of batch verification for SZK asks whether one can prove that $k$ instances $x_1,\ldots,x_k$ all belong to $\Pi$ with a statistical zero-knowledge proof whose communication complexity is better than $k \cdot m$ (which ... more >>>

How difficult is it to compute the communication complexity of a two-argument total Boolean function $f:[N]\times[N]\to\{0,1\}$, when it is given as an $N\times N$ binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard.

In this ... more >>>

We prove that all functions that have low degree torus polynomials approximating them with small error also have $MidBit^+$ circuits computing them. This serves as a partial converse to the result that all $ACC$ functions have low degree torus polynomials approximating them with small error, by Bhrushundi, Hosseini, Lovett and ... more >>>

Shortly after the introduction of zero-knowledge proofs, Goldreich, Micali and Wigderson (CRYPTO '86) demonstrated their wide applicability by constructing zero-knowledge proofs for the NP-complete problem of graph 3-coloring. A long-standing open question has been whether parallel repetition of their protocol preserves zero knowledge. In this work, we answer this question ... more >>>

We show that tree-like resolution is not automatable in time $n^{o(\log n)}$ unless ETH is false. This implies that, under ETH, the algorithm given by Beame and Pitassi (FOCS 1996) that automates tree-like resolution in time $n^{O(\log n)}$ is optimal. We also provide a simpler proof of the result of ... more >>>

We study two notions that refers to asymmetric graphs, which we view as graphs having a unique ordering that can be reconstructed by looking at an unlabeled version of the graph.

A {\em local self-ordering} procedure for a graph $G$ is given oracle access to an arbitrary isomorphic copy of ... more >>>

We study the amortized circuit complexity of boolean functions.

Given a circuit model $\mathcal{F}$ and a boolean function $f : \{0,1\}^n \rightarrow \{0,1\}$, the $\mathcal{F}$-amortized circuit complexity is defined to be the size of the smallest circuit that outputs $m$ copies of $f$ (evaluated on the same input), ... more >>>

In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their ... more >>>

The motivating question for this work is a long standing open problem, posed by Nisan (1991), regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question continues to remain open, we make some progress towards its resolution. To that effect, ... more >>>

We prove that Kilian's four-message succinct argument system is post-quantum secure in the standard model when instantiated with any probabilistically checkable proof and any collapsing hash function (which in turn exist based on the post-quantum hardness of Learning with Errors).

At the heart of our proof is a new ... more >>>

We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $BPTIME$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows.

1. A new pseudorandom generator and its consequences: We build on techniques developed to ... more >>>

We develop a general framework that characterizes strong average-case lower bounds against circuit classes $\mathcal{C}$ contained in $\mathrm{NC}^1$, such as $\mathrm{AC}^0[\oplus]$ and $\mathrm{ACC}^0$. We apply this framework to show:

- Generic seed reduction: Pseudorandom generators (PRGs) against $\mathcal{C}$ of seed length $\leq n -1$ and error $\varepsilon(n) = n^{-\omega(1)}$ can ... more >>>

A probabilistic representation of a string $x \in \{0,1\}^n$ is given by the code of a randomized algorithm that outputs $x$ with high probability [Oliveira, ICALP 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ... more >>>

We show that NP-hardness of approximating Boolean unique games on small set expanders can be amplified to the full Unique Games Conjecture on small set expanders.
The latter conjecture is known to imply hardness results for problems like Balanced-Separator, Minimum-Linear-Rearrangement and Small-Set-Expansion that are not known under the Unique ... more >>>

The Acceptance Probability Estimation Problem (APEP) is to additively approximate the acceptance probability of a Boolean circuit. This problem admits a probabilistic approximation scheme. A central question is whether we can design a pseudodeterministic approximation algorithm for this problem: a probabilistic polynomial-time algorithm that outputs a canonical approximation with high ... more >>>

Finding exact circuit size is a notorious optimization problem in practice. Whereas modern computers and algorithmic techniques allow to find a circuit of size seven in blink of an eye, it may take more than a week to search for a circuit of size thirteen. One of the reasons of ... more >>>

We give new and efficient black-box reconstruction algorithms for some classes of depth-$3$ arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a {\it constant-rank} tensor. ... more >>>

We prove that for every parity decision tree of depth $d$ on $n$ variables, the sum of absolute values of Fourier coefficients at level $\ell$ is at most $d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$.
Our result is nearly tight for small values of $\ell$ and extends a previous Fourier bound ... more >>>

A polynomial threshold function (PTF) $f:\mathbb{R}^n \rightarrow \mathbb{R}$ 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 ... more >>>

Three decades ago, Nisan constructed an explicit pseudorandom generator (PRG) that fools width-$n$ length-$n$ read-once branching programs (ROBPs) with error $\varepsilon$ and seed length $O(\log^2 n + \log n \cdot \log(1/\varepsilon))$ (Combinatorica 1992). Nisan's generator remains the best explicit PRG known for this important model of computation. However, a recent ... more >>>

We call any consistent and sufficiently powerful formal theory that enables to algorithmically in polynomial time verify whether a text is a proof \textbf{efficiently verifiable mathematics} (ev-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of ev-mathematics. Our main results ... more >>>

Motivated in part by applications in lattice-based cryptography, we initiate the study of the size of linear threshold (`$t$-out-of-$n$') secret-sharing where the linear reconstruction function is restricted to coefficients in $\{0,1\}$. We prove upper and lower bounds on the share size of such schemes. One ramification of our results is ... more >>>

Interactive error correcting codes are codes that encode a two party communication protocol to an error-resilient protocol that succeeds even if a constant fraction of the communicated symbols are adversarially corrupted, at the cost of increasing the communication by a constant factor. What is the largest fraction of corruptions that ... more >>>

A secret-sharing scheme allows to distribute a secret $s$ among $n$ parties such that only some predefined ``authorized'' sets of parties can reconstruct the secret, and all other ``unauthorized'' sets learn nothing about $s$.
The collection of authorized/unauthorized sets can be captured by a monotone function $f:\{0,1\}^n\rightarrow \{0,1\}$.
more >>>

We study from the proof complexity perspective the (informal) proof search problem:
Is there an optimal way to search for propositional proofs?
We note that for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal ... more >>>

We show that the Tree Evaluation Problem with alphabet size $k$ and height $h$ can be solved by branching programs of size $k^{O(h/\log h)} + 2^{O(h)}$. This answers a longstanding challenge of Cook et al. (2009) and gives the first general upper bound since the problem's inception.

Let $\mktp[s]$ be the set of strings $x$ such that $K^t(x) \leq s(|x|)$, where $K^t(x)$ denotes the $t$-bounded Kolmogorov complexity of the truthtable described by $x$. Our main theorem shows that for an appropriate notion of mild average-case hardness, for every $\varepsilon>0$, polynomial $t(n) \geq (1+\varepsilon)n$, and every ``nice'' class ... more >>>

Liu and Pass (FOCS'20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity---namely, Levin's notion of Kolmogorov Complexity---whose hardness is closely ... more >>>

A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity K^t is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures:

1. We show, perhaps surprisingly, that the ... more >>>

A long-standing and central open question in the theory of average-case complexity is to base average-case hardness of NP on worst-case hardness of NP. A frontier question along this line is to prove that PH is hard on average if UP requires (sub-)exponential worst-case complexity. The difficulty of resolving this ... more >>>

We present the first natural $\NP$-complete problem whose average-case hardness w.r.t. the uniform distribution over instances implies the existence of one-way functions (OWF). In fact, we prove that the existence of OWFs is \emph{equivalent} to mild average-case hardness of this $\NP$-complete problem. The problem, which originated in the 1960s, is ... more >>>

We study the error resilience of the message exchange task: Two parties, each holding a private input, want to exchange their inputs. However, the channel connecting them is governed by an adversary that may corrupt a constant fraction of the transmissions. What is the maximum fraction of corruptions that still ... more >>>

This paper surveys the development of propositional proof complexity and the seminal contributions of Alasdair Urquhart. We focus on the central role of counting principles, and in particular Tseitin's graph tautologies, to most of the key advances in lower bounds in proof complexity. We reflect on a couple of key ... more >>>

The orbit of an $n$-variate polynomial $f(\mathbf{x})$ over a field $\mathbb{F}$ is the set $\{f(A \mathbf{x} + b)\,\mid\, A\in \mathrm{GL}({n,\mathbb{F}})\mbox{ and }\mathbf{b} \in \mathbb{F}^n\}$, and the orbit of a polynomial class is the union of orbits of all the polynomials in it. In this paper, we give improved constructions of ... more >>>

A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ to subsequences of the $n$ ... more >>>

An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $\Pi$ mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. Given an instance of OCSP$(\Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number ... more >>>

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let $IP$ denote Inner Product on ... more >>>

The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to ... more >>>

We introduce the problem of constructing explicit variety evasive subspace families. Given a family $\mathcal{F}$ of subvarieties of a projective or affine space, a collection $\mathcal{H}$ of projective or affine $k$-subspaces is $(\mathcal{F},\epsilon)$-evasive if for every $\mathcal{V}\in\mathcal{F}$, all but at most $\epsilon$-fraction of $W\in\mathcal{H}$ intersect every irreducible component of $\mathcal{V}$ ... more >>>

We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property $\Pi$ and reject ... more >>>

PPSZ, for long time the fastest known algorithm for k-SAT, works by going through the variables of the input formula in random order; each variable is then set randomly to 0 or 1, unless the correct value can be inferred by an efficiently implementable rule (like small-width resolution; or being ... more >>>

We prove a SOS degree lower bound for the planted clique problem on Erd{\"o}s-R\'enyi random graphs $G(n,1/2)$. The bound we get is degree $d=\Omega(\epsilon^2\log n/\log\log n)$ for clique size $\omega=n^{1/2-\epsilon}$, which is almost tight. This improves the result of \cite{barak2019nearly} on the ``soft'' version of the problem, where the family ... more >>>

We show that given a quantum measurement, for an overwhelming majority of pure states, no meaningful information is produced. This is independent of the number of outcomes of the quantum measurement. Due to conservation inequalities, such random noise cannot be processed into coherent data.

Given polynomials $f,g,h\,\in \mathbb{F}[x_1,\ldots,x_n]$ such that $f=g/h$, where both $g$ and $h$ are computable by arithmetic circuits of size $s$, we show that $f$ can be computed by a circuit of size $\poly(s,\deg(h))$. This solves a special case of division elimination for high-degree circuits (Kaltofen'87 \& WACT'16). The result ... more >>>

Suppose that a distribution $S$ can be approximately sampled by an
efficient cell-probe algorithm. It is shown to be possible to restrict
the input to the algorithm so that its output distribution is still
not too far from $S$, and at the same time many output coordinates
are almost pairwise ... more >>>

We identify two new big clusters of proof complexity measures equivalent up to
polynomial and $\log n$ factors. The first cluster contains, among others,
the logarithm of tree-like resolution size, regularized (that is, multiplied
by the logarithm of proof length) clause and monomial space, and clause
space, both ordinary and ... more >>>

We give an explicit construction of an affine extractor (over $\mathbb{F}_2$) that works for affine sources on $n$ bits with min-entropy $k \ge~ \log n \cdot (\log \log n)^{1 + o(1)}$. This improves prior work of Li (FOCS'16) that requires min-entropy at least $\mathrm{poly}(\log n)$.

Our construction is ... more >>>

Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson \cite{ABRW04} we call a pseudorandom generator $\mathrm{PRG}\colon \{0, 1\}^n \to \{0, 1\}^m$ hard for for a propositional proof system $\mathrm{P}$ if $\mathrm{P}$ cannot efficiently prove the (properly encoded) statement $b \notin \mathrm{Im}(\mathrm{PRG})$ for any string $b \in \{0, 1\}^m$.

In \cite{ABRW04} authors ... more >>>

The stabilizer rank of a quantum state $\psi$ is the minimal $r$ such that $\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$ for $c_j \in \mathbb{C}$ and stabilizer states $\varphi_j$. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the ... more >>>

We give a direct product theorem for the entanglement-assisted interactive quantum communication complexity of an $l$-player predicate $V$. In particular we show that for a distribution $p$ that is product across the input sets of the $l$ players, the success probability of any entanglement-assisted quantum communication protocol for computing $n$ ... more >>>

We prove that there exists an absolute constant $\delta>0$ such any binary code $C\subset\{0,1\}^N$ tolerating $(1/2-\delta)N$ adversarial deletions must satisfy $|C|\le 2^{\poly\log N}$ and thus have rate asymptotically approaching $0$. This is the first constant fraction improvement over the trivial bound that codes tolerating $N/2$ adversarial deletions must have rate ... more >>>

We propose a new approach to the hardness-to-randomness framework and to the promise-BPP=promise-P conjecture. Classical results rely on non-uniform hardness assumptions to construct derandomization algorithms that work in the worst-case, or rely on uniform hardness assumptions to construct derandomization algorithms that work only in the average-case. In both types of ... more >>>

An Algebraic Circuit for a polynomial $P\in F[x_1,\ldots,x_N]$ is a computational model for constructing the polynomial $P$ using only additions and multiplications. It is a \emph{syntactic} model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to ... more >>>

We show that one-way functions exist if and only if there is some samplable distribution D such that it is hard to approximate the Kolmogorov complexity of a string sampled from D. Thus we characterize the existence of one-way functions by the average-case hardness of a natural \emph{uncomputable} problem on ... more >>>

In the coin problem we are asked to distinguish, with probability at least $2/3$, between $n$ $i.i.d.$ coins which are heads with probability $\frac{1}{2}+\beta$ from ones which are heads with probability $\frac{1}{2}-\beta$. We are interested in the space complexity of the coin problem, corresponding to the width of a read-once ... more >>>

Modern cryptography fundamentally relies on the assumption that the adversary trying to break the scheme is computationally bounded. This assumption lets us construct cryptographic protocols and primitives that are known to be impossible otherwise. In this work we explore the effect of bounding the adversary's power in other information theoretic ... more >>>

We present an elementary, self-contained proof of the result of Goldwasser and Rothblum [GR07] that the existence of a (perfect) statistically secure obfuscator implies a collapse of the polynomial hierarchy. In fact, we show that an existence of a weaker object implies a somewhat stronger statement. In addition, we extend ... more >>>

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in $\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $\Omega(n)$ space even on instances with $O(n)$ constraints. We also identify ... more >>>

We show that quantum algorithms of time T and space $S \ge \log T$ with intermediate measurements can be simulated by quantum algorithms of time $T\cdot \mathrm{poly}(S)$ and space $O(S\cdot \log T)$ without intermediate measurements. The best simulations prior to this work required either $\Omega(T)$ space (by the deferred measurement ... more >>>

We briefly discuss a few open problems in the study of various models of testing graph properties, focusing on the query complexity of the various tasks. In the dense graph model, we discuss several open problems, including:

* Determining the complexity of testing triangle-freeness.
* Characterizing the class of properties ... more >>>

Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where:

* MCSP can be solved in deterministic polynomial time, but ... more >>>

Secret-sharing is one of the most basic and oldest primitives in cryptography, introduced by Shamir and Blakely in the 70s. It allows to strike a meaningful balance between availability and confidentiality of secret information. It has a host of applications most notably in threshold cryptography and multi-party computation. All known ... more >>>

Cohen, Peri and Ta-Shma (STOC'21) considered the following question: Assume the vertices of an expander graph are labelled by $\pm 1$. What "test" functions $f : \{\pm 1\}^t \to \{\pm1 \}$ can or cannot distinguish $t$ independent samples from those obtained by a random walk? [CPTS'21] considered only balanced labelling, ... more >>>

We show equivalence between the existence of one-way
functions and the existence of a \emph{sparse} language that is
hard-on-average w.r.t. some efficiently samplable ``high-entropy''
distribution.
In more detail, the following are equivalent:
- The existentence of a $S(\cdot)$-sparse language $L$ that is
hard-on-average with respect to some samplable ... more >>>

A pair of sources $\mathbf{X},\mathbf{Y}$ over $\{0,1\}^n$ are $k$-indistinguishable if their projections to any $k$ coordinates are identically distributed. Can some $\mathit{AC^0}$ function distinguish between two such sources when $k$ is big, say $k=n^{0.1}$? Braverman's theorem (Commun. ACM 2011) implies a negative answer when $\mathbf{X}$ is uniform, whereas Bogdanov et ... more >>>

An Algebraic Formula for a polynomial $P\in F[x_1,\ldots,x_N]$ is an algebraic expression for $P(x_1,\ldots,x_N)$ using variables, field constants, additions and multiplications. Such formulas capture an algebraic analog of the Boolean complexity class $\mathrm{NC}^1.$ Proving lower bounds against this model is thus an important problem.

It is known that, to prove ... more >>>

We investigate randomized LEARN-uniformity, which captures the power of randomness and equivalence queries (EQ) in the construction of Boolean circuits for an explicit problem. This is an intermediate notion between P-uniformity and non-uniformity motivated by connections to learning, complexity, and logic. Building on a number of techniques, we establish the ... more >>>

A card guessing game is played between two players, Guesser and Dealer. At the beginning of the game, the Dealer holds a deck of $n$ cards (labeled $1, ..., n$). For $n$ turns, the Dealer draws a card from the deck, the Guesser guesses which card was drawn, and then ... more >>>

We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and positive depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution.

Using this connection, we ... more >>>

Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\to\{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log n - O(\log \log n)$. This was improved to a tight $(\log n - O(1))$ bound by Chiarelli, Hatami ... more >>>

High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary combinatorial construction with near-optimal expansion is known. In this paper, we introduce an improved combinatorial high-dimensional expander construction, by modifying ... more >>>

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games.

Using this game, we ... more >>>

We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the $n$-fold GHZ game is at most $n^{-\Omega(1)}$. This was first established by Holmgren and ... more >>>

We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\| \hat{f}_\ell \|_1$ is bounded by $d^\ell e^{{\ell+1 \choose 2}} ... more >>>

Over finite fields $F_q$ containing a root of unity of smooth order $n$ (smoothness means $n$ is the product of small primes), the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division, interpolation and multi-point evaluation. These operations can ... more >>>

We show that the QRAT simulation algorithm of $\forall$Exp+Res from [B. Kiesl and M. Seidl, 2019] cannot be lifted to IR-calc.

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d} \in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then ... more >>>

Recently, there has been exciting progress in understanding the complexity of distributions. Here, the goal is to quantify the resources required to generate (or sample) a distribution. Proving lower bounds in this new setting is more challenging than in the classical setting, and has yielded interesting new techniques and surprising ... more >>>

We introduce a new graph parameter called linear upper maximum induced
matching width \textsc{lu-mim width}, denoted for a graph $G$ by $lu(G)$.
We prove that the smallest size of the \textsc{obdd} for $\varphi$,
the monotone 2-\textsc{cnf} corresponding to $G$, is sandwiched
between $2^{lu(G)}$ and $n^{O(lu(G))}$.
The upper bound ... more >>>

The classic Impagliazzo--Nisan--Wigderson (INW) psesudorandom generator (PRG) (STOC `94) for space-bounded computation uses a seed of length $O(\log n \cdot \log(nwd/\varepsilon))$ to fool ordered branching programs of length $n$, width $w$, and alphabet size $d$ to within error $\varepsilon$. A series of works have shown that the analysis of the ... more >>>

Merge Resolution (MRes [Beyersdorff et al. J. Autom. Reason.'2021] ) is a refutational proof system for quantified Boolean formulas (QBF). Each line of MRes consists of clauses with only existential literals, together with information of countermodels stored as merge maps. As a result, MRes has strategy extraction by design. The ... more >>>

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of "structured" $\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at ... more >>>

The influence of a set of variables on a Boolean function has three separate definitions in the literature, the first due to Ben-Or and Linial (1989), the second due to Fischer et al. (2002) and Blais (2009) and the third due to Tal (2017). The goal of the present work ... more >>>

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of ... more >>>

We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $\epsilon$, getting the optimal constant factors in the leading terms in a number of different models.

The following are our results in the randomized model:

1) We give a general technique to convert ... more >>>

A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider ... more >>>

Aaronson and Ambainis (STOC 2015, SICOMP 2018) claimed that the acceptance probability of every quantum algorithm that makes $q$ queries to an $N$-bit string can be estimated to within $\epsilon$ by a randomized classical algorithm of query complexity $O_q((N/\epsilon^2)^{1-1/2q})$. We describe a flaw in their argument but prove that the ... more >>>

In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory---its hardness is mysterious, and a better understanding of its hardness can have ... more >>>

At ITCS 2010, Dziembowski, Pietrzak, and Wichs introduced Non-malleable Codes (NMCs). Non-malleability is one of the strongest and most challenging notions of security considered in cryptography and protects against tampering attacks. In the context of coding schemes, non-malleability requires that it be infeasible to tamper the codeword of a message ... more >>>

We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an
oracle function $f : L^m \rightarrow \mathbb F_q$, where $L\subset \mathbb F_q$, a verifier distinguishes whether $f$ is the evaluation of a ... more >>>

We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call "visible rank." The locality constraints of a linear code are stipulated by a matrix $H$ of $\star$'s and $0$'s (which we ... more >>>

The focus of this survey is the question of *quantified derandomization*, which was introduced by Goldreich and Wigderson (2014): Does derandomization of probabilistic algorithms become easier if we only want to derandomize algorithms that err with extremely small probability? How small does this probability need to be in order for ... more >>>

We study the matroid intersection problem from the parallel complexity perspective. Given
two matroids over the same ground set, the problem asks to decide whether they have a common base and its search version asks to find a common base, if one exists. Another widely studied variant is the weighted ... more >>>

Consider property testing on bounded degree graphs and let $\varepsilon > 0$ denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on $\varepsilon$. Recent advances in testing minor-freeness have proven that ... more >>>

We prove that for every Boolean function, the public-coin zero-error randomized communication complexity and the deterministic communication complexity are polynomially equivalent.

In distributed differential privacy, the parties perform analysis over their joint data while preserving the privacy for both datasets. Interestingly, for a few fundamental two-party functions such as inner product and Hamming distance, the accuracy of the distributed solution lags way behind what is achievable in the client-server setting. McGregor, ... more >>>

How much computational resource do we need for cryptography? This is an important question of both theoretical and practical interests. In this paper, we study the problem on pseudorandom functions (PRFs) in the context of circuit complexity. Perhaps surprisingly, we prove extremely tight upper and lower bounds in various circuit ... more >>>

We show polynomial-time quantum algorithms for the following problems:
(*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a ... more >>>

The inner product function $\langle x,y \rangle = \sum_i x_i y_i \bmod 2$ can be easily computed by a (linear-size) ${AC}^0(\oplus)$ circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on ... more >>>

Similar to the role of Markov decision processes in reinforcement learning, Markov Games (also called Stochastic Games)lay down the foundation for the study of multi-agent reinforcement learning (MARL) and sequential agent interactions. In this paper, we introduce the solution concept, approximate Markov Perfect Equilibrium (MPE), to finite-state Stochastic Games repeated ... more >>>

A distribution is called $m$-grained if each element appears with probability that is an integer multiple of $1/m$.
We prove that, for any constant $c<1$, testing whether a distribution over $[\Theta(m)]$ is $m$-grained requires $\Omega(m^c)$ samples.

Hypercontractive inequalities for real-valued functions over the Boolean cube play an important role in theoretical computer science. In this work, we prove a hypercontractive inequality for matrix-valued functions defined over large alphabets, generalizing the result of Ben-Aroya, Regev, de Wolf (FOCS'08) for the Boolean alphabet. To obtain our result we ... more >>>

Quantified conflict-driven clause learning (QCDCL) is one of the main approaches for solving quantified Boolean formulas (QBF). We formalise and investigate several versions of QCDCL that include cube learning and/or pure-literal elimination, and formally compare the resulting solving models via proof complexity techniques. Our results show that almost all of ... more >>>

Boolean circuits are a model of computation. A class of Boolean circuits is called a polynomial class if the number of nodes is bounded by a polynomial function of the number of input variables. A class $C_n[s(n)]$ of Boolean functions is called learnable if there are algorithms that can approximate ... more >>>

We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings.
Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings).
Accordingly, the ... more >>>

For a function $t : 2^\star \to 1^\star$, let $C_t$ be the set of problems decidable on input $x$ in time at most $t(x)$ almost everywhere. The Union Theorem of Meyer and McCreight asserts that any union $\bigcup_{i < \omega} C_{t_i}$ for a uniformly recursive sequence of bounds $t_i$ is ... more >>>

We suggest a general framework to study dependency schemes for dependency quantified Boolean formulas (DQBF). As our main contribution, we exhibit a new infinite collection of implication-free DQBF dependency schemes that generalise the reflexive resolution path dependency scheme. We establish soundness of all these schemes, implying that they can be ... more >>>

The Alon-Edmonds-Luby distance amplification procedure (FOCS 1995) is an algorithm that transforms a code with vanishing distance to a code with constant distance. AEL was invoked by Kopparty, Meir, Ron-Zewi, and Saraf (J. ACM 2017) for obtaining their state-of-the-art LDC, LCC and LTC. Cohen and Yankovitz (CCC 2021) devised a ... more >>>

We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. ... more >>>

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally.

We give an explicit sequence of CNF formulas $\{\phi_n\}$ such that VNP$\neq$VP iff there are ... more >>>

We prove the existence of Reed-Solomon codes of any desired rate $R \in (0,1)$ that are combinatorially list-decodable up to a radius approaching $1-R$, which is the information-theoretic limit. This is established by starting with the full-length $[q,k]_q$ Reed-Solomon code over a field $\mathbb{F}_q$ that is polynomially larger than the ... more >>>

Assume that $X_0,X_1$ (respectively $Y_0,Y_1$) are $d_X$ (respectively $d_Y$) indistinguishable for circuits of a given size. It is well known that the product distributions $X_0Y_0,\,X_1Y_1$ are $d_X+d_Y$ indistinguishable for slightly smaller circuits. However, in probability theory where unbounded adversaries are considered through statistical distance, it is folklore knowledge that in ... more >>>

We study a natural and quite general model of branch-and-bound algorithms. In this model, an algorithm attempts to minimize (or maximize) a function $f : D \to \mathbb{R}_{\geq 0}$ by making oracle queries to a heuristic $h_f$ satisfying
\[
\min_{x \in S} f(x) \leq h_f(S) \leq \gamma \cdot ... more >>>

In this note, we show the mixing of three-term progressions $(x, xg, xg^2)$ in every finite quasirandom group, fully answering a question of Gowers. More precisely, we show that for any $D$-quasirandom group $G$ and any three sets $A_1, A_2, A_3 \subset G$, we have
\[ \left|\Pr_{x,y\sim G}\left[ x \in ... more >>>

We construct an explicit $\varepsilon$-hitting set generator (HSG) for regular ordered branching programs of length $n$ and $\textit{unbounded width}$ with a single accept state that has seed length
\[O(\log(n)(\log\log(n)+\log(1/\varepsilon))).\]
Previously, the best known seed length for regular branching programs of width $w$ with a single accept state was ... more >>>

We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution.
These results are obtained by taking a technique ... more >>>

A recent work of Li and Wootters (2021) shows a redundancy lower bound of $\Omega(\sqrt{Nk})$ for systematic linear $k$-batch codes of block length $N$ by looking at the $O(k)$ tensor power of the dual code. In this note, we present an alternate proof of their result via a linear independence ... more >>>

Suppose we have random sampling access to a huge object, such as a graph or a database.
Namely, we can observe the values of \emph{random} locations in the object, say random records in the database or random edges in the graph.
We cannot, however, query locations of our choice. Can ... more >>>

We consider the problem of extracting randomness from \textit{sumset sources}, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An $(n,k,C)$-sumset source $\mathbf{X}$ is a distribution on $\{0,1\}^n$ of the form $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_C$, where $\mathbf{X}_i$'s are independent sources on $n$ bits ... more >>>

We describe an explicit and simple subset of the discrete hypercube which cannot be exactly covered by fewer than exponentially many hyperplanes. The proof exploits a connection to communication complexity, and relies heavily on Razborov's lower bound for disjointness.

We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as $P^{NP}$ and $P^{QMA}$, respectively.
The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate ... more >>>

The known constructions of negligible error (non-malleable) two-source extractors can be broadly classified in three categories:

(1) Constructions where one source has min-entropy rate about $1/2$, the other source can have small min-entropy rate, but the extractor doesn't guarantee non-malleability.
(2) Constructions where one source is uniform, and the other ... more >>>

A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester.

LTCs were initially studied as ... more >>>

We introduce the notion of \emph{Min-Entropic Optimality} thereby providing a framework for arguing that a given algorithm computes a function better than any other algorithm. An algorithm is $k(n)$ Min-Entropic Optimal if for every distribution $D$ with min-entropy at least $k(n)$, its expected running time when its input is drawn ... more >>>

The hardness vs.~randomness paradigm aims to explicitly construct pseudorandom generators $G:\{0,1\}^r \to \{0,1\}^m$ that fool circuits of size $m$, assuming the existence of explicit hard functions. A ``high-end PRG'' with seed length $r=O(\log m)$ (implying BPP=P) was achieved in a seminal work of Impagliazzo and Wigderson (STOC 1997), assuming \textsc{the ... more >>>

Since they were first introduced by Schulman (STOC 1993), the construction of tree codes remained an elusive open problem. The state-of-the-art construction by Cohen, Haeupler and Schulman (STOC 2018) has constant distance and $(\log n)^{e}$ colors for some constant $e > 1$ that depends on the distance, where $n$ is ... more >>>

Consider a homogeneous degree $d$ polynomial $f = T_1 + \cdots + T_s$, $T_i = g_i(\ell_{i,1}, \ldots, \ell_{i, m})$ where $g_i$'s are homogeneous $m$-variate degree $d$ polynomials and $\ell_{i,j}$'s are linear polynomials in $n$ variables. We design a (randomized) learning algorithm that given black-box access to $f$, computes black-boxes for ... more >>>

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural.

(*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a balanced bipartite graph with one partition class equipartitioned into k colour classes in ... more >>>

Statistics of small subgraph counts such as triangles, four-cycles, and $s$-$t$ paths of short lengths reveal important structural properties of the underlying graph. These problems have been widely studied in social network analysis. In most relevant applications, the graphs are not only massive but also change dynamically over time. Most ... more >>>

We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, ... more >>>

For a complexity class $C$ and language $L$, a constructive separation of $L \notin C$ gives an efficient algorithm (also called a refuter) to find counterexamples (bad inputs) for every $C$-algorithm attempting to decide $L$. We study the questions: Which lower bounds can be made constructive? What are the consequences ... more >>>

Let $\Pi$ be a protocol over the $n$-party broadcast channel, where in each round, a pre-specified party broadcasts a symbol to all other parties. We wish to design a scheme that takes such a protocol $\Pi$ as input and outputs a noise resilient protocol $\Pi'$ that simulates $\Pi$ over the ... more >>>

A PAC learning model involves two worst-case requirements: a learner must learn all functions in a class on all example distributions. However, basing the hardness of learning on NP-hardness has remained a key challenge for decades. In fact, recent progress in computational complexity suggests the possibility that a weaker assumption ... more >>>

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem is also closely related to fast algorithms for other natural ... more >>>

The multiplicity Schwartz-Zippel lemma asserts that over a field, a low-degree polynomial cannot vanish with high multiplicity very often on a sufficiently large product set. Since its discovery in a work of Dvir, Kopparty, Saraf and Sudan [DKSS13], the lemma has found nu- merous applications in both math and computer ... more >>>

We show that, in the black-box setting, the behavior of quantum polynomial-time (${BQP}$) can be remarkably decoupled from that of classical complexity classes like ${NP}$. Specifically:

-There exists an oracle relative to which ${NP}^{{BQP}}\not \subset {BQP}^{{PH}}$, resolving a 2005 problem of Fortnow. Interpreted another way, we show that ${AC^0}$ circuits ... more >>>

In a Merlin-Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability $1$, and rejects invalid proofs with probability arbitrarily close to $1$. The running time of such a system is defined to be the length of Merlin's proof plus the running time of Arthur. We ... more >>>

What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness ... more >>>

A major difficulty in quantum rewinding is the fact that measurement is destructive: extracting information from a quantum state irreversibly changes it. This is especially problematic in the context of zero-knowledge simulation, where preserving the adversary's state is essential.

In this work, we develop new techniques for ... more >>>

We prove hypercontractive inequalities on high dimensional expanders. As in the settings of the p-biased hypercube, the symmetric group, and the Grassmann scheme, our inequalities are effective for global functions, which are functions that are not significantly affected by a restriction of a small set of coordinates. As applications, we ... more >>>

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the $p$-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of ... more >>>

A language $L$ is random-self-reducible if deciding membership in $L$ can be reduced (in polynomial time) to deciding membership in $L$ for uniformly random instances. It is known that several "number theoretic" languages (such as computing the permanent of a matrix) admit random self-reductions. Feigenbaum and Fortnow showed that NP-complete ... more >>>

Comparator circuits are a natural circuit model for studying bounded fan-out computation whose power sits between nondeterministic branching programs and general circuits. Despite having been studied for nearly three decades, the first superlinear lower bound against comparator circuits was proved only recently by Gál and Robere (ITCS 2020), who established ... more >>>

We show that any nonzero polynomial in the ideal generated by the $r \times r$ minors of an $n \times n$ matrix $X$ can be used to efficiently approximate the determinant. Specifically, for any nonzero polynomial $f$ in this ideal, we construct a small depth-three $f$-oracle circuit that approximates the ... more >>>

Motivated by the goal of showing stronger structural results about the complexity of learning, we study the learnability of strong concept classes beyond P/poly, such as PSPACE/poly and EXP/poly. We show the following:

1. (Unconditional Lower Bounds for Learning) Building on [KKO13], we prove unconditionally that BPE/poly cannot be weakly ... more >>>

Entropy is a fundamental property of both classical and quantum systems, spanning myriad theoretical and practical applications in physics and computer science. We study the problem of obtaining estimates to within a multiplicative factor $\gamma>1$ of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum ... more >>>

This text provides a high-level description of the locally testable code constructed by Dinur, Evra, Livne, Lubotzky, and Mozes (ECCC, TR21-151).
In particular, the group theoretic aspects are abstracted as much as possible.

We show a quadratic separation between resolution and cut-free sequent calculus width. We use this gap to get, for the first time, first, a super-polynomial separation between resolution and cut-free sequent calculus for refuting CNF formulas, and secondly, a quadratic separation between resolution width and monomial space in polynomial calculus ... more >>>

k-median and k-means are the two most popular objectives for clustering algorithms. Despite intensive effort, a good understanding of the approximability of these objectives, particularly in $\ell_p$-metrics, remains a major open problem. In this paper, we significantly improve upon the hardness of approximation factors known in literature for these objectives ... more >>>

We show that for sufficiently large $n\geq 1$ and $d=C n^{3/4}$ for some universal constant $C>0$, a random spectrahedron with matrices drawn from Gaussian orthogonal ensemble has Gaussian surface area $\Theta(n^{1/8})$ with high probability.

This paper aims to derandomize the following problems in the smoothed analysis of Spielman and Teng. Learn Disjunctive Normal Form (DNF), invert Fourier Transforms (FT), and verify small circuits' unsatisfiability. Learning algorithms must predict a future observation from the only $m$ i.i.d. samples of a fixed but unknown joint-distribution $P(G(x),y)$ ... more >>>

Succinct arguments are proof systems that allow a powerful, but untrusted, prover to convince a weak verifier that an input $x$ belongs to a language $L \in NP$, with communication that is much shorter than the $NP$ witness. Such arguments, which grew out of the theory literature, are now drawing ... more >>>

This is a purely pedagogical text.
We advocate using KW-games as a teaser (or ``riddle'') for a complexity theoretic course.
In particular, stating the KW-game for a familiar NP-complete problem such as 3-Colorability and asking to prove that it requires more than polylogarithmic communication poses a seemingly tractable question ... more >>>

The propositional proof system Sherali-Adams (SA) has polynomial-size proofs of the pigeonhole principle (PHP). Similarly, the Nullstellensatz (NS) proof system has polynomial size proofs of the bijective (i.e. both functional and onto) pigeonhole principle (ofPHP). We characterize the strength of these algebraic proof systems in terms of Boolean proof systems ... more >>>