We present a deterministic logspace algorithm for solving s-t connectivity on directed graphs if (i) we are given a stationary distribution for random walk on the graph and (ii) the random walk which starts at the source vertex $s$ has polynomial mixing time. This result generalizes the recent deterministic logspace ... more >>>

Predicting the winner of an election is a favorite problem both for news media pundits and computational social choice theorists. Since it is often infeasible to elicit the preferences of all the voters in a typical prediction scenario, a common algorithm used for winner prediction is to run the election ... 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 study the problem of testing identity against a given distribution (a.k.a. goodness-of-fit) with a focus on the high confidence regime. More precisely, given samples from an unknown distribution $p$ over $n$ elements, an explicitly given distribution $q$, and parameters $0< \epsilon, \delta < 1$, we wish to distinguish, {\em ... more >>>

Blasiok (SODA'18) recently introduced the notion of a subgaussian sampler, defined as an averaging sampler for approximating the mean of functions $f:\{0,1\}^m \to \mathbb{R}$ such that $f(U_m)$ has subgaussian tails, and asked for explicit constructions. In this work, we give the first explicit constructions of subgaussian samplers (and in fact ... more >>>

A circuit $\mathcal{C}$ samples a distribution $\mathbf{X}$ with an error $\epsilon$ if the statistical distance between the output of $\mathcal{C}$ on the uniform input and $\mathbf{X}$ is $\epsilon$. We study the hardness of sampling a uniform distribution over the set of $n$-bit strings of Hamming weight $k$ denoted by $\mathbf{U}^n_k$ ... more >>>

We initiate the study of the following hypergraph sampling problem: Sample a $d$-uniform hypergraph over $n$ vertices and $m$ hyperedges from some pseudorandom distribution $\mathcal{G}$ conditioned on not having some small predefined $t$-size hypergraph $H$ as a subgraph. The algorithm should run in $\mathrm{poly}(n)$-time even when the size of the ... more >>>

We present a novel technique, based on the Jensen-Shannon divergence
from information theory, to prove lower bounds on the query complexity
of sampling algorithms that approximate functions over arbitrary
domain and range. Unlike previous methods, our technique does not
use a reduction from a binary decision problem, but rather ... more >>>

We show that for every small AC$^{0}$ circuit
$C:\{0,1\}^{\ell}\to\{0,1\}^{m}$ there exists a multiset $S$ of
$2^{m-m^{\Omega(1)}}$ restrictions that preserve the output distribution of
$C$ and moreover \emph{polarize min-entropy: }the restriction of $C$ to
any $r\in S$ either is constant or has polynomial min-entropy. This
structural result is then applied to ... more >>>

Given a distribution over $[n]^n$ such that any $k$ coordinates need $k/\log^{O(1)}n$ bits of communication to sample, we prove that any map that samples this distribution from uniform cells requires locality $\Omega(\log(n/k)/\log\log(n/k))$. In particular, we show that for any constant $\delta>0$, there exists $\varepsilon=2^{-\Omega(n^{1-\delta})}$ such that $\Omega(\log n/\log\log n)$ non-adaptive ... more >>>

We give new combinatorial proofs of known almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask [Geom. Funct. Anal. 2010], whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We provide an alternative (and $L^p$-norm free) point of view, which allows for ... more >>>

The Minimum Circuit Size Problem (MCSP) asks, given the truth table of a Boolean function $f$ and an integer $s$, if there is a circuit computing $f$ of size at most $s.$ It has been an open question since Levin's seminal work on NP-completeness whether MCSP is NP-complete. This question ... 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 a \#SAT algorithm for boolean formulas over arbitrary finite bases. Let $B_k$ be the basis composed of all boolean functions on at most $k$ inputs. For $B_k$-formulas on $n$ inputs of size $cn$, our algorithm runs in time $2^{n(1-\delta_{c,k})}$ for $\delta_{c,k} = c^{-O(c^2k2^k)}$. We also show the average-case ... more >>>

Consider the following two-player communication process to decide a language $L$: The first player holds the entire input $x$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $x$; their goal is to cooperatively decide whether $x$ belongs to $L$ at small ... more >>>

A polynomial threshold function (PTF) is defined as the sign of a polynomial $p\colon\bool^n\to\mathbb{R}$. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth.

Satisfiability (#SAT). We give the first zero-error randomized algorithm ... more >>>

The study of the worst-case complexity of the Boolean Satisfiability (SAT) problem has seen considerable progress in recent years, for various types of instances including CNFs \cite{PPZ99, PPSZ05, Sch99, Sch05}, Boolean formulas \cite{San10} and constant-depth circuits \cite{IMP12}. We systematically investigate the complexity of solving {\it mixed} instances, where different parts ... more >>>

Concretely efficient interactive oracle proofs (IOPs) are of interest due to their applications to scaling blockchains, their minimal security assumptions, and their potential future-proof resistance to quantum attacks.

Scalable IOPs, in which prover time scales quasilinearly with the computation size and verifier time scales poly-logarithmically with it, have been known ... more >>>

Scaled dimension has been introduced by Hitchcock et al (2003) in order to quantitatively distinguish among classes such as SIZE(2^{a n}) and SIZE(2^{n^{a}}) that have trivial dimension and measure in ESPACE.

We define a new discrete version of scaled dimension and we find
connections between the scaled dimension of a string and its Kolmogorov
complexity and predictability. We give a new characterization
of constructive scaled dimension by Kolmogorov complexity, and prove
a new result about scaled dimension and prediction.

In this work we relate the deterministic
complexity of factoring polynomials (over
finite
fields) to certain combinatorial objects we
call
m-schemes. We extend the known conditional
deterministic subexponential time polynomial
factoring algorithm for finite fields to get an
underlying m-scheme. We demonstrate ... more >>>

Schur Polynomials are families of symmetric polynomials that have been
classically studied in Combinatorics and Algebra alike. They play a central
role in the study of Symmetric functions, in Representation theory [Sta99], in
Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84,
Sta99]. In recent years, they have ... more >>>

Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the "sculpting problem." In this work, we give a full characterization of sculptable functions in the query complexity setting. ... more >>>

We consider the problem of outputting succinct encodings of lists of generators for invariant rings. Mulmuley conjectured that there are always polynomial sized such encodings for all invariant rings. We provide simple examples that disprove this conjecture (under standard complexity assumptions).

A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem - that is, the problem of determining whether the length of the shortest time-$t$ program generating a given string $x$ is at most $s$.

In this work, we ... more >>>

We show that searching a width k maze is complete for \Pi_k, i.e., for
the k'th level of the AC^0 hierarchy. Equivalently, st-connectivity
for width k grid graphs is complete for \Pi_k. As an application,
we show that there is a data structure solving dynamic st-connectivity
for constant ... more >>>

Given a function $f:\mathbb F_2^n \to [-1,1]$, this work seeks to find a large affine subspace $\mathcal U$ such that $f$, when restricted to $\mathcal U$, has small nontrivial Fourier coefficients.

We show that for any function $f:\mathbb F_2^n \to [-1,1]$ with Fourier degree $d$, there exists an affine subspace ... more >>>

Many real-world optimization problems in, e.g., engineering
or biology have the property that not much is known about
the function to be optimized. This excludes the application
of problem-specific algorithms. Simple randomized search
heuristics are then used with surprisingly good results. In
order to understand the working principles behind such
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 $s$. For over 30 years, it was known that any (monotone) collection of authorized sets can be ... more >>>

Assume that Alice is running a program $P$ on a RAM, and an adversary
Bob would like to get some information about the input or output of the
program. At each time, during the execution of $P$, Bob is able to see
the addresses of the memory cells involved in ... more >>>

This paper takes a fresh look at security notions for steganography --
the art of encoding secret messages into unsuspicious covertexts
such that an adversary cannot distinguish the resulting stegotexts from original covertexts.
Stegosystems that fulfill the security notion used so far, however, are quite inefficient.
This ... more >>>

Let G be a finite cyclic group with generator \alpha and with
an encoding so that multiplication is computable in polynomial time. We
study the security of bits of the discrete log x when given \exp_{\alpha}(x),
assuming that the exponentiation function \exp_{\alpha}(x) = \alpha^x is one-way.
... more >>>

D. Boneh and R. Venkatesan have recently proposed an approach to proving
that a reasonably small portions of most significant bits of the
Diffie--Hellman key modulo a prime are as secure the the whole key. Some
further improvements and generalizations have been obtained by
I. M. Gonzales Vasco ... more >>>

Boneh and Venkatesan have recently proposed a polynomial time
algorithm for recovering a ``hidden'' element $\alpha$ of a
finite field $\F_p$ of $p$ elements from rather short
strings of the most significant bits of the remainder
mo\-du\-lo $p$ of $\alpha t$ for several values of $t$ selected uniformly
at random ... more >>>

We introduce a new type of seeded extractors we dub seed protecting extractors. Informally, a seeded extractor is seed protecting against a class of functions $C$, mappings seeds to seeds, if the seed $Y$ remains close to uniform even after observing the output $\mathrm{Ext}(X,A(Y))$ for every choice of $A \in ... more >>>

We give a self-contained exposition of selected results in additive
combinatorics over the group {0,1}^n. In particular, we prove the
celebrated theorems known as the Balog-Szemeredi-Gowers theorem ('94 and '98) and
the
Freiman-Ruzsa theorem ('73 and '99), leading to the remarkable result
by Samorodnitsky ('07) that linear transformations are efficiently ... more >>>

Improving a long standing result of Sch\"{o}nhage, Paterson
and Pippenger we show that the MEDIAN of a set containing n
elements can always be found using at most 2.95n comparisons.

This is the full version of the paper. An extended abstract
version ... more >>>

A large body of work studies the complexity of selecting the
$j$-th largest element in an arbitrary set of $n$ elements (a.k.a.
the select$(j)$ operation). In this work, we study the
complexity of select in data that is partially structured by
an initial preprocessing stage and in a data structure ... more >>>

Many results in fine-grained complexity reveal intriguing consequences from solving various SAT problems even slightly faster than exhaustive search. We prove a ``self-improving'' (or ``bootstrapping'') theorem for Circuit-SAT, $\#$Circuit-SAT, and its fully-quantified version: solving one of these problems faster for ``large'' circuit sizes implies a significant speed-up for ``smaller'' circuit ... more >>>

We introduce the `binary value principle' which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, ... more >>>

Considerable effort has been devoted to the development of streaming algorithms for analyzing massive graphs. Unfortunately, many results have been negative, establishing that a wide variety of problems require $\Omega(n^2)$ space to solve. One of the few bright spots has been the development of semi-streaming algorithms for a handful of ... more >>>

Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential ... more >>>

The graph homomorphism problem is a canonical $NP$-complete problem.
It generalizes various other well-studied problems such as graph
coloring and finding cliques. To get a better insight into a
combinatorial problem, one often studies relaxations of the problem.
We define fractional homomorphisms and pseudo-homomorphisms ... more >>>

Sensitivity measures how much the output of an algorithm changes, in terms of Hamming distance, when part of the input is modified. While approximation algorithms with low sensitivity have been developed for many problems, no sensitivity lower bounds were previously known for approximation algorithms. In this work, we establish the ... more >>>

Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least the square root of n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we ... more >>>

In this paper, we study the Boolean function parameters sensitivity ($\mathbf{s}$), block sensitivity ($\mathbf{bs}$), and alternation ($\mathbf{alt}$) under specially designed affine transforms and show several applications. For a function $f:\mathbb{F}_2^n \to \{0,1\}$, and $A = Mx+b$ for $M \in \mathbb{F}_2^{n \times n}$ and $b \in \mathbb{F}_2^n$, the result of the ... 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 show that there is a language that is Turing complete for NP but not many-one complete for NP, under a {\em worst-case} hardness hypothesis. Our hypothesis asserts the existence of a non-deterministic, double-exponential time machine that runs in time $O(2^{2^{n^c}})$ (for some $c > 1$) accepting $\Sigma^*$ whose accepting ... more >>>

This work is about the monotone versions of the algebraic complexity classes VP and VNP. The main result is that monotone VNP is strictly stronger than monotone VP.

This work deals with the power of linear algebra in the context of multilinear computation. By linear algebra we mean algebraic branching programs (ABPs) which are known to be computationally equivalent to two basic tools in linear algebra: iterated matrix multiplication and the determinant. We compare the computational power of ... more >>>

We provide a non-explicit separation of the number-on-forehead communication complexity classes RP and NP when the number of players is up to \delta log(n) for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide an explicit separation between these classes when the number of players is only up to o(loglog(n)).

We show (under standard assumptions) that there are NP optimization problems for which estimation is easier than approximation. Namely, one can estimate the value of the optimal solution within a ratio of $\rho$, but it is difficult to find a solution whose value is within $\rho$ of optimal.
As an ... more >>>

We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following:

1. There exists an explicit $n$-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) ... more >>>

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent ... more >>>

We use hypotheses of structural complexity theory to separate various
NP-completeness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is Turing complete but not truth-table complete. We provide fairly thorough analyses of the hypotheses that we introduce. Unlike previous approaches, we ... more >>>

In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate $\gamma_2$-factorization norm is a lower bound for these parameters and asked whether a stronger ... more >>>

The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory.
For example, symmetric functions, that is, functions that are invariant under the action of $S_n$ is an important class of functions in the study of Boolean functions.
A function $f:\{0,1\}^n \to \{0,1\}$ ... more >>>

While exponential separations are known between quantum and randomized communication complexity for partial functions, e.g. Raz [1999], the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized
communication complexity for a ... more >>>

It is well-known that Resolution proofs can be efficiently simulated by Sherali-Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT ... more >>>

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized
query complexity for a total boolean function is given by the function $f$ on $n=2^k$ bits defined by a complete binary tree
of NAND gates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537\ldots})$. ... more >>>

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity ... more >>>

Properties of Boolean functions on the hypercube that are invariant
with respect to linear transformations of the domain are among some of
the most well-studied properties in the context of property testing.
In this paper, we study a particular natural class of linear-invariant
properties, called matroid freeness properties. These properties ... more >>>

We show that assuming the strong exponential-time hypothesis (SETH), there are no non-trivial algorithms for the nearest codeword problem (NCP), the minimum distance problem (MDP), or the nearest codeword problem with preprocessing (NCPP) on linear codes over any finite field. More precisely, we show that there are no NCP, MDP, ... more >>>

We prove that finding the solution of two player Nash Equilibrium is PPAD-complete.

We prove that any non-adaptive algorithm that tests whether an unknown
Boolean function $f\colon \{0, 1\}^n\to\{0, 1\} $ is a $k$-junta or $\epsilon$-far from every $k$-junta must make $\widetilde{\Omega}(k^{3/2} / \epsilon)$ many queries for a wide range of parameters $k$ and $\epsilon$. Our result dramatically improves previous lower ... more >>>

We prove that the Cutting Plane proof system based on
Gomory-Chvatal cuts polynomially simulates the
lift-and-project system with integer coefficients
written in unary. The restriction on coefficients can be
omitted when using Krajicek's cut-free Gentzen-style extension
of both systems. We also prove that Tseitin tautologies
have short proofs in ... more >>>

We introduce the problem of *shadow tomography*: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, ... more >>>

We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of $P$ to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity.

We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case.

First, we demonstrate a hierarchy of modes of shared ... more >>>

We study the problem of {\em generalized uniformity testing}~\cite{BC17} of a discrete probability distribution: Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbf{\Omega}$, we want to distinguish, with probability at least $2/3$, between the case that $p$ is uniform on some {\em subset} of $\mathbf{\Omega}$ ... more >>>

We establish several ``sharp threshold'' results for computational complexity. For certain tasks, we can prove a resource lower bound of $n^c$ for $c \geq 1$ (or obtain an efficient circuit-analysis algorithm for $n^c$ size), there is strong intuition that a similar result can be proved for larger functions of $n$, ... more >>>

We define the sharply bounded hierarchy, SBHQL, a hierarchy of
classes within P, using quasilinear-time computation and
quantification over values of length log n. It generalizes the
limited nondeterminism hierarchy introduced by Buss and Goldsmith,
while retaining the invariance properties. The new hierarchy has
several alternative characterizations.

We define ... more >>>

We study complexity measures on subsets of the boolean hypercube and exhibit connections between algebra (the Hilbert function) and combinatorics (VC theory). These connections yield results in both directions. Our main complexity-theoretic result proves that most linear program feasibility problems cannot be computed by polynomial-sized constant-depth circuits. Moreover, our result ... more >>>

We show how to efficiently compile any given circuit $C$
into a leakage-resistant circuit $\widehat{C}$ such that any
function on the wires of $\widehat{C}$ that leaks information
during a computation $\widehat{C}(x)$ yields advantage in
computing the product of $|\widehat{C}|^{\Omega(1)}$ elements
of the alternating group $A_u$. In combination with new
compression ... more >>>

We study *interactive oracle proofs* (IOPs) (Ben-Sasson, Chiesa, Spooner '16), which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and general techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs ... more >>>

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any universal machine, it is possible to compute in polynomial time on ... more >>>

We survey known results regarding locally testable codes
and locally testable proofs (known as PCPs),
with emphasis on the length of these constructs.
Locally testability refers to approximately testing
large objects based on a very small number of probes,
each retrieving a single bit in the ... more >>>

We construct a PCP for NTIME(2$^n$) with constant
soundness, $2^n \poly(n)$ proof length, and $\poly(n)$
queries where the verifier's computation is simple: the
queries are a projection of the input randomness, and the
computation on the prover's answers is a 3CNF. The
previous upper bound for these two computations was
more >>>

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 >>>

We study arithmetic proof systems $\mathbb{P}_c(\mathbb{F})$ and $\mathbb{P}_f(\mathbb{F})$ operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field $\mathbb{F}$. We establish a series of structural theorems about these proof systems, the main one stating that $\mathbb{P}_c(\mathbb{F})$ proofs can be balanced: if a polynomial identity of ... more >>>

For quantified Boolean formulas (QBF) there are two main different approaches to solving: QCDCL and expansion solving. In this paper we compare the underlying proof systems and show that expansion systems admit strictly shorter proofs than CDCL systems for formulas of bounded quantifier complexity, thus pointing towards potential advantages of ... more >>>

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space.

In this paper we resolve the question by answering ... more >>>

We study bounded depth $(\min, +)$ formulas computing the shortest path polynomial. For depth $2d$ with $d \geq 2$, we obtain lower bounds parametrized by certain fan-in restrictions on all $+$ gates except those at the bottom level. For depth $4$, in two regimes of the parameter, the bounds are ... more >>>

Quantified conflict-driven clause learning (QCDCL) is one of the main solving approaches for quantified Boolean formulas (QBF). One of the differences between QCDCL and propositional CDCL is that QCDCL typically follows the prefix order of the QBF for making decisions.
We investigate an alternative model for QCDCL solving where decisions ... more >>>

We give a new and improved proof that the shrinkage exponent of De Morgan formulae is $2$. Namely, we show that for any Boolean function $f: \{-1,1\}^n \to \{-1,1\}$, setting each variable out of $x_1, \ldots, x_n$ with probability $1-p$ to a randomly chosen constant, reduces the expected formula size ... more >>>

We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the $p$-random restriction $\mathbf R_p$ for all values of $p \in [0,1]$. For a function $f$ with domain $\{0,1\}^n$, let $\mathrm{DL}(f)$ denote the minimum size of a decision list that computes $f$. We show ... more >>>

Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of $O(p^{2})$ under a random restriction that leaves each variable alive independently with probability $p$ [SICOMP, 1998]. Using this result, he gave an $\widetilde{\Omega}(n^{3})$ formula size lower bound for the Andreev function, ... more >>>

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 >>>

We study the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is $3$. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. Similar (slightly less accurate) statements hold for $d>2$ as well. We discuss the tightness of our methods, and describe ... more >>>

The communication class $UPP^{cc}$ is a communication analog of the Turing Machine complexity class $PP$. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.

For a communication problem ... more >>>

Let $\F$ be the field of $q$ elements. An \emph{\afsext{n}{k}} is a mapping $D:\F^n\ar\B$
such that for any $k$-dimensional affine subspace $X\subseteq \F^n$, $D(x)$ is an almost unbiased
bit when $x$ is chosen uniformly from $X$.
Loosely speaking, the problem of explicitly constructing affine extractors gets harder as $q$ gets ... more >>>

We show that a very simple pseudorandom generator fools intersections of $k$ linear threshold functions (LTFs) and arbitrary functions of $k$ LTFs over $n$-dimensional Gaussian space.

The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of ... more >>>

Extending the classical ``hardness-to-randomness'' line-of-works, Doron et al. (FOCS 2020) recently proved that derandomization with near-quadratic time overhead is possible, under the assumption that there exists a function in $\mathcal{DTIME}[2^n]$ that cannot be computed by randomized SVN circuits of size $2^{(1-\epsilon)\cdot n}$ for a small $\epsilon$.

In this work we ... more >>>

In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of LDPC error-correcting codes from expanders, given by
Sipser-Spielman~\cite{SS96} (and Tanner~\cite{Tanner81}), and their analysis. We also ... more >>>

A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier's strategy can be implemented in almost-linear-time.

We present direct constructions of doubly-efficient interactive proof systems for problems in $\cal P$ that are believed to have relatively high complexity.
Specifically, such ... more >>>

Trevisan has shown that constructions of pseudo-random generators from
hard functions (the Nisan-Wigderson approach) also produce extractors.
We show that constructions of pseudo-random generators from one-way permutations
(the Blum-Micali-Yao approach) can be used for building extractors as well.
Using this new technique we build extractors that ... more >>>

We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant $\sum_{i,j,k,l\in[n]} z_{ijkl}x_ix_jx_kx_l-\beta = 0$, for Boolean variables, does not have polynomial-size IPS refutations where the refutations are multilinear and written as ... more >>>

We give a simple explicit hitting set generator for read-once branching programs of width $w$ and length $r$ with known variable order. Our generator has seed length $O\left(\frac{\log(wr) \log r}{\max\{1, \log \log w - \log \log r\}} + \log(1/\varepsilon)\right)$. This seed length improves on recent work by Braverman, Cohen, and ... more >>>

We give constructions of PCPs of length n*polylog(n) (with respect
to circuits of size n) that can be verified by making polylog(n)
queries to bits of the proof. These PCPs are not only shorter than
previous ones, but also simpler. Our (only) building blocks are
Reed-Solomon codes and the bivariate ... more >>>

A set is called NP-simple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha proved that no set which is hard for NP under many-one (Karp) reductions is NP-simple unless the intersection of NP and coNP ... 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 >>>

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 >>>

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khachatrian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs ... more >>>

We show that the deterministic multiparty communication complexity of set disjointness for $k$ parties on a universe of size $n$ is $\Omega(n/4^k)$. We also simplify Sherstov's proof
showing an $\Omega(\sqrt{n}/(k2^k))$ lower bound for the randomized communication complexity of set disjointness.

We give an example of a boolean function whose information complexity is exponentially
smaller than its communication complexity. Our result simplifies recent work of Ganor, Kol and
Raz (FOCS'14, STOC'15).

The Metropolis algorithm is simulated annealing with a fixed temperature.Surprisingly enough, many problems cannot be solved more efficiently by simulated annealing than by the Metropolis algorithm with the best temperature. The problem of finding a natural example (artificial examples are known) where simulated annealing outperforms the Metropolis algorithm for all ... more >>>

We introduce a new technique which enables a learner without access to
hidden information to learn nearly as well as a learner with access to
hidden information. We apply our technique to solve an open problem
of Maass and Turan, showing that for any concept class F the least ... more >>>

Goerdt (1991) considered a weakened version of the Cutting Plane proof system with a restriction on the degree of falsity of intermediate inequalities. (The degree of falsity of an inequality written in the form $\sum a_ix_i+\sum b_i(1-x_i)\ge c,\ a_i,b_i\ge0$ is its constant term $c$.) He proved a superpolynomial lower bound ... more >>>

Dependency quantified Boolean formulas (DQBF) describe an NEXPTIME-complete generalisation of QBF, which in turn generalises SAT. QRAT is a recently proposed proof system for quantified Boolean formulas (QBF), which simulates the full suite of QBF preprocessing techniques and thus forms a uniform proof checking format for solver verification.

In this ... more >>>

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 >>>

We show that $T$ rounds of interaction over the binary symmetric channel $BSC_{1/2-\epsilon}$ with feedback can be simulated with $O(\epsilon^2 T)$ rounds of interaction over a noiseless channel. We also introduce a more general "energy cost'' model of interaction over a noisy channel. We show energy cost to be equivalent ... more >>>

We show that several quantum circuit families can be simulated efficiently classically if it is promised that their output distribution is approximately sparse i.e. the distribution is close to one where only a polynomially small, a priori unknown subset of the measurement probabilities are nonzero. Classical simulations are thereby obtained ... more >>>

We develop a technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC'94) and Miltersen, Nisan, Safra and Wigderson (STOC'95). At the core of our technique is a novel simulation theorem: Alice gets a $p \times n$ ... more >>>

The notion of lifting theorems is a generic method to lift hardness of one-party functions to two-party lower bounds in communication model. It has many applications in different areas such as proof complexity, game theory, combinatorial optimization. Among many lifting results, a central idea is called Raz-McKenize simulation (FOCS 1997). ... more >>>

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context.

Our main result is that ... more >>>

A systematic study of simultaneous optimization of constraint satisfaction problems was initiated in [BKS15]. The simplest such problem is the simultaneous Max-Cut. [BKKST18] gave a $.878$-minimum approximation algorithm for simultaneous Max-Cut which is {\em almost optimal} assuming the Unique Games Conjecture (UGC). For a single instance Max-Cut, [GW95] gave an ... more >>>

In FOCS 2001, Barak, Goldreich, Goldwasser and Lindell conjectured that the existence of ZAPs, introduced by Dwork and Naor in FOCS 2000, could lead to the design of a zero-knowledge proof system that is secure against both resetting provers and resetting verifiers. Their conjecture has been proven true by Deng, ... more >>>

The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: ${\rm SING}_{n,m}$, consisting of all $m$-tuples of $n\times n$ complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in ${\rm SING}_{n,m}$ will imply super-polynomial circuit lower bounds, ... more >>>

We consider the following phenomenon: with just one multiplication
we can compute (3u+2v)(3x+2y)= 3ux+4vy mod 6, while computing the same polynomial modulo 5 needs 2 multiplications. We generalize this observation and we define some vectors, called sixtors, with remarkable zero-divisor properties. Using sixtors, we also generalize our earlier result ... more >>>

We give two results on the size of AC0 circuits computing promise majority. $\epsilon$-promise majority is majority promised that either at most an $\epsilon$ fraction of the input bits are 1, or at most $\epsilon$ are 0.

First, we show super quadratic lower bounds on both monotone and general depth ... more >>>

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of ... more >>>

As a natural extension of the SAT problem, different proof systems for quantified Boolean formulas (QBF) have been proposed. Many of these extend a propositional system to handle universal quantifiers.

By formalising the construction of the QBF proof system from a propositional proof system, by the addition of the ... more >>>

We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first ... more >>>

A celebrated result of Barrington (1985) proved that polynomial size, width-5 branching programs (BP) are equivalent in power to a restricted form of branching programs -- polynomial sized width-5 permutation branching programs (PBP), which in turn capture all of NC1. On the other hand it is known that width-3 PBPs ... more >>>

We consider time-space tradeoffs for exactly computing frequency
moments and order statistics over sliding windows.
Given an input of length $2n-1$, the task is to output the function of
each window of length $n$, giving $n$ outputs in total.
Computations over sliding windows are related to direct sum problems
except ... more >>>

A small-biased function is a randomized function whose distribution of truth-tables is small-biased. We demonstrate that known explicit lower bounds on the size of (1) general Boolean formulas, (2) Boolean formulas of fan-in two, (3) de Morgan formulas, as well as (4) correlation lower bounds against small de Morgan formulas ... more >>>

SBP is a probabilistic promise class located
between MA and AM \cap BPPpath. The first
part of the paper studies the question of whether
SBP has many-one complete sets. We relate
this question to the existence of uniform
enumerations. We construct an oracle relative to
which SBP and AM do ... 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 >>>

A proof system for a language $L$ is a function $f$ such that Range$(f)$ is exactly $L$. In this paper, we look at proofsystems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by ... more >>>

We show that a simple function has small unbounded error communication complexity in the $k$-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with sub-exponential advantage over random guessing has cost essentially $\Omega\left(\frac{\sqrt{n}}{4^k}\right)$ bits. Such a separation was first shown for $k=2$ independently by Buhrman et al. ['07] ... more >>>

Research in the last decade has shown that to prove lower bounds or to derandomize polynomial identity testing (PIT) for general arithmetic circuits it suffices to solve these questions for restricted circuits. In this work, we study the smallest possibly restricted class of circuits, in particular depth-$4$ circuits, which would ... more >>>

We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., ... more >>>

Most known constructions of probabilistically checkable proofs (PCPs)
either blow up the proof size by a large polynomial, or have a high
(though constant) query complexity. In this paper we give a
transformation with slightly-super-cubic blowup in proof size, with a
low query complexity. Specifically, the verifier probes the proof ... more >>>

We show the following Reduction Lemma: any
$\epsilon$-biased sample space with respect to (Boolean) linear
tests is also $2\epsilon$-biased with respect to
any system of independent linear tests. Combining this result with
the previous constructions of $\epsilon$-biased sample space with
respect to linear tests, we obtain the first efficient
more >>>

This paper studies expansion properties of the (generalized) Johnson Graph. For natural numbers
t < l < k, the nodes of the graph are sets of size l in a universe of size k. Two sets are connected if
their intersection is of size t. The Johnson graph arises often ... more >>>

We study the problem of partitioning the unit cube $[0,1]^n$ into $c$ parts so that each $d$-dimensional axis-parallel projection has small volume.

This natural combinatorial/geometric question was first studied by Kopparty and Nagargoje [KN23] as a reformulation of the problem of determining the achievable parameters for seedless multimergers -- which ... more >>>

We prove a parallel repetition theorem for general games with value tending to 0. Previously Dinur and Steurer proved such a theorem for the special case of projection games. We use information theoretic techniques in our proof. Our proofs also extend to the high value regime (value close to 1) ... more >>>

Small-bias probability spaces have wide applications in pseudorandomness which naturally leads to the study of their limitations. Constructing a polynomial complexity hitting set for read-once CNF formulas is a basic open problem in pseudorandomness. We show in this paper that this goal is not achievable using small-bias spaces. Namely, we ... more >>>

The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within $\mathrm{P}$. In this paper, we study the algebraic formula complexity of multiplying $d$ many $2\times 2$ matrices, denoted $\mathrm{IMM}_{d}$, and show ... more >>>

Dinur, Khot, Kindler, Minzer and Safra (2016) recently showed that the (imperfect completeness variant of) Khot's 2 to 2 games conjecture follows from a combinatorial hypothesis about the soundness of a certain ``Grassmanian agreement tester''.
In this work, we show that the hypothesis of Dinur et al follows from a ... more >>>

Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:
- ... more >>>

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 >>>

Binary search trees are a fundamental data structure and their height
plays a key role in the analysis of divide-and-conquer algorithms like
quicksort. Their worst-case height is linear; their average height,
whose exact value is one of the best-studied problems in average-case
complexity, is logarithmic. We analyze their smoothed height ... more >>>

Binary search trees are one of the most fundamental data structures. While the
height of such a tree may be linear in the worst case, the average height with
respect to the uniform distribution is only logarithmic. The exact value is one
of the best studied problems in average case ... more >>>

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 >>>

We construct binary linear codes that are efficiently list-decodable
up to a fraction $(1/2-\eps)$ of errors. The codes encode $k$ bits
into $n = {\rm poly}(k/\eps)$ bits and are constructible and
list-decodable in time polynomial in $k$ and $1/\eps$ (in
particular, in our results $\eps$ need ... more >>>

The solution graph of a Boolean formula on n variables is the subgraph of the hypercube Hn induced by the satisfying assignments of the formula. The structure of solution graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures ... more >>>

The Group Isomorphism problem consists in deciding whether two input
groups $G_1$ and $G_2$ given by their multiplication tables are
isomorphic. We first give a 2-round Arthur-Merlin protocol for the
Group Non-Isomorphism problem such that on input groups $(G_1,G_2)$
of size $n$, Arthur uses ... more >>>

Abstract. We study the decision theory of a maximally risk-averse investor | one whose objec-
tive, in the face of stochastic uncertainties, is to minimize the probability of ever going broke. With
a view to developing the mathematical basics of such a theory, we start with a very simple model
more >>>

Given a system of linear equations $Ax=b$ over the binary field $\mathbb{F}_2$ and an integer $t\ge 1$, we study the following three algorithmic problems:
1. Does $Ax=b$ have a solution of weight at most $t$?
2. Does $Ax=b$ have a solution of weight exactly $t$?
3. Does $Ax=b$ have a ... more >>>

The input to the Tree Evaluation problem is a binary tree of height $h$ in which each internal vertex is associated with a function mapping pairs of $\ell$-bit strings to $\ell$-bit strings,and each leaf is assigned an $\ell$-bit string.
The desired output is the value of the root, where ... more >>>

Error-correcting codes and related combinatorial constructs
play an important role in several recent (and old) results
in computational complexity theory. In this paper we survey
results on locally-testable and locally-decodable error-correcting
codes, and their applications to complexity theory and to
cryptography.

Locally decodable codes are error-correcting codes ... more >>>

We introduce the model of conservative one-way multiparty complexity
and prove lower and upper bounds on the complexity of pointer jumping.

The pointer jumping function takes as its input a directed layered
graph with a starting node and layers of nodes, and a single edge ... more >>>

We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and "lifts". The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The "lift" is a less natural ... more >>>

In a sequence of fundamental results in the 80's, Kaltofen showed that factors of multivariate polynomials with small arithmetic circuits have small arithmetic circuits. In other words, the complexity class $VP$ is closed under taking factors. A natural question in this context is to understand if other natural classes of ... more >>>

We provide a list of new natural VNP-intermediate polynomial
families, based on basic (combinatorial) NP-complete problems that
are complete under \emph{parsimonious} reductions. Over finite
fields, these families are in VNP, and under the plausible
hypothesis $\text{Mod}_pP \not\subseteq P/\text{poly}$, are neither VNP-hard (even under
oracle-circuit reductions) nor in VP. Prior to ... more >>>

Let A(x) be the characteristic function of A. Consider the function
F_k^A(x_1,...,x_k) = A(x_1)...A(x_k). We show that if F_k^A can be
computed with fewer than k queries to some set X, then A can be
computed by polynomial size circuits. A generalization of this result
has applications to bounded query ... more >>>

We introduce a (new) notion of parameterized proof system. For parameterized versions of standard proof systems such as Extended Frege and Substitution Frege we compare their complexity with respect to parameterized simulations.

The computational complexity of learning from binary examples is
investigated for linear threshold neurons. We introduce
combinatorial measures that create classes of infinitely many
learning problems with sample restrictions. We analyze how the
complexity of these problems depends on the values for the measures.
... more >>>

In this tutorial, selected topics of cryptology and of
computational complexity theory are presented. We give a brief overview
of the history and the foundations of classical cryptography, and then
move on to modern public-key cryptography. Particular attention is
paid to cryptographic protocols and the problem of constructing ... more >>>

Denote by $R$ the set of strings with high Kolmogorov complexity. In [E. Allender, H. Buhrman, M. Kouck\'y, D. van Melkebeek, and D. Ronneburger.
Power from random strings.
\emph{SIAM Journal on Computing}, 35:1467--1493, 2006.] the idea of using $R$ as an oracle for resource-bounded computation models was presented. This idea ... more >>>

We exhibit $\epsilon$-biased distributions $D$
on $n$ bits and functions $f\colon \{0,1\}^n
\to \{0,1\}$ such that the xor of two independent
copies ($D+D$) does not fool $f$, for any of the
following choices:

1. $\epsilon = 2^{-\Omega(n)}$ and $f$ is in P/poly;

2. $\epsilon = 2^{-\Omega(n/\log n)}$ and $f$ is ... more >>>

In this paper, we study the structure of set-multilinear arithmetic
circuits and set-multilinear branching programs with the aim of
showing lower bound results. We define some natural restrictions of
these models for which we are able to show lower bound results. Some
of our results extend existing lower bounds, while ... more >>>

We prove that a $O(k \log k)$-probe local algorithm for $k$-Missing String presented by Vyas and Williams is asymptotically optimal among a certain class of algorithms for this problem. The best lower bound we are aware of for the general case is still $\Omega(k)$.

We compare two methods for proving lower bounds on standard two-party model of communication complexity, the Rank method and Fooling set method. We present bounds on the number of functions $f(x,y)$, $x,y\in\{0,1\}^n$, with rank of size $k$ and fooling set of size at least k, $k\in [1,2^n]$. Using these bounds ... more >>>

For a property $P$ and a sub-property $P'$, we say that $P$ is $P'$-partially testable with $q$ queries if there exists an algorithm that distinguishes, with high probability, inputs in $P'$ from inputs $\epsilon$-far from $P$ by using $q$ queries. There are natural properties that require many queries to test, ... more >>>

We survey some recent developments in the study of
the complexity of lattice problems. After a discussion of some
problems on lattices which can be algorithmically solved
efficiently, our main focus is the recent progress on complexity
results of intractability. We will discuss Ajtai's worst-case/
average-case connections, NP-hardness and non-NP-hardness,
more >>>

We establish a 1-1 correspondence between Valiant's
character theory of matchgate/matchcircuit
and his signature theory of planar-matchgate/matchgrid,
thus unifying the two theories in expressibility.
Previously we had established a complete characterization
of general matchgates, in terms of a set of
useful Grassmann-Pl{\"u}cker identities.
With this correspondence,
we give a corresponding ... more >>>

We consider the conjecture stating that a matrix with rank
$o(n)$ and ones on the main diagonal must contain nonzero
entries on a $2\times 2$ submatrix with one entry on the main
diagonal. We show that a slightly stronger conjecture implies
that ... more >>>

This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow?s Theorem and other ideas from the theory of Social Choice; the Bonami-Beckner Inequality as an extension of Chernoff/Hoeffding ... more >>>

Many discrete minimization problems, including various versions of the shortest path problem, can be efficiently solved by dynamic programming (DP) algorithms that are ``pure'' in that they only perform basic operations, as $\min$, $\max$, $+$, but no conditional branchings via if-then-else in their recursion equations. It is known that any ... more >>>

Suppose we want to minimize a polynomial $p(x) = p(x_1, \dots, x_n)$, subject to some polynomial constraints $q_1(x), \dots, q_m(x) \geq 0$, using the Sum-of-Squares (SOS) SDP hierarachy. Assume we are in the "explicitly bounded" ("Archimedean") case where the constraints include $x_i^2 \leq 1$ for all $1 \leq i \leq ... 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 initiate the study of the tradeoff between the {\em length} of a
probabilistically checkable proof of proximity (PCPP) and the
maximal {\em soundness} that can be guaranteed by a $3$-query
verifier with oracle access to the proof. Our main observation is
that a verifier limited to querying a short ... 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 >>>

During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems ... more >>>

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 >>>

We investigate the space complexity of certain perfect matching
problems over bipartite graphs embedded on surfaces of constant genus
(orientable or non-orientable). We show that the problems of deciding
whether such graphs have (1) a perfect matching or not and (2) a
unique perfect matching or not, are in the ... more >>>

We study the space complexity of refuting unsatisfiable random $k$-CNFs in
the Resolution proof system. We prove that for any large enough $\Delta$,
with high probability a random $k$-CNF over $n$ variables and $\Delta n$
clauses requires resolution clause space of
$\Omega(n \cdot \Delta^{-\frac{1+\epsilon}{k-2-\epsilon}})$,
for any $0<\epsilon<1/2$. For constant $\Delta$, ... more >>>

The graph reachability problem, the computational task of deciding whether there is a path between two given nodes in a graph is the canonical problem for studying space bounded computations. Three central open questions regarding the space complexity of the reachability problem over directed graphs are: (1) improving Savitch's $O(\log^2 ... more >>>

Combinatorial property testing deals with the following relaxation
of decision problems: Given a fixed property and an input $x$, one
wants to decide whether $x$ satisfies the property or is ``far''
from satisfying it. The main focus of property testing is in
identifying large families of properties that can be ... more >>>

The subclass of directed series-parallel graphs plays an important role in
computer science. Whether a given graph is series-parallel is a
well studied problem in algorithmic graph theory, for which fast sequential and
parallel algorithms have been developed in a sequence of papers.
Also methods are known to solve ... more >>>

We prove space hierarchy and separation results for randomized and other semantic models of computation with advice. Previous works on hierarchy and separation theorems for such models focused on time as the resource. We obtain tighter results with space as the resource. Our main theorems are the following.

Let <i>s</i>(<i>n</i>) ... more >>>

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a ... more >>>

In 1989, Babai, Nisan and Szegedy [BNS92] gave a construction of a pseudorandom generator for logspace, based on lower bounds for multiparty communication complexity. The seed length of their pseudorandom generator was $2^{\Theta(\sqrt n)}\,\,\,$, because the best lower bounds for multiparty communication complexity are relatively weak. Subsequently, pseudorandom generators for ... more >>>

We introduce two models of space-bounded quantum interactive proof systems, $\mathbf{QIPL}$ and $\mathbf{QIP_\mathrm{U}L}$. The $\mathbf{QIP_\mathrm{U}L}$ model, a space-bounded variant of quantum interactive proofs ($\mathbf{QIP}$) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, $\mathbf{QIPL}$ allows logarithmically many intermediate measurements per ... more >>>

Driven by exploring the power of quantum computation with a limited number of qubits, we present a novel complete characterization for space-bounded quantum computation, which encompasses settings with one-sided error (unitary coRQL) and two-sided error (BQL), approached from a quantum (mixed) state testing perspective:
- The first family of natural ... more >>>

We consider the reachability problem for a certain class of directed acyclic graphs embedded on surfaces. Let ${\cal G}(m,g)$ be the class of directed acyclic graphs with $m = m(n)$ source vertices embedded on a surface (orientable or non-orientable) of genus $g = g(n)$. We give a log-space reduction that ... more >>>

SUBSET SUM is a well known NP-complete problem:
given $t \in Z^{+}$ and a set $S$ of $m$ positive integers, output YES if and only if there is a subset $S^\prime \subseteq S$ such that the sum of all numbers in $S^\prime$ equals $t$. The problem and its search ... more >>>

Quantum query complexity measures the number of input bits that must be read by a quantum algorithm in order to evaluate a function. Hoyer et al. (2007) have generalized the adversary semi-definite program that lower-bounds quantum query complexity. By giving a matching algorithm, we show that the general adversary lower ... more >>>

We show that sparse affine-invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field ${\mathbb{F}}_q$ and an extension field ${\mathbb{F}}_{q^n}$, a property is a set of functions mapping ${\mathbb{F}}_{q^n}$ to ${\mathbb{F}}_q$. The property is said to be affine-invariant if it ... more >>>

We consider the problem of extracting entropy by sparse transformations, namely functions with a small number of overall input-output dependencies. In contrast to previous works, we seek extractors for essentially all the entropy without any assumption on the underlying distribution beyond a min-entropy requirement. We give two simple constructions of ... more >>>

In 1978, Hartmanis conjectured that there exist no sparse complete sets
for P under logspace many-one reductions. In this paper, in support of
the conjecture, it is shown that if P has sparse hard sets under logspace
many-one reductions, then P is included in DPSPACE[log^2 n].

High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex ... more >>>

We show that random sparse binary linear codes are locally testable and locally decodable (under any linear encoding) with constant queries (with probability tending to one). By sparse, we mean that the code should have only polynomially many codewords. Our results are the first to show that local decodability and ... more >>>

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in $EXP^{NP}$, or even ... more >>>

We relate the existence of disjunctive hard sets for NP to
other well studied hypotheses in complexity theory showing
that if an NP-complete set or a coNP-complete set is
polynomial-time disjunctively reducible
to a sparse set then FP$^{\rm NP}_{||}$ = FP$^{\rm NP[log]}$. Using
a similar argument we obtain also that ... more >>>

Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP, as long as one is willing to make a suitable physical assumption: if the provers are spatially isolated, ... more >>>

We present a new algorithm for Unique Games which is based on purely {\em spectral} techniques, in contrast to previous
work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment.
The approximation ... more >>>

Dinitz, Schapira, and Valadarsky (Algorithmica 2017) introduced the intriguing notion of expanding expanders -- a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only ... more >>>

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\leq \log n$ and an error parameter $\varepsilon > 0$, our algorithm runs in space $\tilde{O}(k\log (N\cdot ... more >>>

We give faster approximation algorithms for the
generalization of two NP-hard spanning tree problems. First,
we investigate the problem of minimizing the degree of
minimum spanning forests. The task is to compute for each
number of connected components a minimum spanning forest
whose degree is as small as possible. Fischer
more >>>

We investigate the effect of restricting the mutation operator in
evolutionary algorithms with respect to the runtime behavior.
Considering the Eulerian cycle problem we present runtime bounds on
evolutionary algorithms with a restricted operator that are much
smaller than the best upper bounds for the ... more >>>

Informally, a language L has speedup if, for any Turing machine for L, there exists one that is better. Blum showed that there are computable languages that have almost-everywhere speedup. These languages were unnatural in that they were constructed for the sole purpose of having such speedup. We identify a ... more >>>

We are interested in constructing short two-message arguments for various languages, where the complexity of the verifier is small (e.g. linear in the input size, or even sublinear if the input is coded appropriately).

In 2000 Aiello et al. suggested the tantalizing possibility of obtaining such arguments for all of ... more >>>

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by
branching on an inequality and its "integer negation.'' That is, we can (nondeterministically ... more >>>

--
Scalar product estimates have so far been used in
proving several unweighted threshold lower bounds.
We show that if a basis set of Boolean functions satisfies
certain weak stability conditions, then
scalar product estimates
yield lower bounds for the size of weighted thresholds
of these basis functions.
Stable ... more >>>

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m pricable edges in a graph. The other edges have a fixed cost. Based on the leader's decision one or more followers optimize a polynomial-time solvable combinatorial minimization ... more >>>

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of $t \geq \omega(\log^2 n)$ on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small ... more >>>

We develop a framework for proving lower bounds on computational problems over distributions, including optimization and unsupervised learning. Our framework is based on defining a restricted class of algorithms, called statistical algorithms, that instead of accessing samples from the input distribution can only obtain an estimate of the expectation ... more >>>

The polarization lemma for statistical distance ($\mathrm{SD}$), due to Sahai and Vadhan (JACM, 2003), is an efficient transformation taking as input a pair of circuits $(C_0,C_1)$ and an integer $k$ and outputting a new pair of circuits $(D_0,D_1)$ such that if $\mathrm{SD}(C_0,C_1)\geq\alpha$ then $\mathrm{SD}(D_0,D_1) \geq 1-2^{-k}$ and if $\mathrm{SD}(C_0,C_1) \leq ... more >>>

We prove the first {\em Statistical Query lower bounds} for two fundamental high-dimensional learning problems involving Gaussian distributions: (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown mean Gaussian. In particular, we show a {\em super-polynomial gap} between the (information-theoretic) sample complexity and the ... more >>>

We show that every language in NP has a *statistical* zero-knowledge
argument system under the (minimal) complexity assumption that
one-way functions exist. In such protocols, even a computationally
unbounded verifier cannot learn anything other than the fact that the
assertion being proven is true, whereas a polynomial-time prover
cannot convince ... more >>>

A statistical zero knowledge argument for NP is a cryptographic primitive that allows a polynomial-time prover to convince another
polynomial-time verifier of the validity of an NP statement. It is guaranteed that even an infinitely powerful verifier does not learn any
additional information but the validity of the claim.

Naor ... more >>>

The problem of minimizing the share size of threshold secret-sharing schemes is a basic research question that has been extensively studied. Ideally, one strives for schemes in which the share size equals the secret size. While this is achievable for large secrets (Shamir, CACM '79), no similar solutions are known ... more >>>

A fundamental notion in Algorithmic Statistics is that of a stochastic object, i.e., an object having a simple plausible explanation. Informally, a probability distribution is a plausible explanation for $x$ if it looks likely that $x$ was drawn at random with respect to that distribution. In this paper, we ... more >>>

We consider the following problem that arises in outsourced storage: a user stores her data $x$ on a remote server but wants to audit the server at some later point to make sure it actually did store $x$. The goal is to design a (randomized) verification protocol that has the ... more >>>

In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing.
This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing ... more >>>

In this paper, we give streaming algorithms for some problems which are known to be in deterministic log-space, when the number of passes made on the input is unbounded. If the input data is massive,
the conventional deterministic log-space algorithms may not run efficiently. We study the complexity of the ... more >>>

In this survey we describe progress over the last decade or so in understanding the complexity of solving constraint satisfaction problems (CSPs) approximately in the streaming and sketching models of computation. After surveying some of the results we give some sketches of the proofs and in particular try to explain ... 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 give an $\widetilde{O}(\sqrt{n})$-space single-pass $0.483$-approximation streaming algorithm for estimating the maximum directed cut size (Max-DICUT) in a directed graph on $n$ vertices. This improves over an $O(\log n)$-space $4/9 < 0.45$ approximation algorithm due to Chou, Golovnev, Velusamy (FOCS 2020), which was known to be optimal for $o(\sqrt{n})$-space algorithms.

In graph streaming a graph with $n$ vertices and $m$ edges is presented as a read-once stream of edges. We obtain an $\Omega(n \log n)$ lower bound on the space required to decide graph connectivity. This improves the known bounds of $\Omega(n)$ for undirected and $\Omega(m)$ for sparse directed graphs. ... more >>>

We define the Streaming Communication model that combines the main aspects of communication complexity and streaming. We consider two agents that want to compute some function that depends on inputs that are distributed to each agent. The inputs arrive as data streams and each agent has a bounded memory. Agents ... more >>>

This work is in the line of designing efficient checkers for testing the reliability of some massive data structures. Given a sequential access to the insert/extract operations on such a structure, one would like to decide, a posteriori only, if it corresponds to the evolution of a reliable structure. In ... more >>>

We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely Max-DICUT, for which random ordering makes a provable difference. Whereas a $4/9 \approx 0.445$ approximation of DICUT requires $\Omega(\sqrt{n})$ space with ... more >>>

Motivated by the trend to outsource work to commercial cloud computing services, we consider a variation of the streaming paradigm where a streaming algorithm can be assisted by a powerful helper that can provide annotations to the data stream. We extend previous work on such annotation models by considering a ... more >>>

Frequency estimation in data streams is one of the classical problems in streaming algorithms. Following much research, there are now almost matching upper and lower bounds for the trade-off needed between the number of samples and the space complexity of the algorithm, when the data streams are adversarial. However, in ... more >>>

In the context of language recognition, we demonstrate the superiority of streaming property testers against streaming algorithms and property testers, when they are not combined. Initiated by Feigenbaum et al, a streaming property tester is a streaming algorithm recognizing a language under the property testing approximation: it must distinguish inputs ... more >>>

We study graph computations in an enhanced data streaming setting, where a space-bounded client reading the edge stream of a massive graph may delegate some of its work to a cloud service. We seek algorithms that allow the client to verify a purported proof sent by the cloud service that ... more >>>

We give new algorithms in the annotated data streaming setting---also known as verifiable data stream computation---for certain graph problems. This setting is meant to model outsourced computation, where a space-bounded verifier limited to sequential data access seeks to overcome its computational limitations by engaging a powerful prover, without needing to ... more >>>

In this paper, the author puts forward a variation of Feige's Hypothesis, which claims that it is hard on average refuting Unbalanced Max 3-XOR under biased assignments on a natural distribution. Under this hypothesis, the author strengthens the previous known hardness for approximating Minimum Unique Game, $5/4-\epsilon$, by proving that ... more >>>

We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work (Rudich 1997), and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following:

?*Demi-bit stretch*: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are ... more >>>

The notion of efficient computation is usually identified in cryptography and complexity with probabilistic polynomial time. However, until recently, in order to obtain \emph{constant-round} zero-knowledge proofs and proofs of knowledge, one had to allow simulators and knowledge-extractors to run in time which is only polynomial {\em on the average} (i.e., ... more >>>

Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above self-assemblies tile completely ... more >>>

The sorting by strip exchanges (aka strip exchanging) problem is motivated by applications in optical character recognition and genome rearrangement analysis. While a couple of approximation algorithms have been designed for the problem, nothing has been known about its computational hardness till date. In this work, we show that the ... more >>>

In this paper
we establish a general algorithmic framework between bin packing
and strip packing, with which we achieve the same asymptotic
bounds by applying bin packing algorithms to strip packing. More
precisely we obtain the following results: (1) Any offline bin
packing algorithm can be applied to strip packing ... 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 >>>

We suggest a general framework to study dependency schemes for dependency quantified Boolean formulas (DQBF). As our main contribution, we exhibit a new tautology-free DQBF dependency scheme that generalises the reflexive resolution path dependency scheme. We establish soundness of the tautology-free scheme, implying that it can be used in any ... more >>>

We prove that for all constants a, NQP = NTIME[n^{polylog(n)}] cannot be (1/2 + 2^{-log^a n})-approximated by 2^{log^a n}-size ACC^0 of THR circuits (ACC^0 circuits with a bottom layer of THR gates). Previously, it was even open whether E^NP can be (1/2+1/sqrt{n})-approximated by AC^0[2] circuits. As a straightforward application, ... more >>>

A zero-knowledge proof enables a prover to convince a verifier that $x \in S$, without revealing anything beyond this fact. By running a zero-knowledge proof $k$ times, it is possible to prove (still in zero-knowledge) that $k$ separate instances $x_1,\dots,x_k$ are all in $S$. However, this increases the communication by ... more >>>

A strong direct product theorem (SDPT) states that solving $n$ instances of a problem requires $\Omega(n)$ times the resources for a single instance, even to achieve success probability $2^{-\Omega(n)}.$ We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by ... more >>>

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$,
$\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and ... more >>>

We develop a paradigm for studying multi-player deterministic communication,
based on a novel combinatorial concept that we call a {\em strong fooling
set}. Our paradigm leads to optimal lower bounds on the per-player
communication required for solving multi-player $\textsc{equality}$
problems in a private-message setting. This in turn gives a ... more >>>

Impagliazzo and Levin demonstrated [IL90] that the average-case hardness of any NP-search problem under any P-samplable distribution implies that of another NP-search problem under the uniform distribution. For this they developed a way to define a reduction from an NP-search problem F with ``mild hardness'' under any P-samplable distribution H; ... more >>>

Consider a $K$-uniform hypergraph $H = (V,E)$. A coloring $c : V \rightarrow \{1, 2, \dots, k \}$ with $k$ colors is rainbow if every hyperedge $e$ contains at least one vertex from each color, and is called perfectly balanced when each color appears the same number of times. A ... more >>>

Locally testable codes (LTCs) are error-correcting codes
that admit very efficient codeword tests. An LTC is said to
be strong if it has a proximity-oblivious tester;
that is, a tester that makes only a constant number of queries
and reject non-codewords with probability that depends solely
on their distance from ... more >>>

An error-correcting code $C \subseteq \F^n$ is called $(q,\epsilon)$-strong locally testable code (LTC) if there exists a tester that makes at most $q$ queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords $x\notin C$ with probability at least $\epsilon \cdot \delta(x,C)$, where ... more >>>

An error-correcting code $C \subseteq \F^n$ is called $(q,\epsilon)$-strong locally testable code (LTC) if there exists a randomized algorithm (tester) that makes at most $q$ queries to the input word. This algorithm accepts all codewords with probability 1 and rejects all non-codewords $x\notin C$ with probability at least $\epsilon \cdot ... 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 >>>

In a pair of recent breakthroughs \cite{CHR,Li} it was shown that the classes $S_2^E, ZPE^{NP}$, and $\Sigma_2^E$ require exponential circuit complexity, giving the first unconditional improvements to a classical result of Kannan. These results were obtained by designing a surprising new algorithm for the total search problem Range Avoidance: given ... more >>>

In this paper, we prove a strong XOR lemma for bounded-round two-player randomized communication. For a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow\{0,1\}$, the $n$-fold XOR function $f^{\oplus n}:\mathcal{X}^n\times \mathcal{Y}^n\rightarrow\{0,1\}$ maps $n$ input pairs $(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$ to the XOR of the $n$ output bits $f(X_1,Y_1)\oplus \cdots \oplus f(X_n, Y_n)$. We prove that if every ... more >>>

We associate to each Boolean language complexity class $\mathcal{C}$ the algebraic class $a\cdot\mathcal{C}$ consisting of families of polynomials $\{f_n\}$ for which the evaluation problem over the integers is in $\mathcal{C}$. We prove the following lower bound and randomness-to-hardness results:

1. If polynomial identity testing (PIT) is in $NSUBEXP$ then $a\cdot ... more >>>

We consider the \emph{Hidden Subgroup} in the context of quantum \emph{Ordered Binary Decision Diagrams}.
We show several lower bounds for this function.
In this paper we also consider a slightly more general definition of the
hidden subgroup problem (in contrast to that in \cite{khashsp1}). It turns out that ... more >>>

For a universal constant $\alpha > 0$, we prove size lower bounds of $2^{\alpha N}$ for computing an explicit monotone function in NP in the following models of computation: monotone formulas, monotone switching networks, monotone span programs, and monotone comparator circuits, where $N$ is the number of variables of the ... more >>>

We show that there is a sequence of explicit multilinear polynomials $P_n(x_1,\ldots,x_n)\in \mathbb{R}[x_1,\ldots,x_n]$ with non-negative coefficients that lies in monotone VNP such that any monotone algebraic circuit for $P_n$ must have size $\exp(\Omega(n)).$ This builds on (and strengthens) a result of Yehudayoff (2018) who showed a lower bound of $\exp(\tilde{\Omega}(\sqrt{n})).$

{\bf Abstract}

Isometries on formal power series over the finite field $\ff_2$
or on $2$--adic integers can be
computed by invertible transducers on inputs from $\{0,1\}^\infty$.
We consider the structural complexity of an isometry $f$,
measured as {\it tree complexity} $T(f,h)$, $h$ the tree height
[H.~Niederreiter, M.~Vielhaber, {\it J.~Cpx.}, ... more >>>

We study class AvgBPP that consists of distributional problems that can be solved in average polynomial time (in terms of Levin's average-case complexity) by randomized algorithms with bounded error. We prove that there exists a distributional problem that is complete for AvgBPP under polynomial-time samplable distributions. Since we use deterministic ... more >>>

The study of the approximability properties of NP-hard
optimization problems has recently made great advances mainly due
to the results obtained in the field of proof checking. In a
recent breakthrough the APX-completeness of several important
optimization problems is proved, thus reconciling `two distinct
views of ... more >>>

Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often ... more >>>

Let $f:\{0,1\}^n \to \{0,1\}$ be a boolean function. Its associated XOR function is the two-party function $f_\oplus(x,y) = f(x \oplus y)$.
We show that, up to polynomial factors, the deterministic communication complexity of $f_{\oplus}$ is equal to the parity decision tree complexity of $f$.
This relies on a novel technique ... more >>>

It is well-known that every first-order property on words is expressible
using at most three variables. The subclass of properties expressible with
only two variables is also quite interesting and well-studied. We prove
precise structure theorems that characterize the exact expressive power of
first-order logic with two variables on words. ... more >>>

Some computational problems seem to have a certain "structure" that is manifested in non-trivial algorithmic properties, while others are more "unstructured" in the sense that they are either "very easy" or "very hard". I survey some of the known results and open questions about this classification and its connections to ... more >>>

Cryptography relies on the computational hardness of structured problems. While one-way functions, the most basic cryptographic object, do not seem to require much structure, as we advance up the ranks into public-key cryptography and beyond, we seem to require that certain structured problems are hard. For example, factoring, quadratic residuosity, ... more >>>

We present a deterministic algorithm running in space
O((log^2 n)/loglog n) solving the connectivity problem
on strongly unambiguous graphs. In addition, we present
an O(log n) time-bounded algorithm for this problem
running on a parallel pointer machine.

Given a function f:F^m \rightarrow F over a finite
field F, a low degree tester tests its proximity to
an m-variate polynomial of total degree at most d
over F. The tester is usually given access to an oracle
A providing the supposed restrictions of f to
affine subspaces of ... more >>>

We show a construction of a PCP with both sub-constant error and
almost-linear size. Specifically, for some constant alpha in (0,1),
we construct a PCP verifier for checking satisfiability of
Boolean formulas that on input of size n uses log n + O((log
n)^{1-alpha}) random bits to query a constant ... more >>>

We prove that the Fourier dimension of any Boolean function with
Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof
method yields an improved bound of $\widetilde{O}(\sqrt{s})$
assuming a conjecture of Tsang~\etal~\cite{tsang}, that for every
Boolean function of sparsity $s$ there is an affine subspace of
more >>>

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models.

For depth-3 multilinear formulas, of size $\exp(n^\delta)$, we give a hitting set of size $\exp(\tilde{O}(n^{2/3 + 2\delta/3}))$. ... more >>>

We raise the question of approximating compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and Lempel-Ziv (LZ), and present algorithms and lower bounds for approximating compressibility with respect to ... 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 >>>

Sublinear time algorithms represent a new paradigm
in computing, where an algorithm must give some sort
of an answer after inspecting only a very small portion
of the input. We discuss the types of answers that
one can hope to achieve in this setting.

We show that all non-negative submodular functions have high noise-stability. As a consequence, we obtain a polynomial-time learning algorithm for this class with respect to any product distribution on $\{-1,1\}^n$ (for any constant accuracy parameter $\epsilon$ ). Our algorithm also succeeds in the agnostic setting. Previous work on learning submodular ... more >>>

We study the question of whether the value of mathematical programs such as
linear and semidefinite programming hierarchies on a graph $G$, is preserved
when taking a small random subgraph $G'$ of $G$. We show that the value of the
Goemans-Williamson (1995) semidefinite program (SDP) for \maxcut of $G'$ is
more >>>

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The ... more >>>

In this work we describe an explicit, simple, construction of large subsets of F^n, where F is a finite field, that have small intersection with every k-dimensional affine subspace. Interest in the explicit construction of such sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl (2004) ... more >>>

We study which functions can be computed by efficient circuits whose gate connections are very easy to compute. We give quasilinear-size circuits for sorting whose connections can be computed by decision trees with depth logarithmic in the length of the gate description. We also show that NL has NC$^2$ circuits ... more >>>

This note connects two topics of Complexity Theory: The
topic of succinct circuit representations initiated by
Galperin and Wigderson and the topic of leaf languages
initiated by Bovet, Crescenzi, and Silvestri. It will be
shown for any language that its succinct version is
more >>>

A secret-sharing scheme enables a dealer to share a secret $s$ among $n$ parties such that only authorized subsets of parties, specified by a monotone access structure $f:\{0,1\}^n\to\{0,1\}$, can reconstruct $s$ from their shares. Other subsets of parties learn nothing about $s$.

The question of minimizing the (largest) share size ... more >>>

It is well known that problems encoded with circuits or formulas generally gain an exponential complexity blow-up compared to their original complexity.

We introduce a new way for encoding graph problems, based on $\textrm{CNF}$ or $\textrm{DNF}$ formulas. We show that contrary to the other existing succinct models, there are ... more >>>

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been ... more >>>

Finding a problem that is both hard to solve and hard to solve on many instances is a long standing issue
in theoretical computer science.
In this work, we prove that the Succinct Permanent $\bmod \; p$ is $NEXP$
time hard in the worst case (via randomized polynomial time ... more >>>

In this paper, we present stronger results in the theory of succinct
problem representation by boolean circuits, and establish a close
relationship between succinct problems and leaf languages. As
a major tool, we use quantifierfree projection reductions
from descriptive complexity theory.

In particular, we show that the succinct upgrading ... more >>>

Motivated by questions in property testing, we search for linear
error-correcting codes that have the ``single local orbit'' property:
i.e., they are specified by a single local
constraint and its translations under the symmetry group of the
code. We show that the dual of every ``sparse'' binary code
whose coordinates
more >>>

We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of ``well-behaved'' univariate polynomial inequalities.

We illustrate the technique on ... more >>>

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree $\Omega(s^{1-\epsilon})$ to prove that $f$ does not have circuits of size $s$ (for any $s > \text{poly}(n)$). ... more >>>

Several works have shown unconditional hardness (via integrality gaps) of computing equilibria using strong hierarchies of convex relaxations. Such results however only apply to the problem of computing equilibria that optimize a certain objective function and not to the (arguably more fundamental) task of finding \emph{any} equilibrium.

We present ... more >>>

In order to obtain the best-known guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible ... more >>>

We study the complexity of representing polynomials as a sum of products of polynomials in few variables. More precisely, we study representations of the form $$P = \sum_{i = 1}^T \prod_{j = 1}^d Q_{ij}$$
such that each $Q_{ij}$ is an arbitrary polynomial that depends on at most $s$ variables.

An arithmetic read-once formula (ROF) is a formula (circuit of fan-out
1) over
$+, \times$ where each variable labels at most one leaf.
Every multilinear polynomial can be expressed as the sum of ROFs.
In this work, we prove, for certain multilinear polynomials,
a tight lower bound ... more >>>

The Erdos-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 ... more >>>

A \emph{sunflower} is a collection of sets whose pairwise intersections are identical. In this article, we shall go sunflower-picking. We find sunflowers in several seemingly unrelated fields, before turning to discuss recent progress on the famous sunflower conjecture of Erd\H{o}s and Rado, made by Alweiss, Lovett, Wu and Zhang.

We construct $k$-CNFs with $m$ variables on which the strong version of PPSZ $k$-SAT algorithm, which uses bounded width resolution, has success probability at most $2^{-(1 - (1 + \epsilon)2/k)m}$ for every $\epsilon > 0$. Previously such a bound was known only for the weak PPSZ algorithm which exhaustively searches ... more >>>

Razborov [J. ACM, 2016] exhibited the following surprisingly strong trade-off phenomenon in propositional proof complexity: for a parameter $k = k(n)$, there exists $k$-CNF formulas over $n$ variables, having resolution refutations of $O(k)$ width, but every tree-like refutation of width $n^{1-\epsilon}/k$ needs size $\text{exp}\big(n^{\Omega(k)}\big)$. We extend this result to tree-like ... more >>>

In this paper, we prove a super-cubic lower bound on the size of a communication protocol for generalized Karchmer-Wigderson game for some explicit function $f: \{0,1\}^n\to \{0,1\}^{\log n}$. Lower bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic. ... more >>>

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for ... more >>>

We prove the first time-space lower bound tradeoffs for randomized
computation of decision problems. The bounds hold even in the
case that the computation is allowed to have arbitrary probability
of error on a small fraction of inputs. Our techniques are an
extension of those used by Ajtai in his ... more >>>

We initiate a study of super-perfect zero-knowledge proof systems.
Loosely speaking, these are proof systems for which the interaction can be perfectly simulated in strict probabilistic polynomial-time.
In contrast, the standard definition of perfect zero-knowledge only requires that the interaction can be perfectly simulated
by a strict probabilistic polynomial-time that ... more >>>

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the “short code” of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results.

In particular, we prove quasi-NP-hardness of the following problems on ... more >>>

We show that there are CNF formulas which can be refuted in resolution
in both small space and small width, but for which any small-width
proof must have space exceeding by far the linear worst-case upper
bound. This significantly strengthens the space-width trade-offs in
[Ben-Sasson '09]}, and provides one more ... more >>>

We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the so-called supercritical regime. Our proof is based on an analogous result in proof complexity: ... more >>>

We prove $n^{1+\Omega(1/p)}/p^{O(1)}$ lower bounds for the space complexity of $p$-pass streaming algorithms solving the following problems on $n$-vertex graphs:

* testing if an undirected graph has a perfect matching (this implies lower bounds for computing a maximum matching or even just the maximum matching size),

* testing if two ... more >>>

A {\em supermodel} is a satisfying assignment of a boolean formula
for which any small alteration, such as a single bit flip, can be
repaired by another small alteration, yielding a nearby
satisfying assignment. We study computational problems associated
with super models and some generalizations thereof. For general
formulas, ... 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 >>>

In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree $n$ in $n^2$ variables such that any homogeneous depth 4 arithmetic circuit computing it must have size $n^{\Omega(\log \log n)}$.

Our results extend ... more >>>

Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time $n^{\tilde O(\log\log s)}$ for the classes of $n$-variable size-$s$ DNF, size-$s$ Decision Tree, and $\log s$-Junta by DNF (that returns a DNF hypothesis). Assuming a natural ... more >>>

We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove:

(1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design 3-LCC has the additional property that the correcting sets for every ... more >>>

A superredundant clause is a clause that is redundant in the resolution closure of a formula. The converse concept of superirredundancy ensures membership of the clause in all minimal CNF formulae that are equivalent to the given one. This allows for building formulae where some clauses are fixed when minimizing ... more >>>

We survey recent results on disjoint NP-pairs. In particular, we survey the relationship of disjoint NP-pairs to the theory of proof systems for propositional calculus.

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing ... more >>>

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 >>>

We introduce symmetric Datalog, a syntactic restriction of linear
Datalog and show that its expressive power is exactly that of
restricted symmetric monotone Krom SNP. The deep result of
Reingold on the complexity of undirected
connectivity suffices to show that symmetric Datalog queries can be
evaluated in logarithmic space. We ... more >>>

We show that there is a language in $\mathrm{S}_2\mathrm{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds for the classes $\Sigma_2\mathrm{E}$, $(\Sigma_2\mathrm{E}\cap\Pi_2\mathrm{E})/_1$, and $\mathrm{ZPE}^{\mathrm{NP}}/_1$. Previously, only "half-exponential" circuit lower bounds for these ... more >>>

In a recent breakthrough, Chen, Hirahara and Ren prove that S$_2$E/$_1 \not\subset$ SIZE$[2^n/n]$ by giving a single-valued FS$_2$P algorithm for the Range Avoidance Problem (Avoid) that works for infinitely many input size $n$.

Building on their work, we present a simple single-valued FS$_2$P algorithm for Avoid that works for all ... more >>>

Locally testable codes, i.e., codes where membership in the code is testable with a constant number of queries, have played a central role in complexity theory. It is well known that a code must be a "low-density parity check" (LDPC) code for it to be locally testable, but few LDPC ... more >>>

We present a Logspace, many-one reduction from the undirected
st-connectivity problem to its complement. This shows that
$SL=co-SL$

We study the problem of representing symmetric Boolean functions as symmetric polynomials over Z_m. We show an equivalence between such
representations and simultaneous communication protocols. Computing a function with a polynomial of degree d modulo m=pq is equivalent to a two player protocol where one player is given the first ... more >>>

Suppose that $p$ is a prime number $A$ is a finite set
with $n$ elements
and for each sequence $a=<a_{1},...,a_{k}>$ of length $k$ from the
elements of
$A$, $x_{a}$ is a variable. (We may think that $k$ and $p$ are fixed an
$n$ is sufficiently large.) We will ... more >>>

Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games by considering ... more >>>

We show that every language accepted by a nondeterministic auxiliary pushdown automaton in polynomial time (that is, every language in SAC$^1$ = LogCFL) can be accepted by a symmetric auxiliary pushdown automaton in polynomial time.

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum ... more >>>

We investigate the complexity of the syntactic isomorphism problem of CNF Boolean Formulas (CSFI): given two CNF Boolean formulas $\varphi(a_{1},\ldots,a_{n})$ and $\varphi(b_{1},\ldots,b_{n})$ decide whether there exists a permutation of clauses, a permutation of literals and a bijection between their variables such that $\varphi(a_{1},\ldots,a_{n})$ and $\varphi(b_{1},\ldots,b_{n})$ become syntactically identical. We first ... more >>>

A pseudo-random function is a fundamental cryptographic primitive
that is essential for encryption, identification and authentication.
We present a new cryptographic primitive called pseudo-random
synthesizer and show how to use it in order to get a
parallel construction of a pseudo-random function.
We show an $NC^1$ implementation of synthesizers ... more >>>

We show that the class ${\rm S}_2^p$ is a subclass of
${{\rm ZPP}^{\rm NP}}$. The proof uses universal hashing, approximate counting
and witness sampling. As a consequence, a collapse first noticed by
Samik Sengupta that the assumption NP has small circuits collapses
PH to ${\rm S}_2^p$
becomes the strongest version ... more >>>