We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied ... more >>>

We present an alternate proof of the result by Kabanets and Impagliazzo that derandomizing polynomial identity testing implies circuit lower bounds. Our proof is simpler, scales better, and yields a somewhat stronger result than the original argument.

We consider the notion of a local-characterization of an infinite family of unlabeled bounded-degree graphs.
Such a local-characterization is defined in terms of a finite set of (marked) graphs yielding a generalized notion of subgraph-freeness, which extends the standard notions of induced and non-induced subgraph freeness.

We survey the work ... more >>>

We consider the problem of testing isomorphism to a fixed graph in the bounded-degree graph model. Our main result is that, for almost all $d$-regular $n$-vertex graphs $H$,
testing isomorphism to $H$ can be done using $\tildeO({\sqrt n})$ queries.
This result is shown to be optimal (up to ... more >>>

We revisit the known proof of the lower bound on the length of relaxed locally decodable codes, providing an arguably simpler exposition that yields a slightly better lower bound for the non-adaptive case and a weaker bound in the general case.

Recall that a locally decodable code is an error ... more >>>

We study a new type of separation between quantum and classical communication complexity which is obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits with oracle access to their inputs. More precisely, we give an explicit partial Boolean ... more >>>

The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this work, we connect this problem to a variant of another major problem: the quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the ... more >>>

We present a deterministic algorithm that counts the number of satisfying assignments for any de Morgan formula $F$ of size at most $n^{3-16\epsilon}$ in time $2^{n-n^{\epsilon}}\cdot \mathrm{poly}(n)$, for any small constant $\epsilon>0$. We do this by derandomizing the randomized algorithm mentioned by Komargodski et al. (FOCS, 2013) and Chen et ... more >>>

We prove the following hardness result for a natural promise variant of the classical CNF-satisfiability problem: Given a CNF-formula where each clause has width $w$ and the guarantee that there exists an assignment satisfying at least $g = \lceil \frac{w}{2}\rceil -1$ literals in each clause, it is NP-hard to find ... more >>>

We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{(\log N)^{1/4-o(1)}}$ colors, where $N$ is the number of vertices. There has been much focus on hardness of hypergraph coloring recently. Guruswami, H{\aa}stad, Harsha, Srinivasan and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with ... more >>>

We prove that for an arbitrarily small constant $\eps>0,$ assuming NP$\not \subseteq$DTIME$(2^{{\log^{O(1/\epsilon)} n}})$, the preprocessing versions of the closest vector problem and the nearest codeword problem are hard to approximate within a factor better than $2^{\log ^{1-\epsilon}n}.$ This improves upon the previous hardness factor of $(\log n)^\delta$ for some $\delta ... more >>>

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

A function $f:\Sigma^{*} \rightarrow \Sigma^{*}$ on strings is $AC^0$-pseudorandom if the pair $(x,\hat f(x))$ is $AC^0$-indistinguishable from a uniformly random pair $(y,z)$ when $x$ is chosen uniformly at random. Here $\hat f(x)$ is the string that is obtained from $f(x)$ by discarding some selected bits from $f(x)$.

It is shown ... more >>>

The Minimum Circuit Size Problem (MCSP) asks whether a (given) Boolean function has a circuit of at most a (given) size. Despite over a half-century of study, we know relatively little about the computational complexity of MCSP. We do know that questions about the complexity of MCSP have significant ramifications ... more >>>

Whether $BPL=L$ (which is conjectured to be equal), or even whether $BPL\subseteq NL$, is a big open problem in theoretical computer science. It is well known that $L-NC^1\subseteq L\subseteq NL\subseteq L-AC^1$. In this work we will show that $BPL\subseteq L-AC^1$, which was not known before. Our proof is based on ... more >>>

The $d$-to-$1$ conjecture of Khot asserts that it is hard to satisfy an $\epsilon$ fraction of constraints of a satisfiable $d$-to-$1$ Label Cover instance, for arbitrarily small $\epsilon > 0$. We prove that the $d$-to-$1$ conjecture for any fixed $d$ implies the hardness of coloring a $4$-colorable graph with $C$ ... more >>>

A small-biased distribution of bit sequences is defined as one withstanding $GF(2)$-linear tests for randomness, which are linear combinations of the bits themselves. We consider linear combinations over larger fields, specifically, $GF(2^n)$ for $n$ that divides the length of the bit sequence. Indeed, this means that we partition the bits ... more >>>

Aaronson and Ambainis (SICOMP '18) showed that any partial function on $N$ bits that can be computed with an advantage $\delta$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $\delta/2$ by a randomized decision tree making ${O}_q(N^{1-\frac{1}{2q}}\delta^{-2})$ queries. Moreover, they conjectured ... more >>>

We consider the complexity of LS$_+$ refutations of unsatisfiable instances of Constraint Satisfaction Problems (CSPs) when the underlying predicate supports a pairwise independent distribution on its satisfying assignments. This is the most general condition on the predicates under which the corresponding MAX-CSP problem is known to be approximation resistant.

We ... more >>>

We give a conversion from non-classical polynomials to $\mathit{MidBit}^+$ circuits and vice-versa. This conversion, along with previously known results, shows that torus polynomials, non-classical polynomials and $\mathit{MidBit}^+$ circuits can all be converted to each other. Therefore lower bounds against any of these models lead to lower bounds against all three ... more >>>

$\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuits are $\mathrm{AC}^{0}$ circuits augmented with a layer of parity gates just above the input layer. We study the $\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuit lower bound for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have ... more >>>

We give a three-player game whose non-signaling value is constant (2/3) under any number of parallel repetitions. This is the first known setting where parallel repetition completely fails to reduce the maximum winning probability of computationally unbounded players.

We also show that the best known results on non-signaling ... more >>>

The Local Search problem, which finds a
local minimum of a black-box function on a given graph, is of both
practical and theoretical importance to many areas in computer
science and natural sciences. In this paper, we show that for the
Boolean hypercube $\B^n$, the randomized query complexity of Local
more >>>

A ZAP is a witness-indistinguishable two-message public-coin interactive proof with the following simple structure: the verifier sends a uniformly random string, the prover responds, and the verifier decides in polynomial time whether to accept or reject.

We show that one-way functions imply the existence of ... more >>>

We present the first truly explicit constructions of \emph{non-malleable codes} against tampering by bounded polynomial size circuits. These objects imply unproven circuit lower bounds and our construction is secure provided E requires exponential size nondeterministic circuits, an assumption from the derandomization literature.

Prior works on NMC ... more >>>

One of the major open problems in proof complexity is to prove lower bounds on $AC_0[p]$-Frege proof
systems. As a step toward this goal Impagliazzo, Mouli and Pitassi in a recent paper suggested to prove
lower bounds on the size for Polynomial Calculus over the $\{\pm 1\}$ basis. In this ... more >>>

We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time $2^{(1-\text{poly}(1/c))n}$ where $n$ is the
number of variables and $cn$ is the number of constraints. The key ... more >>>

We give a 1.25 approximation algorithm for the Steiner Tree Problem with distances one and two, improving on the best known bound for that problem.

Analysis of genomes evolving by inversions leads to a general
combinatorial problem of {\em Sorting by Reversals}, MIN-SBR, the problem of
sorting a permutation by a minimum number of reversals.
This combinatorial problem has a long history, and a number of other
motivations. It was studied in a great ... more >>>

The Steiner tree problem requires to find a shortest tree connection
a given set of terminal points in a metric space. We suggest a better
and fast heuristic for the Steiner problem in graphs and in
rectilinear plane. This heuristic finds a Steiner tree at ... more >>>

The 2-D Tucker search problem is shown to be PPA-hard under many-one reductions; therefore it is complete for PPA. The same holds for $k$-D Tucker for all $k\ge 2$. This corrects a claim in the literature that the Tucker search problem is in PPAD.

Yao (in a lecture at DIMACS Workshop on structural complexity and
cryptography) showed that if a language L is 2-locally-random
reducible to a Boolean functio, then L is in PSPACE/poly.
Fortnow and Szegedy quantitatively improved Yao's result to show that
such languages are in fact in NP/poly (Information Processing Letters, ... more >>>

A 2-server Private Information Retrieval (PIR) scheme allows a user to retrieve the $i$th bit of an $n$-bit database replicated among two servers (which do not communicate) while not revealing any information about $i$ to either server. In this work we construct a 1-round 2-server PIR with total communication cost ... more >>>

A basic goal in Property Testing is to identify a
minimal set of features that make a property testable.
For the case when the property to be tested is membership
in a binary linear error-correcting code, Alon et al.~\cite{AKKLR}
had conjectured that the presence of a {\em single} low weight
more >>>

We prove that a variant of 2048, a popular online puzzle game, is PSPACE-Complete. Our hardness result
holds for a version of the problem where the player has oracle access to the computer player's moves.
Specifically, we show that for an $n \times n$ game board $G$, computing a
more >>>

We consider worst case time bounds for NP-complete problems
including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring.
Our algorithms are based on a common generalization of these problems,
called symbol-system satisfiability or, briefly, SSS [R. Floyd &
R. Beigel, The Language of Machines]. 3-SAT is equivalent to
(2,3)-SSS while the other problems ... more >>>

In this paper, we improve a recent result of Daskalakis, Goldberg and Papadimitriou on PPAD-completeness of 4-Nash, showing that 3-Nash is PPAD-complete.

Locally Decodable Codes (LDC) allow one to decode any particular
symbol of the input message by making a constant number of queries
to a codeword, even if a constant fraction of the codeword is
damaged. In recent work ~\cite{Yekhanin08} Yekhanin constructs a
$3$-query LDC with sub-exponential length of size
$\exp(\exp(O(\frac{\log ... more >>>

This paper establishes a randomized algorithm that finds a satisfying assignment for a satisfiable formula $F$ in 3-CNF in $O(1.32793^n)$ expected running time. The algorithms is based on the analysis of so-called strings, which are sequences of 3-clauses where non-succeeding clauses do not share a variable and succeeding clauses share ... more >>>

For a boolean formula \phi on n variables, the associated property
P_\phi is the collection of n-bit strings that satisfy \phi. We prove
that there are 3CNF properties that require a linear number of queries,
even for adaptive tests. This contrasts with 2CNF properties
that are testable with O(\sqrt{n}) ... more >>>

We show that if one can solve 3SUM on a set of size $n$
in time $n^{1+\epsilon}$ then one can list $t$ triangles in a
graph with $m$ edges in time $\tilde
O(m^{1+\epsilon}t^{1/3+\epsilon'})$ for any $\epsilon' > 0$. This is a
reversal of Patrascu's reduction from 3SUM to
listing triangles ... more >>>

In a STOC 1976 paper, Schaefer proved that it is PSPACE-complete to determine the winner of the so-called Maker-Breaker game on a given set system, even when every set has size at most 11. Since then, there has been no improvement on this result. We prove that the game remains ... more >>>

We design a polynomial time 8/7-approximation algorithm for the Traveling Salesman Problem in which all distances are either one or two. This improves over the best known approximation factor of 7/6 for that problem. As a direct application we get a 7/6-approximation algorithm for the Maximum Path Cover Problem, similarily ... more >>>

We prove that MAX-3SAT can be approximated in polynomial time
within a factor 9/8 on random instances.

We express some criticism about the definition of an algorithmic sufficient statistic and, in particular, of an algorithmic minimal sufficient statistic. We propose another definition, which has better properties.

iven a function $f : \{0,1\}^n \to \reals$, its {\em Fourier Entropy} is defined to be $-\sum_S \fcsq{f}{S} \log \fcsq{f}{S}$, where $\fhat$ denotes the Fourier transform of $f$. This quantity arises in a number of applications, especially in the study of Boolean functions. An outstanding open question is a conjecture ... more >>>