The bin packing problem is to find the minimum
number of bins of size one to pack a list of items with sizes
$a_1,\ldots , a_n$ in $(0,1]$. Using uniform sampling, which selects
a random element from the input list each time, we develop a
randomized $O({n(\log n)(\log\log n)\over ... more >>>

Program checking, program self-correcting and program self-testing
were pioneered by [Blum and Kannan] and [Blum, Luby and Rubinfeld] in
the mid eighties as a new way to gain confidence in software, by
considering program correctness on an input by input basis rather than
full program verification. Work in ... more >>>

We reduce the approximation factor for Vertex Cover to $2-\Theta(1/\sqrt{logn})$
(instead of the previous $2-\Theta(loglogn/logn})$, obtained by Bar-Yehuda and Even,
and by Monien and Speckenmeyer in 1985. The improvement of the vanishing
factor comes as an application of the recent results of Arora, Rao, and Vazirani
that improved ... more >>>

We present a candidate counterexample to the
easy cylinders conjecture, which was recently suggested
by Manindra Agrawal and Osamu Watanabe (ECCC, TR09-019).
Loosely speaking, the conjecture asserts that any 1-1 function
in $P/poly$ can be decomposed into ``cylinders'' of sub-exponential
size that can each be inverted by some polynomial-size circuit.
more >>>

Finding an efficient solution to the general problem of polynomial identity testing (PIT) is a challenging task. In this work, we study the complexity of two special but natural cases of identity testing - first is a case of depth-$3$ PIT, the other of depth-$4$ PIT.

Our first problem is ... more >>>

We prove two new multivariate central limit theorems; the first relates the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance matric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central ... more >>>

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as ... more >>>

The Pfaffian of an oriented graph is closely linked to
Perfect Matching. It is also naturally related to the determinant of
an appropriately defined matrix. This relation between Pfaffian and
determinant is usually exploited to give a fast algorithm for
computing Pfaffians.

We present the first completely combinatorial algorithm for ... more >>>

In [FOCS1998],
Paturi, Pudl\'ak, Saks, and Zane proposed a simple randomized algorithm
for finding a satisfying assignment of a $k$-CNF formula.
The main lemma of the paper is as follows:
Given a satisfiable $k$-CNF formula that
has a $d$-isolated satisfying assignment $z$,
the randomized algorithm finds $z$
with probability at ... more >>>

We provide a characterization of the resolution
width introduced in the context of Propositional Proof Complexity
in terms of the existential pebble game introduced
in the context of Finite Model Theory. The characterization
is tight and purely combinatorial. Our
first application of this result is a surprising
proof that the ... more >>>

We show that the Player-Adversary game from a paper
by Pudlak and Impagliazzo played over
CNF propositional formulas gives
an exact characterization of the space needed
in treelike resolution refutations. This
characterization is purely combinatorial
and independent of the notion of resolution.
We use this characterization to give ... more >>>

The current proof of the PCP Theorem (i.e., NP=PCP(log,O(1)))
is very complicated.
One source of difficulty is the technically involved
analysis of low-degree tests.
Here, we refer to the difficulty of obtaining strong results
regarding low-degree tests; namely, results of the type obtained and
used by Arora ... more >>>

The problem of image matching is to find for two given digital images $A$ and $B$
an admissible transformation that converts image $A$ as close as possible to $B$.
This problem becomes hard if the space of admissible transformations is too complex.
Consequently, in many real applications, like the ones ... more >>>

We provide a compendium of problems that are complete for
symmetric logarithmic space (SL). Complete problems are one method
of studying this class for which programming is nonintuitive. A
number of the problems in the list were not previously known to be
complete. A list ... more >>>

Statistical query (SQ) learning model of Kearns (1993) is a natural restriction of the PAC learning model in which a learning algorithm is allowed to obtain estimates of statistical properties of the examples but cannot see the examples themselves. We describe a new and simple characterization of the query complexity ... more >>>

In this paper we study the approximability of boolean constraint
satisfaction problems. A problem in this class consists of some
collection of ``constraints'' (i.e., functions
$f:\{0,1\}^k \rightarrow \{0,1\}$); an instance of a problem is a set
of constraints applied to specified subsets of $n$ boolean
variables. Schaefer earlier studied ... more >>>

We present the first complete problem for SZK, the class of (promise)
problems possessing statistical zero-knowledge proofs (against an
honest verifier). The problem, called STATISTICAL DIFFERENCE, is to
decide whether two efficiently samplable distributions are either
statistically close or far apart. This gives a new characterization
of SZK that makes ... more >>>

We present a cryptosystem which is complete for the class of probabilistic public-key cryptosystems with bounded error. Besides traditional encryption schemes such as RSA and El Gamal, this class contains probabilistic encryption of Goldwasser-Micali as well as Ajtai-Dwork and NTRU cryptosystems. The latter two are known to make errors with ... more >>>

We consider multivariate pseudo-linear functions
over finite fields of characteristic two. A pseudo-linear polynomial
is a sum of guarded linear-terms, where a guarded linear-term is a product of one or more linear-guards
and a single linear term, and each linear-guard is
again a linear term but raised ... more >>>

Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time hierarchy,
$\mathbf{PH}$,
is contained in the class $\mathbf{P}^{\#\mathbf{P}}$,
namely the class of languages that can be
decided by a Turing machine in polynomial time given access to an
oracle with the power to compute a function in the ... more >>>

This paper solves the open problem of exact learning
geometric objects bounded by hyperplanes (and more generally by any constant
degree algebraic surfaces) in the constant
dimensional space from equivalence queries only (i.e., in the on-line learning
model).

We present a novel approach that allows, under certain ... more >>>

We exhibit a new computational-based definition of awareness,
informally that our level of unawareness of an object is the amount
of time needed to generate that object within a certain environment.
We give several examples to show this notion matches our intuition
in scenarios where one organizes, accesses and transfers
more >>>

Given two sets $A,B\subseteq\R^n$, a measure of their dependence, or correlation, is given by the expected squared inner product between random $x\in A $ and $y\in B$. We prove an inequality showing that no two sets of large enough Gaussian measure (at least $e^{-\delta n}$ for some constant $\delta >0$) ... more >>>

A measure $\mu_{n}$ on $n$-dimensional lattices with
determinant $1$ was introduced about fifty years ago to prove the
existence of lattices which contain points from certain sets. $\mu_{n}$
is the unique probability measure on lattices with determinant $1$ which
is invariant under linear transformations with determinant $1$, where a
linear ... more >>>

Recently, Ajtai discovered a fascinating connection
between the worst-case complexity and the average-case
complexity of some well-known lattice problems.
Later, Ajtai and Dwork proposed a cryptosystem inspired
by Ajtai's work, provably secure if a particular lattice
problem is difficult. We show that there is a converse
to the Ajtai-Dwork ... more >>>

The parallel repetition theorem states that for any two-prover game,
with value $1- \epsilon$ (for, say, $\epsilon \leq 1/2$), the value of
the game repeated in parallel $n$ times is at most
$(1- \epsilon^c)^{\Omega(n/s)}$, where $s$ is the answers' length
(of the original game) and $c$ is a universal ... more >>>

Consider a graph obtained by taking edge disjoint union of $k$ complete bipartite graphs.
Alon, Saks and Seymour conjectured that such graph has chromatic number at most $k+1$.
This well known conjecture remained open for almost twenty years.
In this paper, we construct a counterexample to this
conjecture and discuss ... more >>>

In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth ... more >>>

We give a method to decide whether or not an
ordinary finite order linear recurrence with constant, rational
coefficients ever generates zero.

Recent work of [Dasgupta-Kumar-Sarl\'{o}s, STOC 2010] gave a sparse Johnson-Lindenstrauss transform and left as a main open question whether their construction could be efficiently derandomized. We answer their question affirmatively by giving an alternative proof of their result requiring only bounded independence hash functions. Furthermore, the sparsity bound obtained in ... more >>>

We give deterministic $2^{O(n)}$-time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP).
This improves the $n^{O(n)}$ running time of the best previously known algorithms for CVP (Kannan, ... more >>>

We consider pseudorandom generators in which each output bit depends on a constant number of input bits. Such generators have appealingly simple structure: they can be described by a sparse input-output dependency graph and a small predicate that is applied at each output. Following the works of Cryan and Miltersen ... more >>>

P. Gopalan, P. G. Kolaitis, E. N. Maneva and C. H. Papadimitriou
studied in [Gopalan et al., ICALP2006] connectivity properties of the
solution-space of Boolean formulas, and investigated complexity issues
on connectivity problems in Schaefer's framework [Schaefer, STOC1978].
A set S of logical relations is Schaefer if all relations in ... more >>>

This paper provides the first general technique for proving information lower bounds on two-party
unbounded-rounds communication problems. We show that the discrepancy lower bound, which
applies to randomized communication complexity, also applies to information complexity. More
precisely, if the discrepancy of a two-party function $f$ with respect ... more >>>

The superposition (or composition) problem is a problem of
representation of a function $f$ by a superposition of "simpler" (in a
different meanings) set $\Omega$ of functions. In terms of circuits
theory this means a possibility of computing $f$ by a finite circuit
with 1 fan-out gates $\Omega$ of functions.

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $\beta$ is ... more >>>

We present a Fourier-analytic approach to list-decoding Reed-Muller codes over arbitrary finite fields. We prove that the list-decoding radius for quadratic polynomials equals $1 - 2/q$ over any field $F_q$ where $q > 2$. This confirms a conjecture due to Gopalan, Klivans and Zuckerman for degree $2$. Previously, tight bounds ... more >>>

We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. ... more >>>

In the setting of secure multiparty computation, a set of $n$ parties with private inputs wish to jointly compute some functionality of their inputs. One of the most fundamental results of information-theoretically secure computation was presented by Ben-Or, Goldwasser and Wigderson (BGW) in 1988. They demonstrated that any $n$-party functionality ... more >>>

We present the first example of a natural distribution on instances
of an NP-complete problem, with the following properties.
With high probability a random formula from this
distribution (a) is unsatisfiable,
(b) has a short proof that can be found easily, and (c) does not have a short
(general) resolution ... more >>>

We extend Lutz's measure to probabilistic classes, and obtain notions of measure on probabilistic complexity classes
C
such as BPP , BPE and BPEXP. Unlike former attempts,
all our measure notions satisfy all three Lutz's measure axioms, that is
every singleton {L} has measure zero in ... more >>>

We introduce "resource-bounded betting games", and propose
a generalization of Lutz's resource-bounded measure in which the choice
of next string to bet on is fully adaptive. Lutz's martingales are
equivalent to betting games constrained to bet on strings in lexicographic
order. We show that if strong pseudo-random number generators exist,
more >>>

We show that for any reasonable semantic model of computation and for
any positive integer $a$ and rationals $1 \leq c < d$, there exists a language
computable in time $n^d$ with $a$ bits of advice but not in time $n^c$
with $a$ bits of advice. A semantic ... more >>>

A t-private private information retrieval (PIR) scheme allows a user to retrieve the i-th bit of an n-bit string x replicated among k servers, while any coalition of up to t servers learns no information about i. We present a new geometric approach to PIR, and obtain (1) A t-private ... more >>>

We present a new methodology for computing approximate Nash equilibria for two-person non-cooperative games based
upon certain extensions and specializations of an existing optimization approach previously used for the derivation of fixed approximations for this problem. In particular, the general two-person problem is reduced to an indefinite quadratic programming problem ... more >>>

Branching programs (b.p.'s) or decision diagrams are a general
graph-based model of sequential computation. The b.p.'s of
polynomial size are a nonuniform counterpart of LOG. Lower bounds
for different kinds of restricted b.p.'s are intensively
investigated. An important restriction are so called $k$-b.p.'s,
where each computation reads each input ... more >>>

Restricted branching programs are considered in complexity theory in
order to study the space complexity of sequential computations and
in applications as a data structure for Boolean functions. In this
paper (\oplus,k)-branching programs and (\vee,k)-branching
programs are considered, i.e., branching programs starting with a
\oplus- ... more >>>

A hypergraph dictatorship test is first introduced by Samorodnitsky
and Trevisan and serves as a key component in
their unique games based $\PCP$ construction. Such a test has oracle
access to a collection of functions and determines whether all the
functions are the same dictatorship, or all their low degree ... more >>>

We obtain the following full characterization of constructive dimension
in terms of algorithmic information content. For every sequence A,
cdim(A)=liminf_n (K(A[0..n-1])/n.

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovász Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.

Branching programs (b.p.'s) or decision diagrams are a general
graph-based model of sequential computation. B.p.'s of polynomial
size are a nonuniform counterpart of LOG. Lower bounds for
different kinds of restricted b.p.'s are intensively investigated.
An important restriction are so called 1-b.p.'s, where each
computation reads each input bit at ... more >>>

We study semidefinite programming relaxations of Vertex Cover arising from
repeated applications of the LS+ ``lift-and-project'' method of Lovasz and
Schrijver starting from the standard linear programming relaxation.

Goemans and Kleinberg prove that after one round of LS+ the integrality
gap remains arbitrarily close to 2. Charikar proves an integrality ... more >>>

Computing the Hermite Normal Form
of an $n\times n$ matrix using the best current algorithms typically
requires $O(n^3\log M)$ space, where $M$ is a bound on the length of
the columns of the input matrix.
Although polynomial in the input size (which is ... more >>>

One of the crown jewels of complexity theory is Valiant's 1979 theorem that computing the permanent of an n*n matrix is #P-hard. Here we show that, by using the model of linear-optical quantum computing---and in particular, a universality theorem due to Knill, Laflamme, and Milburn---one can give a different and ... more >>>

Designing algorithms that use logarithmic space for graph reachability problems is fundamental to complexity theory. It is well known that for general directed graphs this problem is equivalent to the NL vs L problem. For planar graphs, the question is not settled. Showing that the planar reachability problem is NL-complete ... more >>>

In [Blass, Gurevich, and Shelah, 99] a logic L_Y has been introduced as a possible candidate for a logic capturing the PTIME properties of structures (even in the absence of an ordering of their universe). A reformulation of this problem in terms of a parameterized halting problem p-Acc for nondeterministic ... more >>>

An approximate membership data structure is a randomized data
structure for representing a set which supports membership
queries. It allows for a small false positive error rate but has
no false negative errors. Such data structures were first
introduced by Bloom in the 1970's, and have since had numerous
applications, ... more >>>

We prove an exponential lower bound ($2^{\Omega(n/\log n)}$) on the
size of any randomized ordered read-once branching program
computing integer multiplication. Our proof depends on proving
a new lower bound on Yao's randomized one-way communication
complexity of certain boolean functions. It generalizes to some
other ... more >>>

Recent work by Bernasconi, Damm and Shparlinski
proved lower bounds on the circuit complexity of the square-free
numbers, and raised as an open question if similar (or stronger)
lower bounds could be proved for the set of prime numbers. In
this short note, we answer this question ... more >>>

We extend the lower bounds on the depth of algebraic decision trees
to the case of {\em randomized} algebraic decision trees (with
two-sided error) for languages being finite unions of hyperplanes
and the intersections of halfspaces, solving a long standing open
problem. As an application, among ... more >>>

In this paper, we are concerned with randomized OBDDs and randomized
read-k-times branching programs. We present an example of a Boolean
function which has polynomial size randomized OBDDs with small,
one-sided error, but only non-deterministic read-once branching
programs of exponential size. Furthermore, we discuss a lower bound
technique for randomized ... more >>>

In this note we show that the asymmetric Prover-Delayer game developed by Beyersdorff, Galesi, and Lauria (ECCC TR10-059) for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover-Delayer game to show a lower bound of the form $2^{\Omega(n\log n)}$ for the pigeonhole ... more >>>

We construct an explicit polynomial $f(x_1,...,x_n)$, with
coefficients in ${0,1}$, such that the size of any syntactically
multilinear arithmetic circuit computing $f$ is at least
$\Omega( n^{4/3} / log^2(n) )$. The lower bound holds over any field.

We establish a lower bound of $\Omega{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $\Omega(\sqrt{D(f)})$ is a lower bound for all Boolean functions.Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' ... more >>>

We prove an exponential lower bound on the lengths of resolution proofs of propositions expressing the finite Ramsey theorem for pairs.

One way to quantify how dense a multidag is in long paths is to find
the largest n, m such that whichever &le; n edges are removed, there is still
a path from an original input to an original output with &ge; m edges
- the larger ... more >>>

We present a new lower bound technique for two types of restricted
Branching Programs (BPs), namely for read-once BPs (BP1s) with
restricted amount of nondeterminism and for (1,+k)-BPs. For this
technique, we introduce the notion of (strictly) k-wise l-mixed
Boolean functions, which generalizes the concept of l-mixedness ... more >>>

The Gap-Hamming-Distance problem arose in the context of proving space
lower bounds for a number of key problems in the data stream model. In
this problem, Alice and Bob have to decide whether the Hamming distance
between their $n$-bit input strings is large (i.e., at least $n/2 +
\sqrt n$) ... more >>>

We show that any 1-round 2-server Private Information
Retrieval Protocol where the answers are 1-bit long must ask questions
that are at least $n-2$ bits long, which is nearly equal to the known
$n-1$ upper bound. This improves upon the approximately $0.25n$ lower
bound of Kerenidis and de Wolf while ... more >>>

Recently there was a significant progress in
proving (exponential-time) worst-case upper bounds for the
propositional satisfiability problem (SAT).
MAX-SAT is an important generalization of SAT.
Several upper bounds were obtained for MAX-SAT and
its NP-complete subproblems.
In particular, Niedermeier and Rossmanith recently
... more >>>

We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm; more precisely, it is a constant factor improvement in the base of ... more >>>

We study the problem of constructing affine extractors over $\mathsf{GF(2)}$. Previously the only known construction that can handle sources with arbitrarily linear entropy is due to Bourgain (and a slight modification by Yehudayoff), which relies heavily on the technique of Van der Corput differencing and a careful choice of a ... more >>>

We present a new approach to the composition
of learning algorithms (in various models) for
classes of constant VC-dimension into learning algorithms for
more complicated classes.
We prove that if a class $\CC$ is learnable
in time $t$ from a hypothesis class $\HH$ of constant VC-dimension
then the class $\C^\star$ ... more >>>

Interactive hashing, introduced by Naor et al. [NOVY98], plays
an important role in many cryptographic protocols. In particular, it
is a major component in all known constructions of
statistically-hiding commitment schemes and of zero-knowledge
arguments based on general one-way permutations and on one-way
functions. Interactive hashing with respect to a ... more >>>

The well known dichotomy conjecture of Feder and
Vardi states that for every &#64257;nite family &#915; of constraints CSP(&#915;) is
either polynomially solvable or NP-hard. Bulatov and Jeavons re-
formulated this conjecture in terms of the properties of the algebra
P ol(&#915;), where the latter is ... more >>>

We propose a generalization of the traditional algorithmic space and
time complexities. Using the concept introduced, we derive an
unified proof for the deterministic time and space hierarchy
theorems, now stated in a much more general setting. This opens the
possibility for the unification and generalization of other results
that ... more >>>

We investigate the question of what languages can be decided efficiently with the help of a recursive collision-finding oracle. Such an oracle can be used to break collision-resistant hash functions or, more generally, statistically hiding commitments. The oracle we consider, $Sam_d$ where $d$ is the recursion depth, is based on ... more >>>

We prove a new transference theorem in the geometry of numbers,
giving optimal bounds relating the successive minima of a lattice
with the minimal length of generating vectors of its dual.
It generalizes the transference theorem due to Banaszczyk.
We also prove a stronger bound for the special class of ... more >>>

We prove that for all positive integer $k$ and for all
sufficiently small $\epsilon >0$ if $n$ is sufficiently large
then there is no Boolean (or $2$-way) branching program of size
less than $2^{\epsilon n}$ which for all inputs
$X\subseteq \lbrace 0,1,...,n-1\rbrace $ computes in time $kn$
the parity of ... more >>>

We shall prove a lower bound on the number of edges in some bounded
depth graphs. This theorem is stronger than lower bounds proved on
bounded depth superconcentrators and enables us to prove a lower bound
on certain bounded depth circuits for which we cannot use
superconcentrators: we prove that ... more >>>

We show that in the bounded degree model for graph property testing,
adaptivity is essential. An algorithm is *non-adaptive* if it makes all queries to the input before receiving any answers. We call a property *non-trivial* if it does not depend only on the degree distribution of the nodes. We ... more >>>

We show a generic, simple way to amplify the error-tolerance of locally decodable codes.
Specifically, we show how to transform a locally decodable code that can tolerate a constant fraction of errors
to a locally decodable code that can recover from a much higher error-rate. We also show how to ... more >>>

We design a $0.795$ approximation algorithm for the Max-Bisection problem
restricted to regular graphs. In the case of three regular graphs our
results imply an approximation ratio of $0.834$.

Recently in [Vij09, Corollary 3.7] the complexity class ModL has been shown to be closed under complement assuming NL = UL. In this note we continue to show many other closure properties of ModL which include the following.

1. ModL is closed under $\leq ^L_m$ reduction, $\vee$(join) and $\leq ^{UL}_m$ ... more >>>

We describe a deterministic algorithm that, for constant k,
given a k-DNF or k-CNF formula f and a parameter e, runs in time
linear in the size of f and polynomial in 1/e and returns an
estimate of the fraction of satisfying assignments for f up to ... more >>>

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every $i\geq 0$, $\Ppoly$ has $i$th order scaled $\pthree$-strong dimension $0$. We also show that $\Ppoly^\io$ has $\pthree$-dimension $1/2$, $\pthree$-strong dimension $1$. Our results improve previous measure results of Lutz (1992) and dimension ... more >>>

Building on work of Yekhanin and Raghavendra, Efremenko recently gave an elegant construction of 3-query LDCs which achieve sub-exponential length unconditionally.In this note, we observe that this construction can be viewed in the framework of Reed-Muller codes.

We present an alternate proof of the recent result by Gutfreund and Kawachi that derandomizing Arthur-Merlin games into $P^{NP}$ implies linear-exponential circuit lower bounds for $E^{NP}$. Our proof is simpler and yields stronger results. In particular, consider the promise-$AM$ problem of distinguishing between the case where a given Boolean circuit ... more >>>

Inspired by recent construction of high-rate locally correctable codes with sublinear query complexity due to
Kopparty, Saraf and Yekhanin (2010) we address the similar question for locally testable codes (LTCs).

In this note we show a construction of high-rate LTCs with sublinear query complexity.
More formally, we show that for ... more >>>

We prove that the error-free (Las Vegas) randomized OBDDs
are computationally equivalent to the deterministic OBDDs.
In contrast, it is known the same is not true for the
Las Vegas read-once branching programs.

The deterministic space complexity of approximating the length of the longest increasing subsequence of
a stream of $N$ integers is known to be $\widetilde{\Theta}(\sqrt N)$. However, the randomized
complexity is wide open. We show that the technique used in earlier work to establish the $\Omega(\sqrt
N)$ deterministic lower bound fails ... more >>>

A syntactic read-k times branching program has the restriction
that no variable occurs more than k times on any path (whether or not
consistent). We exhibit an explicit Boolean function f which cannot
be computed by nondeterministic syntactic read-k times branching programs
of size less than exp(\sqrt{n}}k^{-2k}), ... more >>>

It is shown that decomposition via Chinise Remainder does not
yield polynomial size depth 3 threshold circuits for iterated
multiplication of n-bit numbers. This result is achieved by
proving that, in contrast to multiplication of two n-bit
numbers, powering, division, and other related problems, the
... more >>>

In the set cover problem we are given a collection of $m$ sets whose union covers $[n] = \{1,\ldots,n\}$ and must find a minimum-sized subcollection whose union still covers $[n]$. We investigate the approximability of set cover by an approximation ratio that depends only on $m$ and observe that, for ... more >>>

We show that there are infinitely many primes $p$, such
that the subgroup membership problem for PSL(2,p) belongs
to $\NP \cap \coNP$.

This work initiates the study of algorithms
for the testing of monotonicity of mechanisms.
Such testing algorithms are useful for
searching dominant strategy mechanisms.
An $\e$-tester for monotonicity
is given a query access to a mechanism,
accepts if monotonicity is satisfied,
and rejects with high probability if more than $\e$-fraction
more >>>

In this short note we show that for any integer k, there are
languages in the complexity class PP that do not have Boolean
circuits of size $n^k$.

Deciding whether a vertex in a graph is reachable from another
vertex has been studied intensively in complexity theory and is
well understood. For common types of graphs like directed graphs,
undirected graphs, dags or trees it takes a (possibly
nondeterministic) logspace machine to decide the reachability
problem, and ... more >>>

We present a very simple reduction that when given a graph G and an integer k produces a game that has an evolutionary stable strategy if and only if the maximum clique size of G is not exactly k. Formally this shows that existence of evolutionary stable strategies is hard ... more >>>

Given a boolean function, let epsilon_M(f) denote the smallest distance between f and a monotone function on {0,1}^n. Let delta_M(f) denote the fraction of hypercube edges where f violates monotonicity. We give an alternative proof of the tight bound: delta_M(f) >= 2/n eps_M(f) for any boolean function f. This was ... more >>>

In this note, we consider the problem of computing the
coefficients of the characteristic polynomial of a given
matrix, and the related problem of verifying the
coefficents.

Santha and Tan [CC98] show that verifying the determinant
(the constant term in the characteristic polynomial) is
complete for the class C=L, and ... more >>>

In 1957 Markov proved that every circuit in $n$ variables
can be simulated by a circuit with at most $\log(n+1)$ negations.
In 1974 Fischer has shown that this can be done with only
polynomial increase in size.

In this note we observe that some explicit monotone functions
more >>>

We consider the P versus NP\cap coNP question for the classical two-party communication protocols: if both a boolean function and its negation have small nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition case this question was answered by Aho, Ullman and ... more >>>

A language is called k-membership comparable if there exists a
polynomial-time algorithm that excludes for any k words one of
the 2^k possibilities for their characteristic string.
It is known that all membership comparable languages can be
reduced to some P-selective language with polynomially many
adaptive queries. We show however ... more >>>

We consider computations of linear forms over {\bf R} by
circuits with linear gates where the absolute values
coefficients are bounded by a constant. Also we consider a
related concept of restricted rigidity of a matrix. We prove
some lower bounds on the size of such circuits and the
restricted ... more >>>

We consider the problem of traversing skew (unbalanced) Merkle
trees and design an algorithm for traversing a skew Merkle tree
in time O(log n/log t) and space O(log n (t/log t)), for any choice
of parameter t\geq 2.
This algorithm can be of special interest in situations when
more >>>

A very recent paper by Caussinus, McKenzie, Therien, and Vollmer
[CMTV] shows that ACC^0 is properly contained in ModPH, and TC^0
is properly contained in the counting hierarchy. Thus, [CMTV] shows
that there are problems in ModPH that require superpolynomial-size
uniform ACC^0 ... more >>>

Locally Decodable codes(LDC) support decoding of any particular symbol of the input message by reading constant number of symbols of the codeword, even in presence of constant fraction of errors.

In a recent breakthrough, Yekhanin designed $3$-query LDCs that hugely improve over earlier constructions. Specifically, for a Mersenne prime $p ... more >>>

The question whether or not parallel repetition reduces the soundness error is a fundamental question in the theory of protocols. While parallel repetition reduces (at an exponential rate) the error in interactive proofs and (at a weak exponential rate) in special cases of interactive arguments (e.g., 3-message protocols - Bellare, ... more >>>

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class $\mathsf{AM}$. This gives a `PCP characterization' of $\mathsf{AM}$ analogous to the PCP Theorem for $\mathsf{NP}$. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, ... more >>>

The Max-Bisection and Min-Bisection are the problems of finding
partitions of the vertices of a given graph into two equal size subsets so as
to maximize or minimize, respectively, the number of edges with exactly one
endpoint in each subset.
In this paper we design the first ... more >>>

We design a polynomial time approximation scheme (PTAS) for
the problem of Metric MIN-BISECTION of dividing a given finite metric
space into two halves so as to minimize the sum of distances across
that partition. The method of solution depends on a new metric placement
partitioning ... more >>>

We prove that the subdense instances of MAX-CUT of average
degree Omega(n/logn) posses a polynomial time approximation scheme (PTAS).
We extend this result also to show that the instances of general 2-ary
maximum constraint satisfaction problems (MAX-CSP) of the same average
density have PTASs. Our results ... more >>>

The relationship between deterministic and probabilistic computations is one of the central issues in complexity theory. This problem can be tackled by constructing polynomial time hitting set generators which, however, belongs to the hardest problems in computer science even for severely restricted computational models. In our work, we consider read-once ... more >>>

Presented is an algorithm (for learning a subclass of erasing regular
pattern languages) which
can be made to run with arbitrarily high probability of
success on extended regular languages generated by patterns
$\pi$ of the form $x_0 \alpha_1 x_1 ... \alpha_m x_m$
for unknown $m$ but known $c$,
more >>>

We present a fully-polynomial randomized approximation scheme
for computing the permanent of an arbitrary matrix
with non-negative entries.

We prove a simple concentration inequality, which is an extension of the Chernoff bound and Hoeffding's inequality for binary random variables. Instead of assuming independence of the variables we use a slightly weaker condition, namely bounds on the co-moments.

This inequality allows us to simplify and strengthen several known direct-product ... more >>>

Branching programs are a model for representing Boolean
functions. For general branching programs, the
satisfiability and nonequivalence tests are NP-complete.
For read-once branching programs, which can test each
variable at most once in each computation, the satisfiability
test is trivial and there is also a probabilistic polynomial
time test of ... more >>>

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables .
Assume that this polynomial satisfies the property : \\

$|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) : Re(z_i) \geq 0 , 1 \leq i \leq n \}$ . \\

We prove that ... more >>>

In the mid 1980's, Yao presented a constant-round protocol for
securely computing any two-party functionality in the presence of
semi-honest adversaries (FOCS 1986). In this paper, we provide a
complete description of Yao's protocol, along with a rigorous
proof of security. Despite the importance of Yao's protocol to the
field ... more >>>

We present a probabilistic public key cryptosystem which is
secure unless the following worst-case lattice problem can be solved in
polynomial time:
"Find the shortest nonzero vector in an n dimensional lattice
L where the shortest vector v is unique in the sense that any other
vector whose length ... more >>>

Secure computation is one of the most fundamental cryptographic tasks.
It is known that all functions can be computed securely in the
information theoretic setting, given access to a black box for some
complete function such as AND. However, without such a black box, not
all functions can be securely ... more >>>

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real $d$ and every positive real $\epsilon$ ... more >>>

Long Code testing is a fundamental problem in the area of property
testing and hardness of approximation.
Long Code is a function of the form $f(x)=x_i$ for some index $i$.
In the Long Code testing, the problem is, given oracle access to a
collection of Boolean functions, to decide whether ... more >>>

We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input.

We begin by reducing the input linear program to a special form in which we ... more >>>

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

We introduce the polynomial-time tree reducibility
(ptt-reducibility). Our main result states that for
languages $B$ and $C$ it holds that
$B$ ptt-reduces to $C$ if and only if
the unbalanced leaf-language class of $B$ is robustly contained in
the unbalanced leaf-language class of $C$.
... more >>>

We prove that it is not decidable on R-machines if for a fixed finite intervall [a,b) the solution of the initial value problems of systems of ordinary differetial equations have solutions over this interval. This result holds independly from assumptions about differentiability of the right sides of the ODEs. Futhermore ... more >>>

Every pseudorandom generator is in particular a one-way function. If we only consider part of the output of the
pseudorandom generator is this still one-way? Here is a general setting formalizing this question. Suppose
$G:\{0,1\}^n\rightarrow \{0,1\}^{\ell(n)}$ is a pseudorandom generator with stretch $\ell(n)> n$. Let $M_R\in\{0,1\}^{m(n)\times \ell(n)}$ be a linear ... more >>>

We consider the problem of estimating the average of a huge set of values.
That is,
given oracle access to an arbitrary function $f:\{0,1\}^n\mapsto[0,1]$,
we need to estimate $2^{-n} \sum_{x\in\{0,1\}^n} f(x)$
upto an additive error of $\epsilon$.
We are allowed to employ a randomized algorithm which may err ... more >>>

We introduce a second-order system V_1-Horn of bounded arithmetic
formalizing polynomial-time reasoning, based on Graedel's
second-order Horn characterization of P. Our system has
comprehension over P predicates (defined by Graedel's second-order
Horn formulas), and only finitely many function symbols. Other
systems of polynomial-time reasoning either ... more >>>

The investigation of the computational power of randomized computations
is one of the central tasks of current complexity and algorithm theory.
In this paper for the first time a "strong" separation between the power
of determinism, Las Vegas randomization, and nondeterminism for a compu-
ting model is proved. The computing ... more >>>

We prove that NP$\ne$coNP and coNP$\nsubseteq$MA in the number-on-forehead model of multiparty communication complexity for up to $k=(1-\epsilon)\log n$ players, where $\epsilon>0$ is any constant. Specifically, we construct a function $F:(\zoon)^k\to\zoo$ with co-nondeterministic
complexity $O(\log n)$ and Merlin-Arthur
complexity $n^{\Omega(1)}$.
The problem was open for $k\geq3$.

For (1,+k)-branching programs and read-k-times branching
programs syntactic and nonsyntactic variants can be distinguished. The
nonsyntactic variants correspond in a natural way to sequential
computations with restrictions on reading the input while lower bound
proofs are easier or only known for the syntactic variants. In this
paper it is shown ... more >>>

The proof of Toda's celebrated theorem that the polynomial hierarchy is contained in $\P^\numP$ relies on the fact that, under mild technical conditions on the complexity class $\mathcal{C}$, we have $\exists \,\mathcal{C} \subset \BP \cdot \oplus \,\mathcal{C}$. More concretely, there is a randomized reduction which transforms nonempty sets and the ... more >>>

A simple extension of standard neural network models is introduced that
provides a model for neural computations that involve both firing rates and
firing correlations. Such extension appears to be useful since it has been
shown that firing correlations play a significant computational role in
many biological neural systems. Standard ... more >>>

In 1990, Linial and Nisan asked if any polylog-wise independent distribution fools any function in AC^0. In a recent remarkable development, Bazzi solved this problem for the case of DNF formulas.
The aim of this note is to present a simplified version of his proof.

We present a greatly simplified proof of the length-space
trade-off result for resolution in Hertel and Pitassi (2007), and
also prove a couple of other theorems in the same vein. We point
out two important ingredients needed for our proofs to work, and
discuss possible conclusions to be drawn regarding ... more >>>

The k-DNF resolution proof systems are a family of systems indexed by
the integer k, where the kth member is restricted to operating with
formulas in disjunctive normal form with all terms of bounded arity k
(k-DNF formulas). This family was introduced in [Krajicek 2001] as an
extension of the ... more >>>

We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if $X\in\mathcal A$ then $X$ has an infinite ... more >>>

The parallel repetition theorem states that for any Two
Prover Game with value at most $1-\epsilon$ (for $\epsilon<1/2$),
the value of the game repeated $n$ times in parallel is at most
$(1-\epsilon^3)^{\Omega(n/s)}$, where $s$ is the length of the
answers of the two provers. For Projection
Games, the bound on ... more >>>

We present a $2^{\tilde O(\sqrt{n})}$ time exact learning
algorithm for polynomial size
DNF using equivalence queries only. In particular, DNF
is PAC-learnable in subexponential time under any distribution.
This is the first subexponential time
PAC-learning algorithm for DNF under any distribution.

Branching programs (b.p.s) or binary decision diagrams are a
general graph-based model of sequential computation. The b.p.s of
polynomial size are a nonuniform counterpart of LOG. Lower bounds
for different kinds of restricted b.p.s are intensively
investigated. The restrictions based on the number of tests of
more >>>

Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by ... more >>>

Trapdoor permutations (TDPs) are among the most widely studied
building blocks of cryptography. Despite the extensive body of
work that has been dedicated to their study, in many setting and
applications (enhanced) trapdoor permutations behave
unexpectedly. In particular, a TDP may become easy to invert when
the inverter is given ... more >>>

We put forward a general theory of goal-oriented communication, where communication is not an end in itself, but rather a means to achieving some goals of the communicating parties. The goals can vary from setting to setting, and we provide a general framework for describing any such goal. In this ... more >>>

It is known that there exists a PCP characterization of NP
where the verifier makes 3 queries and has a {\em one-sided}
error that is bounded away from 1; and also that 2 queries
do not suffice for such a characterization. Thus PCPs with
3 queries ... more >>>

In [IPL2005],
Frandsen and Miltersen improved bounds on the circuit size $L(n)$ of the hardest Boolean function on $n$ input bits:
for some constant $c>0$:
\[
\left(1+\frac{\log n}{n}-\frac{c}{n}\right)
\frac{2^n}{n}
\leq
L(n)
\leq
\left(1+3\frac{\log n}{n}+\frac{c}{n}\right)
\frac{2^n}{n}.
\]
In this note,
we announce a modest ... more >>>

In this paper, we show how to systematically
improve on parameterized algorithms and their
analysis, focusing on search-tree based algorithms
for d-Hitting Set, especially for d=3.
We concentrate on algorithms which are easy to implement,
in contrast with the highly sophisticated algorithms
which have been elsewhere designed to ... more >>>

Communication complexity is concerned with the question: how much information do the participants of a communication system need to exchange in order to perform certain tasks? The minimum number of bits that must be communicated is the deterministic communication complexity of $f$. This complexity measure was introduced by Yao \cite{1} ... more >>>

Building on known definitions, we present a unified general framework for
defining and analyzing security of cryptographic protocols. The framework
allows specifying the security requirements of a large number of
cryptographic tasks, such as signature, encryption, authentication, key
exchange, commitment, oblivious transfer, zero-knowledge, secret sharing,
general function evaluation, and ... more >>>

The study of the interplay between the testability of properties of Boolean functions and the invariances acting on their domain which preserve the property was initiated by Kaufman and Sudan (STOC 2008). Invariance with respect to F_2-linear transformations is arguably the most common symmetry exhibited by natural properties of Boolean ... more >>>

In this paper, firstly we propose two new concepts concerning the notion of key escrow encryption schemes: provable partiality and independency. Roughly speaking we say that a scheme has provable partiality if existing polynomial time algorithm for recovering the secret knowing escrowed information implies a polynomial time algorithm that can ... more >>>

Branching programs are a well-established computation
model for boolean functions, especially read-once
branching programs (BP1s) have been studied intensively.
A very simple function $f$ in $n^2$ variables is
exhibited such that both the function $f$ and its negation
$\neg f$ can be computed by $\Sigma^3_p$-circuits,
the ... more >>>

Given an LLL-basis $B$ of dimension $n= hk$ we accelerate slide-reduction with blocksize $k$ to run under a reasonable assjmption in \
$\frac1{6} \, n^2 h \,\log_{1+\varepsilon} \, \alpha $ \
local SVP-computations in dimension $k$, where $\alpha \ge \frac 43$
measures the quality of the ... more >>>

We study the notion of learning in an oblivious changing environment. Existing online learning algorithms which minimize regret are shown to converge to the average of all locally optimal solutions. We propose a new performance metric, strengthening the standard metric of regret, to capture convergence to locally optimal solutions, and ... more >>>

We prove that any real matrix $A$ contains a subset of at most
$4k/\eps + 2k \log(k+1)$ rows whose span ``contains" a matrix of
rank at most $k$ with error only $(1+\eps)$ times the error of the
best rank-$k$ approximation of $A$. This leads to an algorithm to
find such ... more >>>

We demonstrate the use of Kolmogorov complexity in average case
analysis of algorithms through a classical example: adding two $n$-bit
numbers in $\ceiling{\log_2{n}}+2$ steps on average. We simplify the
analysis of Burks, Goldstine, and von Neumann in 1946 and
(in more complete forms) of Briley and of Schay.

We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states ... more >>>

We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution $D$ that is $\delta$-dense in a distribution that is ... more >>>

{\em Dispersers} and {\em extractors} for affine sources of dimension $d$ in $\mathbb F_p^n$ --- where $\mathbb F_p$ denotes the finite field of prime size $p$ --- are functions $f: \mathbb F_p^n \rightarrow \mathbb F_p$ that behave pseudorandomly when their domain is restricted to any particular affine space $S \subseteq ... more >>>

An $m$-variate polynomial $f$ is said to be an affine projection of some $n$-variate polynomial $g$ if there exists an $n \times m$ matrix $A$ and an $n$-dimensional vector $b$ such that $f(x) = g(A x + b)$. In other words, if $f$ can be obtained by replacing each variable ... more >>>

In this paper we introduce a new model for computing=20
polynomials - a depth 2 circuit with a symmetric gate at the top=20
and plus gates at the bottom, i.e the circuit computes a=20
symmetric function in linear functions -
$S_{m}^{d}(\ell_1,\ell_2,...,\ell_m)$ ($S_{m}^{d}$ is the $d$'th=20
elementary symmetric polynomial in $m$ ... more >>>

Aggregates are a computational model similar to circuits, but the
underlying graph is not necessarily acyclic. Logspace-uniform
polynomial-size aggregates decide exactly the languages in PSPACE;
without uniformity condition they decide the languages in
PSPACE/poly. As a measure of similarity to boolean circuits we
introduce the parameter component size. We ... more >>>

We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\epsilon)$ of the examples, it is $\mathrm{NP}$-hard to find a halfspace that is correct on $(1/2+\epsilon)$ of the examples, for arbitrary constants $\epsilon > ... more >>>

We consider learning on multi-layer neural nets with piecewise polynomial
activation functions and a fixed number k of numerical inputs. We exhibit
arbitrarily large network architectures for which efficient and provably
successful learning algorithms exist in the rather realistic refinement of
Valiant's model for probably approximately correct learning ("PAC-learning")
where ... more >>>

Ordered binary decision diagrams (OBDDs) and parity ordered binary
decision diagrams have proved their importance as data structures
representing Boolean functions. In addition to parity OBDDs we study
their generalization which we call parity AOBDDs, give the algebraic
characterization theorem and compare their minimal size to the size
more >>>

Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials $\{f_1,\ldots, f_m\} \subset \mathbb{F}[x_1,\ldots, x_n]$ are called algebraically independent if there is no non-zero polynomial $F$ such that $F(f_1, \ldots, f_m) = 0$. The transcendence degree, $\mbox{trdeg}\{f_1,\ldots, f_m\}$, is the maximal ... more >>>

In cryptography, there has been tremendous success in building
primitives out of homomorphic semantically-secure encryption
schemes, using homomorphic properties in a black-box way. A few
notable examples of such primitives include items like private
information retrieval schemes and collision-resistant hash functions. In this paper, we illustrate a general
methodology for ... more >>>

We introduce two algebraic propositional proof systems F-NS
and F-PC. The main difference of our systems from (customary)
Nullstellensatz and Polynomial Calculus is that the polynomials
are represented as arbitrary formulas (rather than sums of
monomials). Short proofs of Tseitin's tautologies in the
constant-depth version of F-NS provide ... more >>>

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege--yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analogue of Frege proofs, different from that given in Buss ... more >>>

We argue that the symmetries of a property being tested play a
central role in property testing. We support this assertion in the
context of algebraic functions, by examining properties of functions
mapping a vector space $\K^n$ over a field $\K$ to a subfield $\F$.
We consider $\F$-linear properties that ... more >>>

This paper is concerned with a new family of error-correcting codes
based on algebraic curves over finite fields, and list decoding
algorithms for them. The basic goal in the subject of list decoding is
to construct error-correcting codes $C$ over some alphabet $\Sigma$
which have good rate $R$, and at ... more >>>

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

In this paper we consider two refined questions regarding
the query complexity of testing graph properties
in the adjacency matrix model.
The first question refers to the relation between adaptive
and non-adaptive testers, whereas the second question refers to
testability within complexity that
is inversely proportional to ... more >>>

An algorithmic meta theorem for a logic and a class $C$ of structures states that all problems expressible in this logic can be solved efficiently for inputs from $C$. The prime example is Courcelle's Theorem, which states that monadic second-order (MSO) definable problems are linear-time solvable on graphs of bounded ... more >>>

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a logical and a
structural component, that is they are results of the form:
"every computational problem that can be formalised in a given logic L ... more >>>

Given a multivariate polynomial f(x) in F[x] as an arithmetic circuit we would like to efficiently determine:

(i) [Identity Testing.] Is f(x) identically zero?

(ii) [Degree Computation.] Is the degree of the
polynomial f(x) at most a given integer d.

(iii) [Polynomial Equivalence.] Upto an invertible ... more >>>

An algorithm is presented for counting the number of maximum weight satisfying assignments of a 2SAT formula. The worst case running time of $O(\mbox{poly($n$)} \cdot 1.2461^n)$ for formulas with $n$ variables improves on the previous bound of $O(\mbox{poly($n$)} \cdot 1.2561^n)$ by Dahll\"of, Jonsson, and Wahlstr\"om . The weighted 2SAT counting ... more >>>

We study multiplayer games in which the participants have access to
only limited randomness. This constrains both the algorithms used to
compute equilibria (they should use little or no randomness) as well
as the mixed strategies that the participants are capable of playing
(these should be sparse). We frame algorithmic ... more >>>

We survey recent algorithms for the propositional
satisfiability problem, in particular algorithms
that have the best current worst-case upper bounds
on their complexity. We also discuss some related
issues: the derandomization of the algorithm of
Paturi, Pudlak, Saks and Zane, the Valiant-Vazirani
Lemma, and random walk ... more >>>

We present a simple randomized algorithm for SAT and prove an upper
bound on its running time. Given a Boolean formula F in conjunctive
normal form, the algorithm finds a satisfying assignment for F
(if any) by repeating the following: Choose an assignment A at
random and ... more >>>

This paper is our second step towards developing a theory of
testing monomials in multivariate polynomials. The central
question is to ask whether a polynomial represented by an
arithmetic circuit has some types of monomials in its sum-product
expansion. The complexity aspects of this problem and its variants
have been ... more >>>

The coin weighing problem is the following: Given $n$ coins for which $m$ of them are counterfeit with the same weight. The problem is to detect the counterfeit coins with minimal number of weighings. This problem has many applications in compressed sensing, multiple access adder channels, etc. The problem was ... more >>>

In 1984 Levin put forward a suggestion for a theory of {\em average
case complexity}. In this theory a problem, called a {\em
distributional problem}, is defined as a pair consisting of a
decision problem and a probability distribution over the instances.
Introducing adequate notions of simple distributions and average
more >>>

Let G=(V,E) be an unweighted undirected graph on n vertices. A simple
argument shows that computing all distances in G with an additive
one-sided error of at most 1 is as hard as Boolean matrix
multiplication. Building on recent work of Aingworth, Chekuri and
Motwani, we describe an \tilde{O}(min{n^{3/2}m^{1/2},n^{7/3}) time
more >>>

We present two new algorithms for solving the {\em All
Pairs Shortest Paths\/} (APSP) problem for weighted directed
graphs. Both algorithms use fast matrix multiplication algorithms.

The first algorithm
solves the APSP problem for weighted directed graphs in which the edge
weights are integers of small absolute value in ... more >>>

Andreev et al.~\cite{ABCR97} give constructions of Boolean
functions (computable by polynomial-size circuits) that require large
read-once branching program (1-b.p.'s): a function in P that requires
1-b.p. of size at least $2^{n-\polylog(n)}$, a function in quasipolynomial
time that requires 1-b.p. of size at least $2^{n-O(\log n)}$, and a
function in LINSPACE ... more >>>

We say that a distribution over $\{0,1\}^n$
is almost $k$-wise independent
if its restriction to every $k$ coordinates results in a
distribution that is close to the uniform distribution.
A natural question regarding almost $k$-wise independent
distributions is how close they are to some $k$-wise
independent distribution. We show that ... more >>>

We constructively prove the existence of almost complete problems under logspace manyone reduction for some small complexity classes by exhibiting a parametrizable construction which yields, when appropriately setting the parameters, an almost complete problem for PSPACE, the class of space efficiently decidable problems, and for SUBEXP, the class of problems ... more >>>

It is well known that $\R^N$ has subspaces of dimension
proportional to $N$ on which the $\ell_1$ norm is equivalent to the
$\ell_2$ norm; however, no explicit constructions are known.
Extending earlier work by Artstein--Avidan and Milman, we prove that
such a subspace can be generated using $O(N)$ random bits.

We give an explicit (in particular, deterministic polynomial time)
construction of subspaces $X
\subseteq \R^N$ of dimension $(1-o(1))N$ such that for every $x \in X$,
$$(\log N)^{-O(\log\log\log N)} \sqrt{N}\, \|x\|_2 \leq \|x\|_1 \leq \sqrt{N}\, \|x\|_2.$$
If we are allowed to use $N^{1/\log\log N}\leq N^{o(1)}$ random bits
and ... more >>>

A family of permutations in $S_n$ is $k$-wise independent if a uniform permutation chosen from the family maps any distinct $k$ elements to any distinct $k$ elements equally likely. Efficient constructions of $k$-wise independent permutations are known for $k=2$ and $k=3$, but are unknown for $k \ge 4$. In fact, ... more >>>

We consider weakly-verifiable puzzles which are challenge-response puzzles such that the responder may not
be able to verify for itself whether it answered the challenge correctly. We consider $k$-wise direct product of
such puzzles, where now the responder has to solve $k$ puzzles chosen independently in parallel.
Canetti et ... more >>>

The Johnson-Lindenstrauss lemma is a fundamental result in probability with several applications in the design and analysis of algorithms in high dimensional geometry. Most known constructions of linear embeddings that satisfy the Johnson-Lindenstrauss property involve randomness. We address the question of explicitly constructing such embedding families and provide a construction ... more >>>

Lattices have received considerable attention as a potential source of computational hardness to be used in cryptography, after a breakthrough result of Ajtai (STOC 1996) connecting the average-case and worst-case complexity of various lattice problems. The purpose of this paper is twofold. On the expository side, we present a rigorous ... more >>>

We observe that many important computational problems in NC^1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC^0 circuits if and only if it has TC^0 circuits of size n^{1+\epsilon} for every \epsilon > 0 (counting the ... more >>>

For a {0,1}-valued matrix $M$ let CC($M$) denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lovasz and Saks [FOCS 1988] states that CC($M$) is at most $\log^c({\mbox{rank}}(M))$ for some absolute constant $c$ where rank($M$) denotes the rank of $M$ over the field ... more >>>

Reducibility concepts are fundamental in complexity theory.
Usually, they are defined as follows: A problem P is reducible
to a problem S if P can be computed using a program or device
for S as a subroutine. However, in the case of such restricted
models as ... more >>>

We obtain a characterization of feasible, Bayesian, multi-item multi-bidder mechanisms with independent, additive bidders as distributions over hierarchical mechanisms. Combined with cyclic-monotonicity our results provide a complete characterization of feasible, Bayesian Incentive Compatible mechanisms for this setting.

Our characterization is enabled by a novel, constructive proof of Border's theorem [Border ... more >>>

We show that the rank of a depth-3 circuit (over any field) that is simple,
minimal and zero is at most O(k^3\log d). The previous best rank bound known was
2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005).
This almost resolves the rank question first posed by ... more >>>

We show a non-inductive proof of the Schwartz-Zippel lemma. The lemma bounds the number of zeros of a multivariate low degree polynomial over a finite field.

A basic property of minimally unsatisfiable clause-sets F is that
c(F) >= n(F) + 1 holds, where c(F) is the number of clauses, and
n(F) the number of variables. Let MUSAT(k) be the class of minimally
unsatisfiable clause-sets F with c(F) <= n(F) + k.

Poly-time decision algorithms are known ... more >>>

Quantum finite automata have been studied intensively since
their introduction in late 1990s as a natural model of a
quantum computer with finite-dimensional quantum memory space.
This paper seeks their direct application
to interactive proof systems in which a mighty quantum prover
communicates with a ... more >>>

We consider the regular languages recognized by weighted threshold circuits with a linear number of wires.
We present a simple proof to show that parity cannot be computed by such circuits.
Our proofs are based on an explicit construction to restrict the input of the circuit such that the value ... more >>>

The bandwidth problem is the problem of numbering the vertices of a
given graph $G$ such that the maximum difference between the numbers
of adjacent vertices is minimal. The problem has a long history and
is known to be NP-complete Papadimitriou [Pa76]. Only few special
cases ... more >>>

Assuming 3-SAT formulas are hard to refute
on average, Feige showed some approximation hardness
results for several problems like min bisection, dense
$k$-subgraph, max bipartite clique and the 2-catalog segmentation
problem. We show a similar result for
max bipartite clique, but under the assumption, 4-SAT formulas
are hard to refute ... more >>>

A new algorithm for learning one-variable pattern languages from positive data
is proposed and analyzed with respect to its average-case behavior.
We consider the total learning time that takes into account all
operations till convergence to a correct hypothesis is achieved.

For almost all meaningful distributions
defining how the ... more >>>

We consider noninteractive zero-knowledge proofs in the shared random
string model proposed by Blum, Feldman and Micali \cite{bfm}. Until
recently there was a sizable polynomial gap between the most
efficient noninteractive proofs for {\sf NP} based on general
complexity assumptions \cite{fls} versus those based on specific
algebraic assumptions \cite{Da}. ... more >>>

We describe a short and easy to analyze construction of
constant-degree expanders. The construction relies on the
replacement-product, which we analyze using an elementary
combinatorial argument. The construction applies the replacement
product (only twice!) to turn the Cayley expanders of \cite{AR},
whose degree is polylog n, into constant degree
expanders.

A Boolean function $f \colon \mathbb{F}^n_2 \rightarrow \mathbb{F}_2$ is called an affine disperser for sources of dimension $d$, if $f$ is not constant on any affine subspace of $\mathbb{F}^n_2$ of dimension at least $d$. Recently Ben-Sasson and Kopparty gave an explicit construction of an affine disperser for $d=o(n)$. The main ... more >>>

Recently there has been much interest in polynomial threshold functions in the context of learning theory, structural results and pseudorandomness. A crucial ingredient in these works is the understanding of the distribution of low-degree multivariate polynomials evaluated over normally distributed inputs. In particular, the two important properties are exponential tail ... more >>>

When we represent a decision problem,like CIRCUIT-SAT, as a language over the binary alphabet,
we usually do not specify how to encode instances by binary strings.
This relies on the empirical observation that the truth of a statement of the form ``CIRCUIT-SAT belongs to a complexity class $C$''
more >>>

We present a simplified proof of Solovay-Calude-Coles theorem
stating that there is an enumerable undecidable set with the
following property: prefix
complexity of its initial segment of length n is bounded by prefix
complexity of n (up to an additive constant).

Cryan and Miltersen recently considered the question
of whether there can be a pseudorandom generator in
NC0, that is, a pseudorandom generator such that every
bit of the output depends on a constant number k of bits
of the seed. They show that for k=3 there ... more >>>

We prove an exponential lower bound on the size of proofs
in the proof system operating with ordered binary decision diagrams
introduced by Atserias, Kolaitis and Vardi. In fact, the lower bound
applies to semantic derivations operating with sets defined by OBDDs.
We do not assume ... more >>>

We prove an exponential lower bound on the size of any
fixed-degree algebraic decision tree for solving MAX, the
problem of finding the maximum of $n$ real numbers. This
complements the $n-1$ lower bound of Rabin \cite{R72} on
the depth of algebraic ... more >>>

We prove lower bounds of the form $exp\left(n^{\epsilon_d}\right),$
$\epsilon_d>0,$ on the length of proofs of an explicit sequence of
tautologies, based on the Pigeonhole Principle, in proof systems
using formulas of depth $d,$ for any constant $d.$ This is the
largest lower bound for the strongest proof system, for which ... more >>>

This paper gives two distinct proofs of an exponential separation
between regular resolution and unrestricted resolution.
The previous best known separation between these systems was
quasi-polynomial.

Satisfiability algorithms have become one of the most practical and successful approaches for solving a variety of real-world problems, including hardware verification, experimental design, planning and diagnosis problems. The main reason for the success is due to highly optimized algorithms for SAT based on resolution. The most successful of these ... more >>>

A construction of Bourgain gave the first 2-source
extractor to break the min-entropy rate 1/2 barrier. In this note,
we write an exposition of his result, giving a high level way to view
his extractor construction.

We also include a proof of a generalization of Vazirani's XOR lemma
that seems ... more >>>

Mergers are functions that transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate $\delta$ then the output has min-entropy rate close to $\delta$. Mergers have proven to be a very useful tool in ... more >>>

Let m,q > 1 be two integers that are co-prime and A be any subset of Z_m. Let P be any multi-linear polynomial of degree d in n variables over Z_m. We show that the MOD_q boolean function on n variables has correlation at most exp(-\Omega(n/(m2^{m-1})^d)) with the boolean function ... more >>>

One of the great challenges of complexity theory is the problem of
analyzing the dependence of the complexity of Boolean functions on the
resources nondeterminism and randomness. So far, this problem could be
solved only for very few models of computation. For so-called
partitioned binary decision diagrams, which are a ... more >>>

We give a randomized algorithm for testing satisfiability of Boolean formulas in conjunctive normal form with no restriction on clause length. Its running time is at most $2^{n(1-1/\alpha)}$ up to a polynomial factor, where $\alpha = \ln(m/n) + O(\ln \ln m)$ and $n$, $m$ are respectively the number of variables ... more >>>

We assume the existence of a function f that is computable in polynomial time but its inverse function is not computable in randomized average-case polynomial time. The cryptographic setting is, however, different: even for a weak one-way function, every possible adversary should fail on a polynomial fraction of inputs. Nevertheless, ... more >>>

Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly
chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$
formed by the intersection of $k$ halfspaces, we prove that
$$\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k \cdot ... more >>>

We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories.

We show that all sets complete for NC$^1$ under AC$^0$
reductions are isomorphic under AC$^0$-computable isomorphisms.

Although our proof does not generalize directly to other
complexity classes, we do show that, for all complexity classes C
closed under NC$^1$-computable many-one reductions, the sets
more >>>

We present a deterministic O(log n log log n) space algorithm for
undirected s,t-connectivity. It is based on the deterministic EREW
algorithm of Chong and Lam (SODA 93) and uses the universal
exploration sequences for trees constructed by Kouck\'y (CCC 01).
Our result improves the O(log^{4/3} n) bound of Armoni ... more >>>

A two server private information retrieval (PIR) scheme
allows a user U to retrieve the i-th bit of an
n-bit string x replicated between two servers while each
server individually learns no information about i. The main
parameter of interest in a PIR scheme is its communication
complexity, namely the ... more >>>

We prove an optimal $\Omega(n)$ lower bound on the randomized
communication complexity of the much-studied
Gap-Hamming-Distance problem. As a consequence, we
obtain essentially optimal multi-pass space lower bounds in the
data stream model for a number of fundamental problems, including
the estimation of frequency moments.

The Gap-Hamming-Distance problem is a ... more >>>

We consider the approximate nearest neighbour search problem on the
Hamming Cube $\b^d$. We show that a randomised cell probe algorithm that
uses polynomial storage and word size $d^{O(1)}$ requires a worst case
query time of $\Omega(\log\log d/\log\log\log d)$. The approximation
factor may be as loose as $2^{\log^{1-\eta}d}$ for any ... more >>>

We prove a number of general theorems about ZK, the class of problems possessing (computational) zero-knowledge proofs. Our results are unconditional, in contrast to most previous works on ZK, which rely on the assumption that one-way functions exist.

We establish several new characterizations of ZK, and use these characterizations to ... more >>>

We prove an unexpected upper bound on a communication game proposed
by Jeff Edmonds and Russell Impagliazzo as an approach for
proving lower bounds for time-space tradeoffs for branching programs.
Our result is based on a generalization of a construction of Erdos,
Frankl and Rodl of a large 3-hypergraph with ... more >>>

We prove that, with high probability, the space complexity of refuting
a random unsatisfiable boolean formula in $k$-CNF on $n$
variables and $m = \Delta n$ clauses is
$O(n \cdot \Delta^{-\frac{1}{k-2}})$.

We consider recurrent analog neural nets where the output of each
gate is subject to Gaussian noise, or any other common noise
distribution that is nonzero on a large set.
We show that many regular languages cannot be recognized by
networks of this type, and
more >>>

In this paper the $R$-machines defined by Blum, Shub and Smale
are generalized by allowing infinite convergent computations.
The description of real numbers is infinite.
Therefore, considering arithmetic operations on real numbers should
also imply infinite computations on {\em analytic machines}.
We prove that $\R$-computable functions are $\Q$-analytic.
We show ... more >>>

Mergers are functions that transform k (possibly dependent)
random sources into a single random source, in a way that ensures
that if one of the input sources has min-entropy rate $\delta$
then the output has min-entropy rate close to $\delta$. Mergers
have proven to be a very useful tool in ... more >>>

We provide another proof of the Sipser--Lautemann Theorem
by which $BPP\subseteq MA$ ($\subseteq PH$).
The current proof is based on strong
results regarding the amplification of $BPP$, due to Zuckerman.
Given these results, the current proof is even simpler than previous ones.
Furthermore, extending the proof leads to ... more >>>

Ant Colony Optimization (ACO) is a kind of randomized search heuristic that has become very popular for solving problems from combinatorial optimization. Solutions for a given problem are constructed by a random walk on a so-called construction graph. This random walk can be influenced by heuristic information about the problem. ... more >>>

Many BDD (binary decision diagram) models are motivated
by CAD applications and have led to complexity theoretical
problems and results. Motivated by applications in genetic
programming Krause, Savick\'y, and Wegener (1999) have shown
that for the inner product function IP$_n$ and the direct
storage access function DSA$_n$ ... more >>>

We give a polynomial time approximation scheme (PTAS) for dense
instances of the NEAREST CODEWORD problem.

We prove that the problems of minimum bisection on k-uniform
hypergraphs are almost exactly as hard to approximate,
up to the factor k/3, as the problem of minimum bisection
on graphs. On a positive side, our argument gives also the
first approximation ... more >>>

Given a set of monomials, the Minimum AND-Circuit problem asks for a
circuit that computes these monomials using AND-gates of fan-in two and
being of minimum size. We prove that the problem is not polynomial time
approximable within a factor of less than 1.0051 unless P = NP, even if
more >>>

We give a randomized approximation algorithm taking
$O(k^{O(k)}n^{b+O(1)})$ time to count the number of copies of a
$k$-vertex graph with treewidth at most $b$ in an $n$ vertex graph
$G$ with approximation ratio $1/k^{O(k)}$ and error probability
inverse exponential in $n$. This algorithm is based on ... more >>>

Let A_1,...,A_n be events in a probability space. The
approximate inclusion-exclusion problem, due to Linial and
Nisan (1990), is to estimate Pr[A_1 OR ... OR A_n] given
Pr[AND_{i\in S}A_i] for all |S|<=k. Kahn et al. (1996) solve
this problem optimally for each k. We study the following more
general question: ... more >>>

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex (typically aperiodic) DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in ... more >>>

A discrete distribution $p$, over $[n]$, is a $k$-histogram if its probability distribution function can be
represented as a piece-wise constant function with $k$ pieces. Such a function
is
represented by a list of $k$ intervals and $k$ corresponding values. We consider
the following problem: given a collection of samples ... more >>>

Inspired by Feige ({\em 36th STOC}, 2004),
we initiate a study of sublinear randomized algorithms
for approximating average parameters of a graph.
Specifically, we consider the average degree of a graph
and the average distance between pairs of vertices in a graph.
Since our focus is on sublinear algorithms, these ... more >>>

We present some of the recent results on computational complexity
of approximating bounded degree combinatorial optimization problems. In
particular, we present the best up to now known explicit nonapproximability
bounds on the very small degree optimization problems which are of
particular importance on the intermediate stages ... more >>>

In the buy-at-bulk $k$-Steiner tree (or rent-or-buy
$k$-Steiner tree) problem we are given a graph $G(V,E)$ with a set
of terminals $T\subseteq V$ including a particular vertex $s$ called
the root, and an integer $k\leq |T|$. There are two cost functions
on the edges of $G$, a buy cost $b:E\longrightarrow ... more >>>

The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical results on this subject.
We consider the approximation ability of randomized search for the class of ... more >>>

This paper shows finding the closest vector in a lattice
to be NP-hard to approximate to within any factor up to
$2^{(\log{n})^{1-\epsilon}}$ where
$\epsilon = (\log\log{n})^{-\alpha}$
and $\alpha$ is any positive constant $<{1\over 2}$.

We study dense instances of several covering problems. An instance of
the set cover problem with $m$ sets is dense if there is $\epsilon>0$
such that any element belongs to at least $\epsilon m$ sets. We show
that the dense set cover problem can be approximated with ... more >>>

In this paper we present some new results on the approximate parallel
construction of Huffman codes. Our algorithm achieves linear work
and logarithmic time, provided that the initial set of elements
is sorted. This is the first parallel algorithm for that problem
with the optimal time and ... more >>>

The PAC learning of rectangles has been studied because they have
been found experimentally to yield excellent hypotheses for several
applied learning problems. Also, pseudorandom sets for rectangles
have been actively studied recently because (i) they are a subproblem
common to the derandomization of depth-2 (DNF) ... more >>>

We study constraint satisfaction problems on the domain $\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form $\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>>

We investigate the hardness of approximating the longest path and
the longest cycle in directed graphs on $n$ vertices. We show that
neither of these two problems can be polynomial time approximated
within $n^{1-\epsilon}$ for any $\epsilon>0$ unless
$\text{P}=\text{NP}$. In particular, the result holds for
... more >>>

We consider the following optimization problem:
given a system of m linear equations in n variables over a certain field,
a feasible solution is any assignment of values to the variables, and the
minimized objective function is the number of equations that are not
satisfied. For ... more >>>

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a $\Pi\Sigma\Pi$ polynomial.
We first prove ... more >>>

A model for parallel and distributed programs, the dynamic process graph (DPG),
is investigated under graph-theoretic and complexity aspects.
Such graphs embed constructors for parallel programs,
synchronization mechanisms as well as conditional branches.
They are capable of representing all possible executions of a
parallel or distributed program ... more >>>

We show that given oracle access to a subroutine which
returns approximate closest vectors in a lattice, one may find in
polynomial-time approximate shortest vectors in a lattice.
The level of approximation is maintained; that is, for any function
$f$, the following holds:
Suppose that the subroutine, on input a ... more >>>

This paper shows SVP_\infty and CVP_\infty to be NP-hard to approximate
to within any factor up to $n^{1/\log\log n}$. This improves on the
best previous result \cite{ABSS} that showed quasi-NP-hardness for
smaller factors, namely $2^{\log^{1-\epsilon}n}$ for any constant
$\epsilon>0$. We show a direct reduction from SAT to these
problems, that ... more >>>

We consider the problem of approximating the entropy of a discrete distribution P on a domain of size q, given access to n independent samples from the distribution. It is known that n > q is necessary, in general, for a good additive estimate of the entropy. A problem of ... more >>>

In this paper, we consider the weighted online set k-multicover problem. In this problem, we have an universe V of elements, a family SS of subsets of V with a positive real cost for every S\in SS, and a ``coverage factor'' (positive integer) k. A subset \{i_0,i_1,\ldots\ \subseteq V of ... more >>>

Recently Ajtai showed that
to approximate the shortest lattice vector in the $l_2$-norm within a
factor $(1+2^{-\mbox{\tiny dim}^k})$, for a sufficiently large
constant $k$, is NP-hard under randomized reductions.
We improve this result to show that
to approximate a shortest lattice vector within a
factor $(1+ \mbox{dim}^{-\epsilon})$, for any
$\epsilon>0$, ... more >>>

We consider <i>minimum equivalent digraph</i> (<i>directed network</i>) problem (also known as the <i>strong transitive reduction</i>), its maximum optimization variant, and some extensions of those two types of problems. We prove the existence of polynomial time approximation algorithms with ratios 1.5 for all the minimization problems and 2 for all the ... more >>>

We present a c.k/2^k approximation algorithm for the Max k-CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which has an approximation guarantee of Omega(k/(2^k log k)). Our approximation guarantee matches the upper bound of Samorodnitsky and Trevisan up ... more >>>

The max-bisection problem is to find a partition of the vertices of a
graph into two equal size subsets that maximizes the number of edges with
endpoints in both subsets.
We obtain new improved approximation ratios for the max-bisection problem on
the low degree $k$-regular graphs for ... more >>>

Khot formulated in 2002 the "Unique Games Conjectures" stating that, for any epsilon > 0, given a system of constraints of a certain form, and the promise that there is an assignment that satisfies a 1-epsilon fraction of constraints, it is intractable to find a solution that satisfies even an ... more >>>

We prove existence of approximation schemes for instances of MAX-CUT with $\Omega(\frac{n^2}{\Delta})$ edges which work in $2^{O^\thicksim(\frac{\Delta}{\varepsilon^2})}n^{O(1)}$ time. This entails in particular existence of quasi-polynomial approximation schemes (QPTASs) for mildly sparse instances of MAX-CUT with $\Omega(\frac{n^2}{\operatorname{polylog} n})$ edges. The result depends on new sampling method for smoothed linear programs that ... more >>>

We develop an analytic framework based on
linear approximation and point out how a number of complexity
related questions --
on circuit and communication
complexity lower bounds, as well as
pseudorandomness, learnability, and general combinatorics
of Boolean functions --
fit neatly into this framework. This ... more >>>

We study approximation hardness and satisfiability of bounded
occurrence uniform instances of SAT. Among other things, we prove
the inapproximability for SAT instances in which every clause has
exactly 3 literals and each variable occurs exactly 4 times,
and display an explicit ... more >>>

The paper contributes to the systematic study (started by Berman and
Karpinski) of explicit approximability lower bounds for small occurrence optimization
problems. We present parametrized reductions for some packing and
covering problems, including 3-Dimensional Matching, and prove the best
known inapproximability results even for highly restricted versions of ... more >>>

We consider bounded occurrence (degree) instances of a minimum
constraint satisfaction problem MIN-LIN2 and a MIN-BISECTION problem for
graphs. MIN-LIN2 is an optimization problem for a given system of linear
equations mod 2 to construct a solution that satisfies the minimum number
of them. E3-OCC-MIN-E3-LIN2 is ... more >>>

We prove approximation hardness of short symmetric instances
of MAX-3SAT in which each literal occurs exactly twice, and
each clause is exactly of size 3. We display also an explicit
approximation lower bound for that problem. The bound two on
the number ... more >>>

The general asymmetric (and metric) TSP is known to be approximable
only to within an O(log n) factor, and is also known to be
approximable within a constant factor as soon as the metric is
bounded. In this paper we study the asymmetric and symmetric TSP
problems with bounded metrics ... more >>>

This paper deals with the number of monochromatic combinatorial
rectangles required to approximate a Boolean function on a constant
fraction of all inputs, where each rectangle may be defined with
respect to its own partition of the input variables. The main result
of the paper is that the number of ... more >>>

We study the problem of absolute approximability of MAX-CSP problems with the global constraints. We prove existence of an efficient sampling method for the MAX-CSP class of problems with linear global constraints and bounded feasibility gap. It gives for the first time a polynomial in epsilon^-1 sample complexity bound for ... more >>>

We study the approximability of predicates on $k$ variables from a
domain $[q]$, and give a new sufficient condition for such predicates
to be approximation resistant under the Unique Games Conjecture.
Specifically, we show that a predicate $P$ is approximation resistant
if there exists a balanced pairwise independent distribution over
more >>>

We present the first approximation schemes for minimizing weighted flow time
on a single machine with preemption. Our first result is an algorithm that
computes a $(1+\eps)$-approximate solution for any instance of weighted flow
time in $O(n^{O(\ln W \ln P/\eps^3)})$ time; here $P$ is the ratio of
more >>>

We consider computationally-efficient incentive-compatible
mechanisms that use the VCG payment scheme, and study how well they
can approximate the social welfare in auction settings. We obtain a
$2$-approximation for multi-unit auctions, and show that this is
best possible, even though from a purely computational perspective
an FPTAS exists. For combinatorial ... more >>>

Ordered binary decision diagrams (OBDDs) and their variants
are motivated by the need to represent Boolean functions
in applications. Research concerning these applications leads
also to problems and results interesting from theoretical
point of view. In this paper, methods from communication
complexity and information theory ... more >>>

Complexity theory typically studies the complexity of computing a function $h(x) : \{0,1\}^n \to \{0,1\}^m$ of a given input $x$. We advocate the study of the complexity of generating the distribution $h(x)$ for uniform $x$, given random bits.

Our main results are:

\begin{itemize}
\item There are explicit $AC^0$ circuits of ... more >>>

Starting with Kilian (STOC `92), several works have shown how to use probabilistically checkable proofs (PCPs) and cryptographic primitives such as collision-resistant hashing to construct very efficient argument systems (a.k.a. computationally sound proofs), for example with polylogarithmic communication complexity. Ishai et al. (CCC `07) raised the question of whether PCPs ... more >>>

We introduce the notion of a stable instance for a discrete
optimization problem, and argue that in many practical situations
only sufficiently stable instances are of interest. The question
then arises whether stable instances of NP--hard problems are
easier to solve. In particular, whether there exist algorithms
that solve correctly ... more >>>

Let $\F\{x_1,x_2,\cdots,x_n\}$ be the noncommutative polynomial
ring over a field $\F$, where the $x_i$'s are free noncommuting
formal variables. Given a finite automaton $\A$ with the $x_i$'s as
alphabet, we can define polynomials $\f( mod A)$ and $\f(div A)$
obtained by natural operations that we call ... more >>>

Functions in arithmetic NC1 are known to have equivalent constant
width polynomial degree circuits, but the converse containment is
unknown. In a partial answer to this question, we show that syntactic
multilinear circuits of constant width and polynomial degree can be
depth-reduced, though the resulting circuits need not be ... more >>>

We show that proving exponential lower bounds on depth four arithmetic
circuits imply exponential lower bounds for unrestricted depth arithmetic
circuits. In other words, for exponential sized circuits additional depth
beyond four does not help.

We then show that a complete black-box derandomization of Identity Testing problem for depth four ... more >>>

The aim of this paper is to use formal power series techniques to
study the structure of small arithmetic complexity classes such as
GapNC^1 and GapL. More precisely, we apply the Kleene closure of
languages and the formal power series operations of inversion and
root ... more >>>

The boolean circuit complexity classes
AC^0 \subseteq AC^0[m] \subseteq TC^0 \subseteq NC^1 have been studied
intensely. Other than NC^1, they are defined by constant-depth
circuits of polynomial size and unbounded fan-in over some set of
allowed gates. One reason for interest in these classes is that they
contain the ... more >>>

The parallel complexity class NC^1 has many equivalent models such as
polynomial size formulae and bounded width branching
programs. Caussinus et al. \cite{CMTV} considered arithmetizations of
two of these classes, #NC^1 and #BWBP. We further this study to
include arithmetization of other classes. In particular, we show that
counting paths ... more >>>

We study some problems solvable in deterministic polynomial time given oracle access to the (promise version of) the Arthur-Merlin class.
Our main results are the following: (i) $BPP^{NP}_{||} \subseteq P^{prAM}_{||}$; (ii) $S_2^p \subseteq P^{prAM}$. In addition to providing new upperbounds for the classes $S_2^p$ and $BPP^{NP}_{||}$, these results are interesting ... more >>>

It is well known that probabilistic boolean decision trees
cannot be much more powerful than deterministic ones (N.~Nisan, SIAM
Journal on Computing, 20(6):999--1007, 1991). Motivated by a question
if randomization can significantly speed up a nondeterministic
computation via a boolean decision tree, we address structural
properties of Arthur-Merlin games in ... more >>>

Algebraic codes that achieve list decoding capacity were recently
constructed by a careful ``folding'' of the Reed-Solomon code. The
``low-degree'' nature of this folding operation was crucial to the list
decoding algorithm. We show how such folding schemes conducive to list
decoding arise out of the Artin-Frobenius automorphism at primes ... more >>>

We show that the asymmetric $k$-center problem is
$\Omega(\log^* n)$-hard to approximate unless
${\rm NP} \subseteq {\rm DTIME}(n^{poly(\log \log n)})$.
Since an $O(\log^* n)$-approximation algorithm is known
for this problem, this essentially resolves the approximability
of this problem. This is the first natural problem
whose approximability threshold does not polynomially ... more >>>

Ordered binary decision diagrams (OBDDs) are nowadays the
most common dynamic data structure or representation type
for Boolean functions. Among the many areas of application
are verification, model checking, and computer aided design.
For many functions it is easy to estimate the OBDD ... more >>>

We introduce new algorithms for lattice basis reduction that are
improvements of the LLL-algorithm. We demonstrate the power of
these algorithms by solving random subset sum problems of
arbitrary density with 74 and 82 many weights, by breaking the
Chor-Rivest cryptoscheme in dimensions 103 and 151 ... more >>>

We study attribute efficient learning in the PAC learning model with
membership queries. First, we give an {\it attribute efficient}
PAC-learning algorithm for DNF with membership queries under the
uniform distribution. Previous algorithms for DNF have sample size
polynomial in the number of attributes $n$. Our algorithm is the
first ... more >>>

We show the following results regarding complete sets:

NP-complete sets and PSPACE-complete sets are many-one
autoreducible.

Complete sets of any level of PH, MODPH, or
the Boolean hierarchy over NP are many-one autoreducible.

EXP-complete sets are many-one mitotic.

NEXP-complete sets are weakly many-one mitotic.

PSPACE-complete sets are weakly Turing-mitotic.

If ... more >>>

We survey the theory of average-case complexity, with a
focus on problems in NP.

Both average-case complexity and the study of the approximability properties of NP-optimization problems are well established and active fields of research. By applying the notion of average-case complexity to approximation problems we provide a formal framework that allows the classification of NP-optimization problems according to their average-case approximability. Thus, known ... more >>>

We use the assumption that all sets in NP (or other levels
of the polynomial-time hierarchy) have efficient average-case
algorithms to derive collapse consequences for MA, AM, and various
subclasses of P/poly. As a further consequence we show for
C in {P(PP), PSPACE} that ... more >>>

Separating different propositional proof systems---that is, demonstrating that one proof system cannot efficiently simulate another proof system---is one of the main goals of proof complexity. Nevertheless, all known separation results between non-abstract proof systems are for specific families of hard tautologies: for what we know, in the average case all ... more >>>

It is a trivial observation that every decidable set has strings of length $n$ with Kolmogorov complexity $\log n + O(1)$ if it has any strings of length $n$ at all. Things become much more interesting when one asks whether a similar property holds when one
considers *resource-bounded* Kolmogorov complexity. ... more >>>