ECCC Reports 2008

The Power of Unentanglement, by Aaronson, Scott (aaronson@csail.mit.edu, MIT 32-G638,Cambridge MA 02139 USA)Beigi, SalmanDrucker, AndrewFefferman, BillShor, Peter

The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given ~O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on Dinur's version of the PCP Theorem and is inherently non-relativizing. Second, we show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we give evidence that QMA(2) is contained in PSPACE, by showing that this would follow from "strong amplification" of QMA(2) protocols. Finally, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.

Cryptographic Complexity of Multi-party Computation Problems: Classifications and Separations, by Manoj Prabhakaran, Mike Rosulek

We develop new tools to study the relative complexities of secure multi-party computation tasks (functionalities) in the Universal Composition framework. When one task can be securely realized using another task as a black-box, we interpret this as a qualitative, complexity-theoretic reduction between the two tasks. Virtually all previous characterizations of MPC functionalities, in the UC model or otherwise, focus exclusively on secure function evaluation. In comparison, the tools we develop do not rely on any special internal structure of the functionality, thus applying to functionalities with arbitrary behavior. Our tools additionally apply uniformly to both the PPT and unbounded computation models. Our first main tool is the notion of {em splittability}, which is an exact characterization of realizability in the UC framework with respect to a large class of communication channel functionalities. Using this characterization, we can rederive all previously-known impossibility results as immediate and simple corollaries. We also complete the combinatorial characterization of 2-party secure function evaluation initiated by cite{CanettiKuLi03} and partially extend the combinatorial conditions to the multi-party setting. Our second main tool is the notion of {em deviation-revealing} functionalities, which allows us to translate complexity separations in simpler MPC settings (such as the honest-but-curious corruption model) to the standard (malicious) setting. Applying this tool, we demonstrate the existence of functionalities which are neither realizable nor complete, in the unbounded computation model.

Derandomizing the Isolation Lemma and Lower Bounds for Noncommutative Circuit Size, by Vikraman Arvind, Partha Mukhopadhyay

We give a randomized polynomial-time identity test for noncommutative circuits of polynomial degree based on the isolation lemma. Using this result, we show that derandomizing the isolation lemma implies noncommutative circuit size lower bounds. More precisely, we consider two restricted versions of the isolation lemma and show that derandomizing each of them implies nontrivial circuit size lower bounds for noncommutative circuits. These restricted versions of the isolation lemma are natural and would suffice for the standard applications of the isolation lemma.

Arithmetic circuits, syntactic multilinearity, and thelimitations of skew formulae, by Meena Mahajan and B. V. Raghavendra Rao

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 syntactic multilinear. We then focus specifically on polynomial-size syntactic multilinear circuits, and study relationships between classes of functions obtained by imposing various resource (width, depth, degree) restrictions on these circuits. Along the way, we obtain a characterisation of NC1 (and its arithmetic counterparts) in terms of log width restricted planar branching programs. We also study the power of skew formulae, and show that even exponential sums of these are unlikely to suffice to express the determinant function.

Continuity Properties of Equilibria in SomeFisher and Arrow-Debreu Market Models, by Vijay Vazirani, Wang Lei

Following up on the work of Megiddo and Vazirani cite{MV.2007}, who determined continuity properties of equilibrium prices and allocations for perhaps the simplest market model, Fisher's linear case, we do the same for: begin{itemize} item Fisher's model with piecewise-linear, concave utilities item Fisher's model with spending constraint utilities item Arrow-Debreu's model with linear utilities item Arrow-Debreu's model with piecewise-linear, concave utilities end{itemize}

A Computational Theory of Awareness and Decision Making, by Nikhil Devanur, Lance Fortnow

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 information. We also give a formal process-independent definition of awareness based on Levin's universal enumeration. We show the usefulness of computational awareness by showing how it relates to decision making, and how others can manipulate our decision making with appropriate advertising, in particular, connections to sponsored search and brand awareness. Understanding awareness can also help rate the effectiveness of various user interfaces designed to access information.

Dense Subsets of Pseudorandom Sets, by Omer Reingold, Luca Trevisan, Madhur Tulsiani, Salil Vadhan

A theorem of Green, Tao, and Ziegler can be stated (roughly) as follows: if R is a pseudorandom set, and D is a dense subset of R, then D may be modeled by a set M that is dense in the entire domain such that D and M are indistinguishable. (The precise statement refers to``measures'' or distributions rather than sets.) The proof of this theorem is very general, and it applies to notions of pseudorandomness and indistinguishability defined in terms of any family of distinguishers with some mild closure properties. The proof proceeds via iterative partitioning and an energy increment argument, in the spirit of the proof of the weak Szemeredi regularity lemma. The ``reduction'' involved in the proof has exponential complexity in the distinguishing probability. We present a new proof inspired by Nisan's proof of Impagliazzo's hardcore set theorem. The reduction in our proof has polynomial complexity in the distinguishing probability and provides a new characterization of the notion of ``pseudoentropy'' of a distribution. We also follow the connection between the two theorems and obtain a new proof of Impagliazzo's hardcore set theorem via iterative partitioning and energy increment. While our reduction has exponential complexity in some parameters, it has the advantage that the hardcore set is efficiently recognizable.

Complexity of Counting the Optimal Solutions, by Miki Hermann, Reinhard Pichler

Following the approach of Hemaspaandra and Vollmer, we can define counting complexity classes #.C for any complexity class C of decision problems. In particular, the classes #.Pi_{k}P with k >= 1 corresponding to all levels of the polynomial hierarchy have thus been studied. However, for a large variety of counting problems arising from optimization problems, a precise complexity classification turns out to be impossible with these classes. In order to remedy this unsatisfactory situation, we introduce a hierarchy of new counting complexity classes #.Opt_{k}P and #.Opt_{k}P[log n] with k >= 1. We prove several important properties of these new classes, like closure properties and the relationship with the #.Pi_{k}P-classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.

Schemes for Deterministic Polynomial Factoring, by Gábor Ivanyos, Marek Karpinski, NitinSaxena

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n-1) is a smooth number.

Deterministic Extractors for Algebraic Sources, by Zeev Dvir

An algebraic source is a random variable distributed uniformly over the set of common zeros of one or more multivariate polynomials defined over a finite field $F$. Our main result is the construction of an explicit deterministic extractor for algebraic sources over exponentially large prime fields. More precisely, we give an explicit (and arguably simple) function $E : F^n --> {0,1}^m$ such that the output of $E$ on any algebraic source in $F^n$ is close to the uniform distribution, provided that the degrees of the defining polynomials are not too high and that the algebraic source contains `enough' points. This extends previous works on extraction from affine sources (sources distributed over subspaces) and from polynomial sources (sources defined as the image of a low degree polynomial mapping). We also give an additional construction of a deterministic extractor for algebraic sources with support larger than $|F|^{n/2}$. This construction works over fields as small as $d^O(1)$, where $d$ is the maximal degree of a polynomial used to define the source.

On Proximity Oblivious Testing, by Oded Goldreich and Dana Ron

We initiate a systematic study of a special type of property testers. These testers consist of repeating a basic test for a number of times that depends on the proximity parameters, whereas the basic test is oblivious of the proximity parameter. We refer to such basic tests by the term proximity-oblivious testers. While proximity-oblivious testers were studied before - most notably in the algebraic setting - the current study seems to be the first one to focus on graph properties. We provide a mix of positive and negative results, and in particular characterizations of the graph properties that have constant-query proximity-oblivious testers in the two standard models (i.e., the adjacency matrix and the bounded-degree models). Furthermore, we show that constant-query proximity-oblivious testers do not exist for many easily testable properties, and that even when proximity-oblivious testers exist repeating them does not necessarily yield the best standard testers for the corresponding property.

Property Testing of Equivalence under a Permutation Group Action, by Sourav Chakraborty, Laszlo Babai

For a permutation group $G$ acting on the set $Omega$ we say that two strings $x,y,:,Omegatoboole$ are {em $G$-isomorphic} if they are equivalent under the action of $G$, ie, if for some $piin G$ we have $x(i^{pi})=y(i)$ for all $iinOmega$. Cyclic Shift, Graph Isomorphism and Hypergraph Isomorphism are special cases, and subcases corresponding to certain classes of groups have been central to the design of efficient isomorphism testing for subclasses of graphs (Luks 1982). We study the complexity of $G$-isomorphism in the context of property testing: we want to find the randomized decision tree complexity of distinguishing the cases when $x$ and $y$ are $G$-isomorphic from the cases when they are at least $delta$-far from being $G$-isomorphic (in normalized Hamming distance). Error can be 1-sided or 2-sided. In each case we consider two models. In the query-1 model we assume $y$ is known and only $x$ needs to be queried. In the query-2 model we have to query both $x$ and $y$. We give various upper and lower bounds for the four combinations of models considered in terms of $n=|Omega|$ and $|G|$. In many cases, substantial gaps remain between the upper and lower bounds. However, for {em primitive permutation groups}, we obtain a tight (up to polylog($n$) factors) bound of $wti{Theta}(sqrt{nlog |G|})$ for the 1-sided error query complexity in the query-2 model and a tight bound of $wti{Theta}(log |G|)$ for the 1-sided error query complexity in the query-1 model. This result extends results of Fischer and Matsliah (2006) on Graph Isomorphism to a surprisingly general class of groups which also includes isomorphism of uniform hypergraphs of any rank. Besides the fact that they include Graph Isomorphism, primitive permutation groups are significant because they form the ``building blocks'' of all permutations groups, providing the base cases of a natural divide-and-conquer approach successfully exploited in algorithm design (Luks, 1982). While all our bounds are in terms of the order of $G$, it seems likely that tighter bounds will depend on the finer structure of $G$; our result on primitive groups is a first step in this direction.

Algorithmic Aspects of Property Testing in the Dense Graphs Model, by Oded Goldreich and Dana Ron

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 the proximity parameter, denoted $e$. The study of these questions reveals the importance of algorithmic design (also) in this model. The highlights of our study are:

A gap between the complexity of adaptive and non-adaptive testers. Specifically, there exists a (natural) graph property that can be tested using $tildeO(e^{-1})$ adaptive queries, but cannot be tested using $o(e^{-3/2})$ non-adaptive queries.
In contrast, there exist natural graph properties that can be tested using $tildeO(e^{-1})$ non-adaptive queries, whereas $Omega(e^{-1})$ queries are required even in the adaptive case.

We mention that the properties used in the foregoing conflicting results have a similar flavor, although they are of course different.

Amplifying Lower Bounds by Means of Self-Reducibility, by Eric Allender, Michal Koucký

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 number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC^1. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC^0 circuits of size n^{1+epsilon_d}. If one were able to improve this lower bound to show that there is some constant epsilon>0 such that every TC^0 circuit family recognizing BFE has size n^{1+epsilon}, then it would follow that TC^0 not= NC^1. We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC^0 and AC^0[6] circuits of size n^{1+c} for some constant c depending on d.

Curves That Must Be Retraced, by Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo

We exhibit a polynomial time computable plane curve GAMMA that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization f of GAMMA and every positive integer n, there is some positive-length subcurve of GAMMA that f retraces at least n times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem.

Soft decoding, dual BCH codes, and better list-decodable eps-biased codes, by Venkatesan Guruswami, Atri Rudra

We construct binary linear codes that are efficiently list-decodable up to a fraction $(1/2-eps)$ of errors. The codes encode $k$ bits into $n = {rm poly}(k/eps)$ bits and are constructible and list-decodable in time polynomial in $k$ and $1/eps$ (in particular, in our results $eps$ need not be constant and can even be polynomially small in $n$). Our results give the best known polynomial dependence of $n$ on $k$ and $1/eps$ for such codes. Specifically, we are able to achieve $n le O(k^3/eps^{3+gamma})$ or, if a linear dependence on $k$ is required, $n le O(k/eps^{5+gamma})$, where $gamma > 0$ is an arbitrary constant. The best previously known constructive bounds in this setting were $n le O(k^2/eps^4)$ and $n le O(k/eps^6)$. Non-constructively, a random linear encoding of length $n = O(k/eps^2)$ suffices, but no sub-exponential algorithm is known for list decoding random codes. Our construction with a cubic dependence on $eps$ is obtained by concatenating the recent Parvaresh-Vardy (PV) codes with dual BCH codes, and crucially exploits the soft decoding algorithm for PV codes. This result yields better hardness results for the problem of approximating NP witnesses in the model of Kumar and Sivakumar. Our result with the linear dependence on $k$ is based on concatenation of the PV code with an arbitrary inner code of good minimum distance. In addition to being a basic question in coding theory, codes that are list-decodable from a fraction $(1/2-eps)$ of errors for $eps to 0$ have found many uses in complexity theory. In addition, our codes have the property that all nonzero codewords have relative Hamming weights in the range $(1/2-eps ,1/2+eps)$; this {em $eps$-biased} property is a fundamental notion in pseudorandomness.

Strict Self-Assembly of Discrete Sierpinski Triangles, by James I. Lathrop, Jack H. Lutz, Scott M. Summers

Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2007).

The Complexity of Local List Decoding, by Dan Gutfreund, Guy Rothblum

We study the complexity of locally list-decoding binary error correcting codes with good parameters (that are polynomially related to information theoretic bounds). We show that computing majority over $Theta(1/eps)$ bits is essentially equivalent to locally list-decoding binary codes from relative distance $1/2-eps$ with list size $poly(1/eps)$. That is, a local-decoder for such a code can be used to construct a circuit of roughly the same size and depth that computes majority on $Theta(1/eps)$ bits. On the other hand, there is an explicit locally list-decodable code with these parameters that has a very efficient (in terms of circuit size and depth) local-decoder that uses majority gates of fan-in $Theta(1/eps)$. Using known lower bounds for computing majority by constant depth circuits, our results imply that every constant-depth decoder for such a code must have size almost exponential in $1/eps$. This shows that the list-decoding radius of the constant-depth local-list-decoders of Goldwasser {em et al.} [STOC07] is essentially optimal. Using the tight connection between locally-list-decodable codes and hardness amplification, we obtain similar limitations on the complexity of uniform (and even somewhat non-uniform) fully-black-box worst-case to average-case reductions. Very recently, Shaltiel and Viola [ECCC07] obtained similar limitations for completely non-uniform fully-black-box worst-case to average-case reductions, but only for the special case that the reduction is {em non-adaptive}.

2-Transitivity is Insufficient for Local Testability, by Elena Grigorescu, Tali Kaufman, Madhu Sudan

A basic goal in Property Testing is to identify a minimal set of features that make a property testable. For the case when the property to be tested is membership in a binary linear error-correcting code, Alon et al.~cite{AKKLR} had conjectured that the presence of a {em single} low weight code in the dual, and ``2-transitivity'' of the code (i.e., the code is invariant under a 2-transitive group of permutations on the coordinates of the code) suffice to get local testability. We refute this conjecture by giving a family of error correcting codes where the coordinates of the codewords form a large field of characteristic two, and the code is invariant under affine transformations of the domain. This class of properties was introduced by Kaufman and Sudan~cite{Kauf-Sudan} as a setting where many results in algebraic property testing generalize. Our result shows a complementary virtue: this family also can be useful in producing counterexamples to natural conjectures.

Lower Bounds for Boolean Circuits with Finite Depth and Arbitrary Gates, by Dmitriy Cherukhin

We consider bounded depth circuits over an arbitrary field $K$. If the field $K$ is finite, then we allow arbitrary gates $K^nto K$. For instance, in the case of field $GF(2)$ we allow any Boolean gates. If the field $K$ is infinite, then we allow only polinomials. For every fixed depth $d$, we prove a lower bound $Omega(nlambda_{d-1}(n))$ for the size (i.e. the number of wires) of any circuit for computing the cyclic convolution over the field $K$. In particular, for $d=2,3,4$, our bounds are $Omega(n^{1.5})$, $Omega(nlog n)$ and $Omega(nloglog n)$ respectively; for $dge 5$, the function $lambda_{d-1}(n)$ is slowly growing. On the Boolean model, our bounds are the best known for all even $d$ and for $d=3$. For $d=2,3$, we prove these bounds in previous papers.

Fixed Point and Aperiodic Tilings, by James I. Lathrop, Jack H. Lutz, Matthew J. Patitz, Scott M. =SummersTR08-030Bruno Durnad, Alexander Shen, Andrei Romashchenko

An aperiodic tile set was first constructed by R.Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals) We present a new construction of an aperiodic tile set. The flexibility of this construction simplifies proofs of some known results and allows us to construct a ``robust'' aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. Our construction of an aperiodic self-similar tile set is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J.von Neumann self-reproducing automata; similar ideas were also used by P.Gacs in the context of error-correcting computations.

The Shrinking Property for NP and coNP, by Christian Glaßer, Christian Reitwießner, Victor Selivanov

We study the shrinking and separation properties (two notions well-known in descriptive set theory) for NP and coNP and show that under reasonable complexity-theoretic assumptions, both properties do not hold for NP and the shrinking property does not hold for coNP. In particular we obtain the following results. 1. NP and coNP do not have the shrinking property, unless PH is finite. In general, Sigma_n and Pi_n do not have the shrinking property, unless PH is finite. This solves an open question from [Selivanov 94]. 2. The separation property does not hold for NP, unless UP subseteq coNP. 3. The shrinking property does not hold for NP, unless there exist NP-hard disjoint NP-pairs (existence of such pairs would contradict a conjecture by Even, Selman, and Yacobi). 4. The shrinking property does not hold for NP, unless there exist complete disjoint NP-pairs. Moreover, we prove that the assumption NP neq coNP is too weak to refute the shrinking property for NP in a relativizable way. For this we construct an oracle relative to which P = NP cap coNP, NP neq coNP, and NP has the shrinking property. This solves an open question by Blass and Gurevich who explicitly ask for such an oracle.

The Tractability of Model-Checking for LTL: The Good, the Bad, and the Ugly Fragments, by Michael Bauland, Martin Mundhenk, Thomas Schneider, Henning Schnoor, Ilka Schnoor, Heribert Vollmer

In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in PTIME) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators.

Generalizations of the Hartmanis-Immerman-Sewelson Theorem and Applications to Infinite Subsets of P-Selective Sets, by Till Tantau

The Hartmanis--Immerman--Sewelson theorem is the classical link between the exponential and the polynomial time realm. It states that NE = E if, and only if, every sparse set in NP lies in P. We establish similar links for classes other than sparse sets: 1. E = UE if, and only if, all functions f: {1}^* to Sigma^* in NPSV_g lie in FP. 2. E = NE if, and only if, all functions f: {1}^* to Sigma^* in NPFewV lie in FP. 3. E = E^NP if, and only if, all functions f: {1}^* to Sigma^* in OptP lie in FP. 4. E = E^NP if, and only if, all standard left cuts in NP lie in P. 5. E = EH if, and only if, PH cap P/poly = P. We apply these results to the immunity of P-selective sets. It is known that they can be bi-immune, but not Pi_2^p/1-immune. Their immunity is closely related to top-Toda languages, whose complexity we link to the exponential realm, and also to king languages. We introduce the new notion of superkings, which are characterized in terms of existsforall-predicates rather than forallexists-predicates, and show that king languages cannot be Sigma_2^p-immune. As a consequence, P-selective sets cannot be Sigma_2^/1-immune and, if E^NP^NP = E, not even P/1-immune.

Towards an Optimal Separation of Space and Length in Resolution, by Jakob Nordström, Johan Håstad

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Theta(sqrt(n)) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width in (Nordstr�m 2006) from logarithmic to polynomial. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.

New results on Noncommutative and Commutative Polynomial Identity Testing, by Vikraman Arvind, Partha Mukhopadhyay, Srikanth Srinivasan

Using ideas from automata theory we design a new efficient (deterministic) identity test for the emph{noncommutative} polynomial identity testing problem (first introduced and studied by Raz-Shpilka in 2005 and Bogdanov-Wee in 2005). More precisely, given as input a noncommutative circuit $C(x_1,cdots,x_n)$ computing a polynomial in $F{x_1,cdots,x_n}$ of degree $d$ with at most $t$ monomials, where the variables $x_i$ are noncommuting, we give a deterministic polynomial identity test that checks if $Cequiv 0$ and runs in time polynomial in $d, n, |C|$, and $t$. The same methods works in a black-box setting: Given a noncommuting black-box polynomial $finF{x_1,cdots,x_n}$ of degree $d$ with $t$ monomials we can, in fact, reconstruct the entire polynomial $f$ in time polynomial in $n,d$ and $t$. Indeed, we apply this idea to the reconstruction of black-box noncommuting algebraic branching programs (the ABPs considered by Nisan in 1991 and Raz-Shpilka in 2005). Assuming that the black-box model allows us to query the ABP for the output at any given gate then we can reconstruct an (equivalent) ABP in deterministic polynomial time. Finally, we turn to commutative identity testing and explore the complexity of the problem when the coefficients of the input polynomial come from an arbitrary finite commutative ring with unity whose elements are uniformly encoded as strings and the ring operations are given by an oracle. We show that several algorithmic results for polynomial identity testing over fields also hold when the coefficients come from such finite rings.

On the Value of Multiple Read/Write Streams for Approximating Frequency Moments, by Paul Beame, Dang-Trinh Huynh-Ngoc

Recently, an extension of the standard data stream model has been introduced in which an algorithm can create and manipulate multiple read/write streams in addition to its input data stream. Like the data stream model, the most important parameter for this model is the amount of internal memory used by such an algorithm. The other key parameters are the number of streams the algorithm uses and the number of passes it makes on these streams. We consider how the addition of these multiple read/write streams impacts the ability of algorithms to approximate the frequency moments of the input stream. We show that any randomized read/write stream algorithm with a fixed number of streams and a sub-logarithmic number of passes that approximates the k-th frequency moment F_k of an input sequence of length of at most N from {1,...,N} within a constant factor requires space Omega(N^{1-4/k-delta}) for any delta > 0. For comparison, it is known that with a single read-only data stream there is a randomized constant-factor approximation for F_k using tilde O(N^{1-2/k}) space and that there is a deterministic algorithm computing F_k exactly using 3 read/write streams, O(log N) passes, and O(log N) space. Our lower bounds also apply to (1+epsilon)-approximations of F_k for epsilon ge 1/N.

NP-Hard Sets are Exponentially Dense Unless NP is contained in coNP/poly, by Harry Buhrman, John Hitchcock

We show that hard sets S for NP must have exponential density, i.e. |S_=n| ≥ 2^{n^ε} for some ε > 0 and infinitely many n, unless coNP ⊆ NPpoly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n^1-ε queries. In addition we study the instance complexity of NP-hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n^δ for some δ > 0. This result also holds for Turing reductions that make n^1-ε queries.

The Complexity of Rationalizing Matchings, by Shankar Kalyanaraman, Christopher Umans

Given a set of observed economic choices, can one infer preferences and/or utility functions for the players that are consistent with the data? Questions of this type are called {em rationalization} or {em revealed preference} problems in the economic literature, and are the subject of a rich body of work. From the computer science perspective, it is natural to study the complexity of rationalization in various scenarios. We consider a class of rationalization problems in which the economic data is expressed by a collection of matchings, and the question is whether there exist preference orderings for the nodes under which all the matchings are {em stable}. We show that the rationalization problem for one-one matchings is NP-complete. We propose two natural notions of approximation, and show that the problem is hard to approximate to within a constant factor, under both. On the positive side, we describe a simple algorithm that achieves a $3/4$ approximation ratio for one of these approximation notions. We also prove similar results for a version of many-one matching.

Decodability of Group Homomorphisms beyond the Johnson Bound, by Irit Dinur, Elena Grigorescu, Swastik Kopparty, Madhu Sudan

Given a pair of finite groups $G$ and $H$, the set of homomorphisms from $G$ to $H$ form an error-correcting code where codewords differ in at least $1/2$ the coordinates. We show that for every pair of {em abelian} groups $G$ and $H$, the resulting code is (locally) list-decodable from a fraction of errors arbitrarily close to its distance. At the heart of this result is the following combinatorial result: There is a fixed polynomial $p(cdot)$ such that for every pair of abelian groups $G$ and $H$, if the maximum fraction of agreement between two distinct homomorphisms from $G$ to $H$ is $Lambda$, then for every $epsilon> 0$ and every function $f:Gto H$, the number of homomorphisms that have agreement $Lambda + epsilon$ with $f$ is at most $p(1/epsilon)$. We thus give a broad class of codes whose list-decoding radius exceeds the ``Johnson bound''. Examples of such codes are rare in the literature, and for the ones that do exist, ``combinatorial'' techniques to analyze their list-decodability are limited. Our work is an attempt to add to the body of such techniques. We use the fact that abelian groups decompose into simpler ones and thus codes derived from homomorphisms over abelian groups may be viewed as certain ``compositions'' of simpler codes. We give techniques to lift list-decoding bounds for the component codes to bounds for the composed code. We believe these techniques may be of general interest.

Entropy of operators or why matrix multiplication is hard forsmall depth circuits, by Stasys Jukna

In this note we consider unbounded fanin depth-2 circuits with arbitrary boolean functions as gates. We define the entropy of an operator f:{0,1}^n --> {0,1}^m is as the logarithm of the maximum number of vectors distinguishable by at least one special subfunction of f. Then we prove that every depth-2 circuit for f requires at least entropy(f) wires. This generalizes and substantially simplifies the argument used by Cherukhin in 2005 to derive the highest known lower bound n^{3/2} for the operator of cyclic convolutions. We then show that the multiplication of two n^{1/2} by n^{1/2} matrices over any finite field has entropy at least n^{3/2}. All proofs are elementary.

A Counterexample to Strong Parallel Repetition, by Ran Raz

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 constant. Several researchers asked wether this bound could be improved to $(1- epsilon)^{Omega(n/s)}$; this question is usually referred to as the strong parallel repetition problem. We show that the answer for this question is negative. More precisely, we consider the odd cycle game of size $m$; a two-prover game with value $1-1/2m$. We show that the value of the odd cycle game repeated in parallel $n$ times is at least $1- (1/m) cdot O(sqrt{n})$. This implies that for large enough $n$ (say, $n geq Omega(m^2)$), the value of the odd cycle game repeated in parallel $n$ times is at least $(1- 1/4m^2)^{O(n)}$. Thus: 1) For parallel repetition of general games: the bounds of $(1- epsilon^c)^{Omega(n/s)}$ given in~cite{R,Hol} are of the right form, up to determining the exact value of the constant $c geq 2$. 2) For parallel repetition of XOR games, unique games and projection games: the bounds of $(1- epsilon^2)^{Omega(n)}$ given in~cite{FKO} (for XOR games) and in~cite{Rao} (for unique and projection games) are tight. 3) For parallel repetition of the odd cycle game: the bound of $1- (1/m) cdot tilde{Omega}(sqrt{n})$ given in~cite{FKO} is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least $1 - epsilon^2$ satisfiable from instances that are at most $1 - (2/pi) cdot epsilon$ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.

A Quantum Time-Space Lower Bound for the Counting Hierarchy, by Dieter van Melkebeek, Thomas Watson

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 $epsilon$ there exists a real $c>1$ such that either: begin{itemize} item MajMajSAT does not have a quantum algorithm with bounded two-sided error that runs in time $n^c$, or item MajSAT does not have a quantum algorithm with bounded two-sided error that runs in time $n^d$ and space $n^{1-epsilon}$. end{itemize} In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded two-sided error running in time $n^{1+o(1)}$ and space $n^{1-epsilon}$ for any $epsilon>0$. The key technical novelty is a time- and space-efficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation.

The Sign-Rank of AC^0, by Alexander Razborov, Alexander Sherstov

The sign-rank of a matrix A=[A_{ij}] with +/-1 entries is the least rank of a real matrix B=[B_{ij}] with A_{ij}B_{ij}>0 for all i,j. We obtain the first exponential lower bound on the sign-rank of a function in AC^0. Namely, let f(x,y)=bigwedge_{i=1}^mbigvee_{j=1}^{m^2} (x_{ij}wedge y_{ij}). We show that the matrix [f(x,y)]_{x,y} has sign-rank 2^{Omega(m)}. This in particular implies that Sigma_2^{cc}notsubseteq UPP^{cc}, which solves a long-standing open problem posed by Babai, Frankl, and Simon (1986). Our result additionally implies a lower bound in learning theory. Specifically, let phi_1,...,phi_r : {0,1}^n -> R be functions such that every DNF formula f:{0,1}^n->{+1,-1} of polynomial size has the representation f=sign(a_1*phi_1+...+a_r*phi_r) for some reals a_1,...,a_r. We prove that then r> 2^{Omega(n^{1/3})}, which essentially matches an upper bound of 2^{tilde O(n^{1/3})} due to Klivans and Servedio (2001). Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in AC^0. This substantially generalizes and strengthens the results of Krause and Pudlak (1997).

Extractors for Low-Weight Affine Sources, by Anup Rao

We give polynomial time computable extractors for low-weight affine sources. A distribution is affine if it samples a random point from some unknown low dimensional subspace of F^n_2 . A distribution is low weight affine if the corresponding linear space has a basis of low-weight vectors. Low-weight ane sources are thus a generalization of the well studied models of bit-fixing sources (which are just weight 1 affine sources). For universal constants c,e , our extractors can extract almost all the entropy from weight k affine sources of dimension k, as long as k > log^c n, with error 2^{-k^Omega(1)} . This gives new extractors for low entropy bit-xing sources with exponentially small error, a parameter that is important for the application of these extractors to cryptography. Our techniques involve constructing new condensers for affine somewhere random sources.

Separating NOF communication complexity classes RP and NP, by Matei David, Toniann Pitassi

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

Parallel Repetition in Projection Games and a Concentration Bound, by Anup Rao

In a two player game, a referee asks two cooperating players (who are not allowed to communicate) questions sampled from some distribution and decides whether they win or not based on some predicate of the questions and their answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than (1-e), what's the best they can do in the repeated game? We improve earlier results of Raz and Holenstein, which showed that the players cannot win all copies in the repeated game with probability better than (1-e^3)^{Omega(n/c)} (here c is the length of the answers in the game), in the following ways: We prove the bound (1-e^2)^{Omega(n)} as long as n = Omega(log(1/e)/e^2) and the game is a ``projection game'', the type of game most commonly used in hardness of approximation results. Our bound is independent of the answer length and has a better dependence on e. By the recent work of Raz, this bound is essentially tight (we might still hope to get rid of the restriction on n). A consequence of this bound is that the Unique Games Conjecture of Khot is equivalent to: [Unique Games Conjecture] For every delta,e > 0, there exists an alphabet size M(e) such that it is NP-hard to distinguish a Unique Game with alphabet size M for which a 1-e^2 fraction of the constraints can be satisfied from one in which a 1-e^{1-delta} fraction of the constraints can be satisfied. We prove a concentration bound for parallel repetition (of general games) showing that for any constant 0

A Note on the Distance to Monotonicity of Boolean Functions, by Arnab Bhattacharyya

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 already shown by Raskhodnikova earlier.

The Complexity of the Hajos Calculus for Planar Graphs, by Kazuo Iwama, Suguru Tamaki

The planar Hajos calculus is the Hajos calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. We prove that the planar Hajos calculus is polynomially bounded iff the Haj�Los calculus is polynomially bounded.

Every Minor-Closed Property of Sparse Graphs is Testable, by Itai Benjamini, Oded Schramm, Asaf Shapira

Testing a property P of graphs in the bounded degree model deals with the following problem: given a graph G of bounded degree d we should distinguish (with probability 0.9, say) between the case that G satisfies P and the case that one should add/remove at least epsilon d n edges of G to make it satisfy P. In sharp contrast to property testing of dense graphs, which is relatively well understood, very few properties are known to be testable in bounded degree graphs with a constant number of queries. In this paper we identify for the first time a large (and natural) family of properties that can be efficiently tested in bounded degree graphs, by showing that every minor-closed graph property can be tested with a constant number of queries. As a special case, we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.

Approximation Resistant Predicates From Pairwise Independence, by Per Austrin, Elchanan Mossel

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 $[q]^k$ whose support is contained in the set of satisfying assignments to $P$. Using constructions of pairwise indepenent distributions this result implies that: For general $k ge 3$ and $q ge 2$, the Max $k$-CSP$_q$ problem is UG-hard to approximate within $q^{lceil log_2 k +1 rceil - k} + epsilon$. For $k geq 3$ and $q$ prime power, the hardness ratio is improved to $kq(q-1)/q^k + epsilon$. For the special case of $q = 2$, i.e., boolean variables, we can sharpen this bound to $(k + O(k^{0.525}))/2^k + epsilon$, improving upon the best previous bound of $2k/{2^k} + epsilon$ (Samorodnitsky and Trevisan, STOC'06) by essentially a factor $2$. Finally, for $q=2$, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the $O(k^{0.525})$ term can be replaced by the constant $4$.

Constraint Satisfaction over a Non-Boolean Domain: Approximation algorithms and Unique-Games hardness, by Venkatesan Guruswami, Prasad Raghavendra

We study the approximability of the maxcsp problem over non-boolean domains, more specifically over ${0,1,ldots,q-1}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to obtain a UGC hardness result when $q$ is a prime. More precisely, assuming the Unique Games Conjecture, we show that it is NP-hard to approximate the problem to a ratio greater than $q^2k/q^k$. Except for constant factors depending on $q$, the algorithm and the UGC hardness result have the same dependence on the arity $k$.

Limitations of Hardness vs. Randomness under Uniform Reductions, by Dan Gutfreund, Salil Vadhan

We consider (uniform) reductions from computing a function f to the task of distinguishing the output of some pseudorandom generator G from uniform. Impagliazzo and Wigderson (FOCS `98, JCSS `01) and Trevisan and Vadhan (CCC `02, CC `07) exhibited such reductions for every function f in PSPACE. Moreover, their reductions are "black box," showing how to use *any* distinguisher T, given as oracle, in order to compute f (regardless of the complexity of T). The reductions are also adaptive, but only "mildly" (queries of the same length do not occur in different levels of adaptivity). Impagliazzo and Wigderson also exhibited such reductions for every function f in EXP, but those reductions are not black-box, because they only work when the oracle T has small circuits. Our main results are that: 1. *Nonadaptive* black-box reductions as above can only exist for functions f in BPP^{NP} (and thus are unlikely to exist for all of PSPACE). 2. *Mildly adaptive* black-box reductions as above can only exist for functions f in PSPACE (and thus are unlikely to exist for all of EXP).

Lower Bounds and Separations for Constant Depth Multilinear Circuits, by Ran Raz, Amir Yehudayoff

We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth $d$ and product-depth $d+1$ multilinear circuits (where $d$ is constant). That is, there exists a polynomial $f$ such that (1) There exists a multilinear circuit of product-depth $d+1$ and of polynomial size computing $f$. (2) Every multilinear circuit of product-depth $d$ computing $f$ has super-polynomial size.

Algebrization: A New Barrier in Complexity Theory, by Scott Aaronson, Avi Wigderson

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 is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to the low-degree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization -- both inclusions such as IP=PSPACE and MIP=NEXP, and separations such as MA-EXP not in P/poly -- do indeed algebrize. Second, we show that almost all of the major open problems -- including P versus NP, P versus RP, and NEXP versus P/poly -- will require non-algebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O(sqrt(n) log(n)) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL!=NP.

Noisy Interpolating Sets for Low Degree Polynomials, by Zeev Dvir, Amir Shpilka

A Noisy Interpolating Set (NIS) for degree $d$ polynomials is a set $S subseteq F^n$, where $F$ is a finite field, such that any degree $d$ polynomial $q in F[x_1,ldots,x_n]$ can be efficiently interpolated from its values on $S$, even if an adversary corrupts a constant fraction of the values. In this paper we construct explicit NIS for every prime field $F_p$ and any degree $d$. Our sets are of size $O(n^d)$ and have efficient interpolation algorithms that can recover $q$ from a fraction $exp(-O(d))$ of errors. Our construction is based on a theorem which roughly states that if $S$ is a NIS for degree 1 polynomials then $d cdot S= { a_1 + ldots + a_d ,|, a_i in S}$ is a NIS for degree $d$ polynomials. Furthermore, given an efficient interpolation algorithm for $S$, we show how to use it in a black-box manner to build an efficient interpolation algorithm for $d cdot S$. As a corollary we get an explicit family of punctured Reed-Muller codes that is a family of good codes that have an efficient decoding algorithm from a constant fraction of errors. To the best of our knowledge no such construction was known previously.

Disjointness is hard in the multi-party number-on-the-forehead model, by Troy Lee, Adi Shraibman

We show that disjointness requires randomized communication Omega(frac{n^{1/2k}}{(k-1)2^{k-1}2^{2^{k-1}}}) in the general k-party number-on-the-forehead model of complexity. The previous best lower bound was Omega(frac{log n}{k-1}). By results of Beame, Pitassi, and Segerlind, this implies 2^{n^{Omega(1)}} lower bounds on the size of tree-like Lovasz-Schrijver proof systems needed to refute certain unsatisfiable CNFs, and super-polynomial lower bounds on the size of any tree-like proof system whose terms are degree-d polynomial inequalities for d = log log n - O(log log log n). To prove our bound, we develop a new technique for showing lower bounds in the number-on-the-forehead model which is based on the norm induced by cylinder intersections. This bound naturally extends the linear program bound for rank useful in the two-party case to the case of more than two parties, where the fundamental concept of monochromatic rectangles is replaced by monochromatic cylinder intersections. Previously, the only general method known for showing lower bounds in the unrestricted number-on-the-forehead model was the discrepancy method, which can only show bounds of size O(log n) for disjointness. To analyze the bound given by our new technique for the disjointness function, we extend an elegant framework developed by Sherstov in the two-party case which relates polynomial degree to communication complexity. Using this framework we are able to obtain bounds for any tensor of the form F(x_1,ldots,x_k) = f(x_1 wedge ldots wedge x_k) where f is a function which only depends on the number of ones in the input.

Multiparty Communication Complexity of Disjointness, by Arkadev Chattopadhyay, Anil Ada

We extend the 'Generalized Discrepancy' technique suggested by Sherstov to the `Number on the Forehead' model of multiparty communication. This allows us to prove strong lower bounds of n^{Omega(1)} on the communication needed by k players to compute the Disjointness function, provided $k$ is a constant. In general, our method yields strong bounds for functions induced by a symmetric predicate if the approximation degree of the predicate is n^{Omega(1)}. Similar bounds have been independently obtained recently by Lee and Shraibman.

Elusive Functions and Lower Bounds for Arithmetic Circuits, by Ran Raz

A basic fact in linear algebra is that the image of the curve $f(x)=(x^1,x^2,x^3,...,x^m)$, say over $C$, is not contained in any $m-1$ dimensional affine subspace of $C^m$. In other words, the image of $f$ is not contained in the image of any polynomial-mapping $G:C^{m-1} ---> C^m$ of degree 1(that is, an affine mapping). Can one give an explicit example for a polynomial curve $f:C ---> C^m$, such that, the image of $f$ is not contained in the image of any polynomial-mapping $G:C^{m-1} ---> C^m$ of degree 2 ? We show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit $f$ as above of degree up to $2^{m^{o(1)}}$, implies super-polynomial lower bounds for computing the permanent over $C$. More generally, we say that a polynomial-mapping $f:F^n ---> F^m$ is $(s,r)$-elusive, if for every polynomial-mapping $G:F^s ---> F^m$ of degree $r$, $Image(f)$ is not contained in Image(G)$. We show that for many settings of the parameters $n,m,s,r$, explicit constructions of elusive polynomial-mappings imply strong (up to exponential) lower bounds for general arithmetic circuits. Finally, for every $r < log n$, we give an explicit example for a polynomial-mapping $f:F^n ---> F^{n^2}$, of degree $O(r)$, that is $(s,r)$-elusive for $s = n^{1+Omega(1/r)}$. We use this to construct for any $r$, an explicit example for an $n$-variate polynomial of total-degree $O(r)$, such that, any depth $r$ arithmetic circuit for this polynomial (over any field) is of size $> n^{1+Omega(1/r)}$. In particular, for any constant $r$, this gives a constant degree polynomial, such that, any depth $r$ arithmetic circuit for this polynomial is of size $> n^{1+Omega(1)}$. Previously, only lower bounds of the type $Omega(n cdot lambda_r (n))$, where $lambda_r (n)$ are extremely slowly growing functions (e.g., $lambda_5(n) = log^{*}n$, and $lambda_7(n) = log^* log^{*}n$), were known for constant-depth arithmetic circuits for polynomials of constant degree.