ECCC Reports 2009

A NEW LINE OF ATTACK ON THE DICHOTOMY CONJECTURE, by Gábor Kun, Mario Szegedy

The well known dichotomy conjecture of Feder and Vardi states that for every ﬁnite family Γ of constraints CSP(Γ) is either polynomially solvable or NP-hard. Bulatov and Jeavons re- formulated this conjecture in terms of the properties of the algebra P ol(Γ), where the latter is the collection of those n-ary operations (n = 1, 2, . . .) that keep all constraints in Γ invariant. We show that the algebraic condition boils down to whether there are arbi- trarily resilient functions in P ol(Γ). Equivalently, we can express this in the terms of the PCP theory: CSP(Γ) is NP-hard iﬀ all long code tests created from Γ that passes with zero error admits only juntas1. Then, using this characterization and a result of Dinur, Friedgut and Regev, we give an entirely new and transparent proof to the Hell-Neˇsetˇril theorem, which states that for a simple, con- nected and undirected graph H , the problem CSP(H ) is NP-hard if and only if H is non-bipartite. We also introduce another notion of resilience (we call it strong resilience), and we use it in the investigation of CSP problems that �do not have the ability to count.� The complexity of this class is unknown. Several authors conjectured that CSP problems without the ability to count have bounded width, or equivalently, that they can be characterized by existential k-pebble games. The resolution of this conjecture would be a ma jor step towards the resolution of the dichotomy conjecture. We show that CSP problems without the ability to count are exactly the ones with strongly resilient term operations, which might give a handier tool to attack the conjecture than the known algebraic characterizations. Finally, we show that a yet stronger notion of resilience, when the term operation is asymptotically constant, allows us to char- acterize the class of width one CSPs. What emerges from our research, is that certain important al- gebraic conditions that are usually expressed via identities have equivalent analytic deﬁnitions that rely on asymptotic properties of term operations.

Learning parities in the mistake-bound model., by Harry Buhrman, David García-Soriano, Arie Matsliah

We study the problem of learning parity functions that depend on at most $k$ variables ($k$-parities) attribute-efficiently in the mistake-bound model. We design simple, deterministic, polynomial-time algorithms for learning $k$-parities with mistake bound $O(n^{1-frac{c}{k}})$, for any constant $c > 0$. These are the first polynomial-time algorithms that learn $omega(1)$-parities in the mistake-bound model with mistake bound $o(n)$. Using the standard conversion techniques from the mistake-bound model to the PAC model, our algorithms can also be used for learning $k$-parities in the PAC model. In particular, this implies a slight improvement over the results of Klivans and Servedio for learning $k$-parities in the PAC model. We also show that the $widetilde{O}(n^{k/2})$ time algorithm from Klivans and Servedio's paper that PAC-learns $k$-parities with optimal sample complexity can be extended to the mistake-bound model.

Deterministic Polynomial Time Algorithms for Matrix Completion Problems, by Gábor Ivanyos, Marek Karpinski, Nitin Saxena

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to the fundamental problems in computational complexity with numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings (Harvey et al SODA 2005, SODA 2006). We design efficient deterministic algorithms for common generalizations of the results of Lovasz and Geelen on this problem by allowing linear functions in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. We present also a deterministic polynomial time algorithm for finding the minimal number of generators of a given module structure given by matrices. We establish further several hardness results related to matrix algebras and modules. As a result we connect the classical problem of polynomial identity testing with checking surjectivity (or injectivity) between two given modules. One of the elements of our algorithm is a construction of a greedy algorithm for finding a maximum rank element in the more general setting of the problem. The proof methods used in this paper could be also of independent interest.

Are stable instances easy?, by Yonatan Bilu, Nathan Linial

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 and in polynomial time all sufficiently stable instances of some NP--hard problem. The paper focuses on the Max--Cut problem, for which we show that this is indeed the case.

Speedup for Natural Problems and coNP?=NP, by Hunter Monroe

Informally, a language L has speedup if, for any Turing machine for L, there exists one that is better. Blum showed that there are computable languages that have almost-everywhere speedup. These languages were unnatural in that they were constructed for the sole purpose of having such speedup. We identify a condition apparently only slightly stronger than P!=NP which implies that accepting any coNP-complete language has an infinitely-often (i.o.) superpolynomial speedup and NP!=coNP. We also exhibit a natural problem which unconditionally has a weaker type of i.o. speedup based upon whether the full input is read. Neither speedup pertains to the worst case.

Arthur and Merlin as Oracles, by Venkatesan Chakaravarthy, Sambuddha Roy

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 from a derandomization perspective. In conjunction with the hitting set generator construction of Miltersan and Vinodchandran [Computational Complexity, 2005], we get that $S_2^p = P^{NP}$ and $BPP^{NP}_{||} = P^{NP}_{||}$, under the hardness hypothesis associated with derandomizing the class $AM$. This gives an alternative proof of the same result obtained by Shaltiel and Umans [Computational Complexity, 2007]. We also show that if $NP$ has polynomial size circuits then the polynomial time hierarchy ($PH$) collapses as $PH = P^{prMA}$. Under the same hypothesis, we also derive an $FP^{prMA}$ algorithm for learning circuits for SAT; this improves the $ZPP^{NP}$ algorithm for the same problem by Bshouty et al.[JCSS, 1996]. Finally, we design a $FP^{prAM}$ algorithm for the problem of finding near-optimal strategies for succinctly presented zero-sum games. For the same problem, Fortnow et al. [Computational Complexity, 2008] described a $ZPP^{NP}$ algorithm. One advantage of our $FP^{prAM}$ algorithm is that it can be derandomized using the construction of Miltersen and Vinodchandran yielding a $FP^{NP}$ algorithm, under a hardness hypothesis used for derandomizing $AM$.

Cell-Probe Lower Bounds for Prefix Sums, by Emanuele Viola, Emanuele Viola

We prove that to store n bits x so that each prefix-sum query Sum(i) := sum_{k < i} x_k can be answered by non-adaptively probing q cells of log n bits, one needs memory > n + n/log^{O(q)} n. Our bound matches a recent upper bound of n + n/log^{Omega(q)} n by Patrascu (FOCS 2008), also non-adaptive. We also obtain a n + n/log^{2^{O(q)}} n lower bound for storing a string of balanced brackets so that each Match(i) query can be answered by non-adaptively probing q cells. To obtain these bounds we show that a too efficient data structure allows us to break the correlations between query answers.

The Isomorphism Problem for k-Trees is Complete for Logspace, by Johannes Koebler, Sebastian Kuhnert

We show that k-tree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. This improves over the previous StUL upper bound and matches the lower bound. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. We also show that even simple structural properties of k-trees are complete for logspace.

Planar Graph Isomorphism is in Log-space, by Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, Fabian Wagner

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell�s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in log-space.

The Pervasive Reach of Resource-Bounded Kolmogorov Complexity in Computational Complexity Theory, by Eric Allender, Michal Koucký, Detlef Ronneburger, Sambuddha Roy

We continue an investigation into resource-bounded Kolmogorov complexity cite{abkmr}, which highlights the close connections between circuit complexity and Levin's time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced have many advantages over other approaches to defining resource-bounded Kolmogorov complexity (such as much greater independence from the underlying choice of universal machine that is used to define the measure) [ABKMR]. Here, we study the properties of other measures that arise naturally in this framework. The motivation for introducing yet more notions of resource-bounded Kolmogorov complexity are two-fold: * to demonstrate that other complexity measures such as branching-program size and formula size can also be discussed in terms of Kolmogorov complexity, and * to demonstrate that notions such as nondeterministic Kolmogorov complexity and distinguishing complexity [BFL] also fit well into this framework. The main theorems that we provide using this new approach to resource-bounded Kolmogorov complexity are: * A complete set ($RKNt$) for NEXP/poly defined in terms of strings of high Kolmogorov complexity. * A lower bound, showing that $RKNt$ is not in NP intersect coNP. * New conditions equivalent to the conditions ``NEXP is contained in nonuniform NC1'' and ``NEXP is contained in L/poly''. * Theorems showing that ``distinguishing complexity'' is closely connected to both FewEXP and to EXP. * Hardness results for the problems of approximating formula size and branching program size.

Logspace reduction of directed reachability for bounded genus graphs to the planar case, by Jan Kyncl, Tomas Vyskocil

Directed reachability (or briefly reachability) is the following decision problem: given a directed graph G and two of its vertices s,t, determine whether there is a directed path from s to t in G. Directed reachability is a standard complete problem for the complexity class NL. Planar reachability is an important restricted version of the reachability problem, where the input graph is planar. Planar reachability is hard for L and is contained in NL but is not known to be NL-complete or contained in L. Allender et al. showed that reachability for graphs embedded on the torus is logspace-reducible to the planar case. We generalize this result to graphs embedded on a fixed surface of arbitrary genus.

A log-space algorithm for reachability in planar DAGs with few sources, by Derrick Stolee, Derrick Stolee, Chris Bourke, N.V. Vinodchandran

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 would show that nondeterministic log-space computations can be made unambiguous. On the other hand, very little is known about classes of planar graphs that admit log-space algorithms. We make progress in this direction. We show that reachability in planar DAGs with O(log n) number of sources can be solved in log-space. We use a new decomposition technique for planar DAGs as a basis for our algorithm.

On the Complexity of Boolean Functions in Different Characteristics, by Parikshit Gopalan, Shachar Lovett, Amir Shpilka

Every Boolean function on $n$ variables can be expressed as a unique multivariate polynomial modulo $p$ for every prime $p$. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree modulo $p$ must have high complexity in every other characteristic $q$. More precisely, we show the following results about Boolean functions $f:{0,1}^n rightarrow {0,1}$ which depend on all $n$ variables, and distinct primes $p,q$: begin{itemize} item If $f$ has degree $o(log n)$ modulo $p$, then it must have degree $Omega(n^{1-o(1)})$ modulo $q$. Thus a Boolean function has degree $o(log n)$ in only one characteristic. This result is essentially tight as there exist functions that have degree $log n$ in every characteristic. item If $f$ has degree $d = o(log n)$ modulo $p$, it cannot be computed correctly on more than $ 1- p^{-O(d)}$ fraction of the hypercube by polynomials of degree $n^{frac{1}{2} - eps}$ modulo $q$. end{itemize} As a corollary of the above results it follows that if $f$ has degree $o(log n)$ modulo $p$, then it requires super-polynomial size $AC_0[q]$ circuits. This gives a lower bound for a broad and natural class of functions.

A Space Hierarchy for k-DNF Resolution, by Eli Ben-Sasson, Jakob Nordström

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 well-studied resolution proof system. A number of lower bounds have been proven on k-DNF resolution proof length and space, and it has also been shown that (k+1)-DNF resolution is exponentially more powerful than k-DNF resolution for all k with respect to length. For proof space, however, no corresponding hierarchy has been known except for the (very weak) subsystems restricted to tree-like proofs. In this work, we establish a strict space hierarchy for the general, unrestricted k-DNF resolution proof systems.

Transitive-Closure Spanners of the Hypercube and the Hypergrid, by Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff

Given a directed graph $G = (V,E)$ and an integer $k geq 1$, a $k$-transitive-closure-spanner ($k$-TC-spanner) of $G$ is a directed graph $H = (V, E_H)$ that has (1) the same transitive-closure as $G$ and (2) diameter at most $k$. Transitive-closure spanners were introduced in cite{tc-spanners-soda} as a common abstraction for applications in access control, property testing and data structures. In this work we study the number of edges in the sparsest 2-TC-spanners for the directed hypercube ${0,1}^d$ and hypergrid ${1,2,dots,m}^d$ with the usual partial order, $preceq$, defined by: $x_1dots x_d preceq y_1dots y_d$ if and only if $x_i leq y_i$ for all $iin{1,...,d}$. We show that the number of edges in the sparsest 2-TC-spanner of the hypercube is $2^{cd+Theta(log d)}$, where $capprox 1.1620.$ We also present upper and lower bounds on the size of the sparsest 2-TC-spanner of the directed hypergrid. Our first pair of upper and lower bounds for the hypergrid is in terms of an expression with binomial coefficients. The bounds differ by a factor of $O(d^{2m})$ and, in particular, give tight (up to a $poly(d)$ factor) bounds for constant $m$. We also give a second set of bounds, which show that the number of edges in the sparsest 2-TC-spanner of the hypergrid is at most $m^d log^d m$ and at least $Omega (max {m^d (log^d m)/((2dloglog m)^{d-1}) , (m-1)^d 2^{(c+alpha-1)d} } )$, where $c approx 1.1620$, as above, and $alpha > 0$ satisfies $1+H_b(alpha) < c$. The two sets of bounds are, in general, incomparable. Our results rule out a class of approaches to monotonicity testing of functions of the form $f:{0,1}^dto R$ and, more generally, $f:{1,2,dots, m}^dto R$, where $R$ is an arbitrary range. cite{tc-spanners-soda} showed that sparse 2-TC-spanners imply fast monotonicity testers, and used this connection to improve existing monotonicity testers for planar and other $H$-minor-free graphs. It left open the question, which was again raised at the 2008 Dagstuhl seminar on Sublinear Algorithms, of whether the 2-TC-spanner approach can improve monotonicity testers on the hypercube and hypergrid. We show that a fundamentally new approach is required.

Inaccessible Entropy, by Iftach Haitner, Omer Reingold, Salil Vadhan, Hoeteck Wee

We put forth a new computational notion of entropy, which measures the (in)feasibility of sampling high entropy strings that are consistent with a given protocol. Specifically, we say that the i'th round of a protocol (A, B) has _accessible entropy_ at most k, if no polynomial-time strategy A^* can generate messages for A such that the entropy of its message in the i'th round has entropy greater than k when conditioned both on prior messages of the protocol and on prior coin tosses of A^*. We say that the protocol has _inaccessible entropy_ if the total accessible entropy (summed over the rounds) is noticeably smaller than the real entropy of A's messages, conditioned only on prior messages (but not the coin tosses of A). As applications of this notion, we * Give a much simpler and more efficient construction of statistically hiding commitment schemes from arbitrary one-way functions. * Prove that constant-round statistically hiding commitments are necessary for constructing constant-round zero-knowledge proof systems for NP that remain secure under parallel composition (assuming the existence of one-way functions).

Direct Sums in Randomized Communication Complexity, by Boaz Barak, Mark Braverman, Xi Chen, Anup Rao

Does computing n copies of a function require n times the computational effort? In this work, we give the first non-trivial answer to this question for the model of randomized communication complexity. We show that: 1. Computing n copies of a function requires sqrt{n} times the communication. 2. For average case complexity, given any distribution mu on inputs, computing n copies of the function on n independent inputs sampled according to mu requires at least sqrt{n} times the communication for computing one copy. 3. If mu is a product distribution, computing n copies on n independent inputs sampled according to mu requires n times the communication. We also study the complexity of computing the parity of n evaluations of f, and obtain results analogous to those above. Our results are obtained by designing compression schemes for communication protocols that can be used to compress the communication in a protocol that does not transmit a lot of information about its inputs.

Succinct Representation of Codes with Applications to Testing, by Elena Grigorescu, Tali Kaufman, Madhu Sudan

Motivated by questions in property testing, we search for linear error-correcting codes that have the ``single local orbit'' property: i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every ``sparse'' binary code whose coordinates are indexed by elements of F_{2^n} for prime n, and whose symmetry group includes the group of non-singular affine transformations of F_{2^n}, has the single local orbit property. (A code is said to be {em sparse} if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known. Our result gives the first short (O(n)-bit, as opposed to the natural exp(n)-bit) description of a low-weight basis for BCH codes. The interest in the ``single local orbit'' property comes from the recent result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that have the single local orbit property under the affine symmetry group are locally testable. When combined with our main result, this shows that all sparse affine-invariant codes over the coordinates F_{2^n} for prime n are locally testable. If, in addition to n being prime, if 2^n-1 is also prime (i.e., 2^n-1 is a Mersenne prime), then we get that every sparse {em cyclic} code also has the single local orbit. In particular this implies that BCH codes of Mersenne prime length are generated by a single low-weight codeword and its cyclic shifts. In retrospect, the single local orbit property has been central to most previous results in algebraic property testing. However, in the previous cases, the single local property was almost ``evident'' for the code in question (the single local constraint was explicitly known, and it is a simple algebraic exercise to show that its translations under the symmetry group completely characterize the code). Our work gives an alternate proof of the single local orbit property, effectively by counting, and its effectiveness is demonstrated by the fact that we are able to analyze it in cases where even the local constraint is not ``explicitly'' known. Our techniques involve the use of recent results from additive number theory to prove that the codes we consider, and related codes emerging from our proofs, have high distance. We then combine these with the MacWilliams identities and some careful analysis of the invariance properties to derive our results.

Composition of low-error 2-query PCPs using decodable PCPs, by Irit Dinur, Prahladh Harsha

The main result of this paper is a simple, yet generic, composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored to the specific PCPs that were being composed), resulting in complicated constructions of PCPs. Furthermore, until recently, composition in the low error regime suffered from incurring an extra `consistency' query, resulting in PCPs that are not `two-query' and hence, much less useful for hardness-of-approximation reductions. In a recent breakthrough, Moshkovitz and Raz [In {em Proc. 49th IEEE Symp. on Foundations of Comp. Science (FOCS)}, 2008] constructed almost linear-sized low-error 2-query PCPs for every language in NP. Indeed, the main technical component of their construction is a novel composition of certain specific PCPs. We give an alternate, modular and, considerably simpler proof of their result by repeatedly applying the new composition theorem to known PCP components. To facilitate the new modular composition, we introduce a new variant of PCP, which we call a {em decodable PCP (dPCP)}. A dPCP is an {em encoding} of an NP witness that is both locally checkable and locally decodable. The dPCP verifier in addition to verifying the validity of the given proof like a standard PCP verifier, also locally decodes the original NP witness. Our composition is generic in the sense that it works regardless of the way the component PCPs are constructed.

Reducibility Among Fractional Stability Problems, by Shiva Kintali, Laura J Poplawski, Rajmohan Rajaraman, Ravi Sundaram, Shang-Hua Teng

"As has often been the case with NP-completeness proofs, PPAD-completeness proofs will be eventually refined to cover simpler and more realistic looking classes of games. And then researchers will strive to identify even simpler classes." --Papadimitriou (chapter 2 of Algorithmic Game Theory book) In a landmark paper, Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results [12, 19, 6, 7, 8, 9] has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A ma jor goal of this work is to expand the universe of known PPAD-complete problems. We resolve the computational complexity of a number of outstanding open problems with practical applications. Here is the list of problems we show to be PPAD-complete, along with the domains of practical significance: Fractional Stable Paths Problem (FSPP) [21] - Internet routing; Core of Balanced Games [41] - Economics and Game theory; Scarf's Lemma [41] - Combinatorics; Hypergraph Matching [1]- Social Choice and Preference Systems; Fractional Bounded Budget Connection Games (FBBC) [30] - Social networks; and Strong Fractional Kernel [2]- Graph Theory. In fact, we show that no fully polynomial-time approximation schemes exist (unless PPAD is in FP). This paper is entirely a series of reductions that build in nontrivial ways on the framework established in previous work. In the course of deriving these reductions, we created two new concepts - preference games and personalized equilibria. The entire set of new reductions can be presented as a lattice with the above problems sandwiched between preference games (at the "easy" end) and personalized equilibria (at the "hard" end). Our completeness results extend to natural approximate versions of most of these problems. On a technical note, we wish to highlight our novel "continuous-to-discrete" reduction from exact personalized equilibria to approximate personalized equilibria using a linear program augmented with an exponential number of "min" constraints of a specific form. In addition to enhancing our repertoire of PPAD-complete problems, we expect the concepts and techniques in this paper to find future use in algorithmic game theory.

On convex complexity measures, by Pavel Hrubes, Stasys Jukna, Alexander Kulikov, Pavel Pudlak

Khrapchenko's classical lower bound $n^2$ on the formula size of the parity function~$f$ can be interpreted as designing a suitable measure of subrectangles of the combinatorial rectangle $f^{-1}(0)times f^{-1}(1)$. Trying to generalize this approach we arrived at the concept of emph{convex measures}. We prove the negative result that convex measures are bounded by $O(n^2)$ and show that several measures considered for proving lower bounds on the formula size are convex. We also prove quadratic upper bounds on a class of measures that are not necessarily convex.

Polynomial Time with Restricted Use of Randomness, by Matei David, Periklis Papakonstantinou, Anastasios Sidiropoulos

We define a hierarchy of complexity classes that lie between P and RP, yielding a new way of quantifying partial progress towards the derandomization of RP. A standard approach in derandomization is to reduce the number of random bits an algorithm uses. We instead focus on a model of computation that allows us to quantify the extent to which random bits are being used. More specifically, we consider Stack Machines (SMs), which are log-space Turing Machines that have access to an unbounded stack, an input tape of length N, and a random tape of length N^O(1). We parametrize these machines by allowing at most r(N)-1 reversals on the random tape, thus obtaining the r(N)-th level of our hierarchy, denoted by RPdL[r]. It follows by a result of Cook [Coo71] that RPdL[1]=P, and of Ruzzo [Ruz81] that RPdL[exp(N)]=RP. Our main results are the following. - For every i>=1, derandomizing RPdL[2^{O(log^i N)}] implies the derandomization of RNC^i. Thus, progress towards the P vs RP question along our hierarchy implies also progress towards derandomizing RNC. Perhaps more surprisingly, we also prove a partial converse: Pseurorandom generators (PRGs) for RNC^(i+1) are sufficient to derandomize RPdL[2^{O(log^i N)}]; i.e. derandomizing using PRGs a class believed to be strictly inside P, we derandomize a class containing P. More generally, we introduce Randomness Compilers, a model equivalent to Stack Machines. In this model a polynomial time algorithm gets an input x and it outputs a circuit C_x, which takes random inputs. Acceptance of x is determined by the acceptance probability of C_x. When C_x is of polynomial size and depth O(log^i N) the corresponding class is denoted by P+RNC^i, and we show that RPdL[2^{O(log^i N)}]subseteq P+RNC^i subseteq RPdL[2^{O(log^{i+1} N)}]. - We show an unconditional N^Omega(1) lower bound on the number of reversals required by a SM for Polynomial Evaluation. This in particular implies that known Schwartz-Zippel-like algorithms for Polynomial Identity Testing cannot be implemented in the lowest levels of our hierarchy. - We show that in the 1-st level of our hierarchy, machines with one-sided error are as powerful as machines with two-sided and unbounded error.

Toward a Model for Backtracking and Dynamic Programming, by Michael Alekhnovich, Allan Borodin, Joshua Buresh-Oppenheim, Russell Impagliazzo, Avner Magen, Toniann Pitassi

We propose a model called priority branching trees (pBT ) for backtracking and dynamic programming algorithms. Our model generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model due to Woeginger, and hence spans a wide spectrum of algorithms. After witnessing the strength of the model, we then show its limitations by providing lower bounds for algorithms in this model for several classical problems such as Interval Scheduling, Knapsack and Satisfiability.

A Fourier-analytic approach to Reed-Muller decoding, by Parikshit Gopalan

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 for quadratic polynomials were known only for $q =2,3$; the best bound known for other fields was the Johnson radius which is roughly $1 - 1/sqrt{q}$. We say that a polynomial over $F_q$ is $k$-dimensional if it can be expressed as a function of $k$ linear functions. We reduce the Reed-Muller list-decoding problem to list-decoding low-dimensional polynomials and present a new Fourier-based algorithm for the low-dimensional case. The list-decoding radius achieved by this approach for degree $3$ and higher depends on questions regarding the weight-distribution of the Reed-Muller code. We propose a conjecture in this regard, which if true, improves on the best known bounds for the list-decoding radius for all $d$ and $q$. The conjecture holds true for $F_2$, giving an alternate proof of the main result of GKZ. Departing from previous work on Reed-Muller decoding which relies on some form of self-corrector, our work applies ideas from Fourier analysis of Boolean functions to low-degree polynomials over finite fields. We believe that the techniques used here could find other applications, we present applications to testing and learning.

The Power of Depth 2 Circuits over Algebras, by Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

We study the problem of polynomial identity testing (PIT) for depth 2 arithmetic circuits over matrix algebra. We show that identity testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (Pi-Sigma) arithmetic circuits over U_2(F), the algebra of upper- triangular 2 x 2 matrices with entries from F. Such a connection is a bit surprising since we also show that, as computational models, Pi-Sigma circuits over U_2(F) are strictly `weaker' than Sigma-Pi-Sigma circuits over F. The equivalence further shows that PIT of depth 3 arithmetic circuits reduces to PIT of width-2 planar commutative Algebraic Branching Programs (ABP). Thus, identity testing for commutative ABPs is interesting even in the case of width-2. Further, we give a deterministic polynomial time identity testing algorithm for a Pi-Sigma circuit over any constant dimensional commutative algebra over F. While over commutative algebras of polynomial dimension, identity testing is at least as hard as that of Sigma-Pi-Sigma circuits over F.

On the Automatizability of Polynomial Calculus, by Nicola Galesi, Massimo Lauria

We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W[P]-hard problems are fixed parameter tractable by one-side error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Computing, 38(4), 2008).

Understanding Space in Resolution: Optimal Lower Bounds and Exponential Trade-offs, by Eli Ben-Sasson, Jakob Nordström

For current state-of-the-art satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. Understanding time and memory consumption, and how they are related to one another, is therefore a question of considerable practical importance. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are trade-offs between these two measures, has remained essentially open apart from a few results in very limited settings suffering from various technical restrictions. In this paper, we remedy this situation by proving a host of length-space trade-off results for resolution in a completely general setting. Our collection of trade-offs cover space ranging over the whole interval from constant to O(n/log n), and most of them are superpolynomial or even exponential. Our key technical contribution is the following, somewhat surprising, theorem: Any CNF formula F can be transformed by simple substitution into a new formula F' such that if F has the right properties, F' can be proven in essentially the same length as F while the minimal space needed for F' is lower-bounded by the number of variables mentioned simultaneously in any proof for F. Applying this theorem to so-called pebbling formulas defined in terms of pebble games on directed acyclic graphs, and then using known results from the pebbling literature as well as a proving a couple of new ones, we obtain our resolution trade-off theorems.

Random Graphs and the Parity Quantifier, by Phokion G. Kolaitis, Swastik Kopparty

The classical zero-one law for first-order logic on random graphs says that for every first-order property $varphi$ in the theory of graphs and every $p in (0,1)$, the probability that the random graph $G(n, p)$ satisfies $varphi$ approaches either $0$ or $1$ as $n$ approaches infinity. It is well known that this law fails to hold for any formalism that can express the parity quantifier: for certain properties, the probability that $G(n,p)$ satisfies the property need not converge, and for others the limit may be strictly between $0$ and $1$. In this work, we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier, FO[parity], over $G(n,p)$, thus eluding the above hurdles. Specifically, we establish the following ``modular convergence law": "For every FO[parity] sentence $varphi$, there are two explicitly computable rational numbers $a_0$, $a_1$, such that for $i in {0,1}$, as $n$ approaches infinity, the probability that the random graph $G(2n+i, p)$ satisfies $varphi$ approaches $a_i$." Our results also extend appropriately to $FO$ equipped with $Mod_q$ quantifiers for prime $q$. In the process of deriving the above theorem, we explore a new question that may be of interest in its own right. Specifically, we study the joint distribution of the subgraph statistics modulo $2$ of $G(n,p)$: namely, the number of copies, mod $2$, of a fixed number of graphs $F_1, ldots, F_ell$ of bounded size in $G(n,p)$. We first show that every $FOP$ property $varphi$ is almost surely determined by subgraph statistics modulo $2$ of the above type. Next, we show that the limiting joint distribution of the subgraph statistics modulo $2$ depends only on $n mod 2$, and we determine this limiting distribution completely. Interestingly, both these steps are based on a common technique using multivariate polynomials over finite fields and, in particular, on a new generalization of the Gowers norm. The first step above is analogous to the Razborov-Smolensky method for lower bounds for AC0 with parity gates, yet stronger in certain ways. For instance, it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every FO[parity] sentence, which is something that is not known for AC0 with parity gates.

Blackbox Polynomial Identity Testing for Depth 3 Circuits, by Neeraj Kayal, Shubhangi Saraf

We study depth three arithmetic circuits with bounded top fanin. We give the first deterministic polynomial time blackbox identity test for depth three circuits with bounded top fanin over the field of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for depth three circuits with bounded top fanin that compute the zero polynomial. In particular we show that if a circuit C with real coefficients is simple, minimal and computes the zero polynomial, then the rank of C can be upper bounded by a function only of the top fanin. This proves a weak form of a conjecture of Dvir and Shpilka (STOC 2005) on the structure of identically zero depth three arithmetic circuits. Our blackbox identity test follows from this structural theorem by combining it with a construction of Karnin and Shpilka (CCC 2008). Our proof of the structure theorem exploits the geometry of finite point sets in R^n. We identify the linear forms appearing in the circuit C with points in R^n. We then show how to apply high dimensional versions of the Sylvester--Gallai Theorem, a theorem from incidence-geometry, to identify a special linear form appearing in C, such that on the subspace where the linear form vanishes, C restricts to a simpler circuit computing the zero polynomial. This allows us to build an inductive argument bounding the rank of our circuit. While the utility of such theorems from incidence geometry for identity testing has been hinted at before, our proof is the first to develop the connection fully and utilize it effectively.

From absolute distinguishability to positive distinguishability, by Zvika Brakerski and Oded Goldreich

We study methods of converting algorithms that distinguish pairs of distributions with a gap that has an absolute value that is noticeable into corresponding algorithms in which the gap is always positive. Our focus is on designing algorithms that, in addition to the tested string, obtain a fixed number of samples from each distribution. Needless to say, such algorithms can not provide a very reliable guess for the sign of the original distinguishability gap, still we show that even guesses that are noticeably better than random are useful in this setting.

The density of weights of Generalized Reed-Muller codes, by Shachar Lovett

We study the density of the weights of Generalized Reed--Muller codes. Let $RM_p(r,m)$ denote the code of multivariate polynomials over $F_p$ in $m$ variables of total degree at most $r$. We consider the case of fixed degree $r$, when we let the number of variables $m$ tend to infinity. We prove that the set of relative weights of codewords is quite sparse: for every $alpha in [0,1]$ which is not rational of the form $frac{ell}{p^k}$, there exists an interval around $alpha$ in which no relative weight exists, for any value of $m$. This line of research is to the best of our knowledge new, and complements the traditional lines of research, which focus on the weight distribution and the divisibility properties of the weights. Equivalently, we study distributions taking values in a finite field, which can be approximated by distributions coming from constant degree polynomials, where we do not bound the number of variables. We give a complete characterization of all such distributions.

Reachability in K_{3,3}-free Graphs and K_5-free Graphs is in Unambiguous Log-Space, by Fabian Wagner, Thomas Thierauf

We show that the reachability problem for directed graphs that are either K_{3,3}-free or K_5-free is in unambiguous log-space, UL cap coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL cap coUL. Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The non-planar components are replaced by planar components that maintain the reachabilty properties. For K_5-free graphs we also need a decomposition into fourconnected components. A careful analysis finally gives a polynomial size planar graph which can be computed in log-space. We show the same upper bound for computing distances in K_{3,3}-free and K_5-free directed graphs and for computing longest paths in K_{3,3}-free and K_5-free directed acyclic graphs.

A Candidate Counterexample to the Easy Cylinders Conjecture, by Oded Goldreich,

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. Although all popular one-way functions have such easy (to invert) cylinders, we suggest a possible counterexample. Our suggestion builds on the candidate one-way function based on expander graphs (see ECCC, TR00-090), and essentially consists of iterating this function polynomially many times.

A Parallel Repetition Theorem for Any Interactive Argument, by Iftach Haitner

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, Impagliazzo and Naor [FOCS '97], and constant-round public-coin protocols - Pass and Venkitasubramaniam [STOC '07]), Bellare et. al gave example of interactive arguments for which parallel repetition does not reduce the soundness error at all. We show that by slightly modifying any interactive argument, in a way that preserves its completeness and only slightly deteriorates its soundness, we get a protocol for which parallel repetition does reduce the error at a weak exponential rate. In this modified version, the verifier flips at the beginning of each round an (1 - frac1{4m}, frac1{4m}) biased coin (i.e., 1 is tossed with probability 1/4m), where m is the round complexity of the (original) protocol. If the coin is one, the verifier halts the interaction and accepts, otherwise it sends the same message that the original verifier would. At the end of the protocol (if reached), the verifier accepts if and only if the original verifier would.

Arithmetic Circuit Size, Identity Testing, and Finite Automata, by V. Arvind, Pushkar Joglekar

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 emph{intersecting} and emph{quotienting} the polynomial $f$ by $A$. Related to intersection, we also define the emph{Hadamard product} $fcirc g$ of two polynomials $f$ and $g$. In this paper we study the circuit and algebraic branching program (ABP) complexities of the polynomials $f( mod A)$, $f( div A)$, and $fcirc g$ in terms of the corresponding complexities of $f$ and $g$ and size of the automaton $A$. We show upper and lower bound results. Our results have consequences in new polynomial identity testing algorithms (and algorithms for its corresponding search version of finding a nonzero monomial). E.g. we show the following: begin{itemize} item[(a)] A deterministic $NC^2$ identity test for noncommutative ABPs over rationals. In fact, we tightly classify the problem as complete for the logspace counting class $C_=L$. item[(b)] Randomized $NC^2$ algorithms for finding a nonzero monomial in both noncommutative and commutative ABPs. item[(c)] Over monomial algebras $F{x_1,cdots,x_n}/I$ we derive an exponential size lower bound for ABPs computing the Permanent. We also obtain deterministic polynomial identity testing for ABPs over such algebras. end{itemize} We also study analogous questions in the emph{commutative} case and obtain some results. E.g. we show over any commutative monomial algebra $Q[x_1,cdots,x_n]/I$ such that the ideal $I$ is generated by $o(n/lg n)$ monomials, the Permanent requires exponential size monotone circuits.

A New Look at Some Classical Results in Computational Complexity, by Arnaldo Moura, Igor Carboni Oliveira

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 apply to both the time and space complexities. As an example, we present a similar approach for the gap theorems.

On the Power of Isolation in Planar Structures, by Raghav Kulkarni

The purpose of this paper is to study the deterministic {em isolation} for certain structures in directed and undirected planar graphs. The motivation behind this work is a recent development on this topic. For example, cite{btv07} isolate a directed path in planar graphs and cite{dkr08} isolate a perfect matching in bipartite planar graphs. One natural question raised by their work is: ``How far is the reach of the deterministic isolation in planar structures ?'' Our first observation is that the restriction of planarity is in fact, fairly general, in the sense that efficiently isolating certain planar structures would significantly bring down the complexities of some fundamental problems in general graphs. For example, we show that efficiently isolating a cycle cover in directed planar graphs would imply {em {sf Bipartite-Matching} $in$ NC}, efficiently isolating a minimum weight perfect matching in undirected planar graphs would imply that non-deterministic log-space computations can be made unambiguous, i.e., {em NL = UL} and efficiently isolating a {em Red-Blue path} in directed planar graphs would imply $NP subseteq oplus P.$ Further, we show that such efficient isolations are indeed possible in {em bipartite} planar graphs, thus leaving non-bipartiteness as the only bottleneck to break. A deceptively simple combinatorial puzzle comes out of our investigations where a positive solution to the puzzle would have strong consequences like $NL = UL.$ Our main tools are some new simple bijections, which might be of an independent interest combinatorially.

Strong Hardness Preserving Reduction from a P-Samplable Distribution to the Uniform Distribution for NP-Search Problems, by Akinori Kawachi, Osamu Watanabe

Impagliazzo and Levin demonstrated [IL90] that the average-case hardness of any NP-search problem under any P-samplable distribution implies that of another NP-search problem under the uniform distribution. For this they developed a way to define a reduction from an NP-search problem F with ``mild hardness'' under any P-samplable distribution H; more specifically, F is a problem with positive hard instances with probability 1/poly(n) under H. In this paper we show a similar reduction for an NP-search problem $F$ with ``strong hardness'', that is, F with positive hard instances with probability 1-1/poly(n) under H in its positive domain (i.e., the set of positive instances). Our reduction defines from this pair of F and H, some NP-search problem $G$ with a similar hardness under the uniform distribution U; more precisely, (i) G has positive hard instances with probability 1-1/poly(n) under U in its positive domain, and (ii) the positive domain itself occupies 1/poly(n) of {0,1}^n.

Inseparability and Strong Hypotheses for Disjoint NP Pairs, by Jack H. Lutz, Elvira Mayordomo

This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2^(n^k)-inseparable. We also relate these conditions to strong hypotheses concerning randomness and genericity of disjoint pairs.

How to Play Unique Games on Expanders, by Konstantin Makarychev, Yury Makarychev

In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders. Given a (1 - epsilon)-satisfiable instance of Unique Games with the constraint graph G, our algorithm finds an assignments satisfying at least a (1 - C epsilon/h) fraction of all constraints if epsilon < c lambda where h is the edge expansion of G, lambda is the second smallest eigenvalue of the Laplacian of G, and C and c are some absolute constants.

Hardness of Solving Sparse Overdetermined Linear Systems: A 3-Query PCP over Integers., by Venkatesan Guruswami, Prasad Raghavendra

A classic result due to Hastad established that for every constant eps > 0, given an overdetermined system of linear equations over a finite field F_q where each equation depends on exactly 3 variables and at least a fraction (1-eps) of the equations can be satisfied, it is NP-hard to satisfy even a fraction 1/q+eps of the equations. In this work, we prove the analog of Hastad's result for equations over the integers (as well as the reals). Formally, we prove that for every eps,delta > 0, given a system of linear equations with integer coefficients where each equation is on 3 variables, it is NP-hard to distinguish between the following two cases: (i) There is an assignment of integer values to the variables that satisfies at least a fraction (1-eps) of the equations, and (ii) No assignment even of real values to the variables satisfies more than a fraction delta of the equations.

One-Way Functions and the Isomorphism Conjecture, by Agrawal Manindra, Osamu Watanabe

We study the Isomorphism Conjecture proposed by Berman and Hartmanis. It states that all sets complete for NP under polynomial-time many-one reductions are P-isomorphic to each other. From previous research it has been widely believed that all NP-complete sets are reducible each other by one-to-one and length-increasing polynomial-time reductions, but we may not hope for the full p-isomorphism due to the existence of one-way functions. Here we showed two results on the relation between one-way functions and the Isomorphism Conjecture. Firstly, we imporve the result of Agrawal [Agrawal, CCC'02] to show that if regular one-way functions exist, then all NP-complete sets are indeed reducible each other by one-to-one, length-increasing and P/poly-reductions. A consequence of this result is the complete description of the structure of many-one complete sets of NP relative to a random oracle: all NP-complete sets are reducible each other by one-one and length-increasing polynomial-time reductions but (as already shown by [Kurtz etal, JACM 95]) they are not P-isomorphic. Neverthless, we also conjecture that (different from the random oracle world) all one-way functions should have some dense easy parts, which we call P/poly-easy cylinders, where they are P/poly-invertible. Then as our second result we show that if regular one-way functions exist and furthermore all one-one, length-increasing and P/poly-computable functions have P/poly-easy cylinders, then all many-one complete sets for NP are P/poly-isomorphic.

$GF(2^n)$-Linear Tests versus $GF(2)$-Linear Tests, by Yoav Tzur

A small-biased distribution of bit sequences is defined as one withstanding $GF(2)$-linear tests for randomness, which are linear combinations of the bits themselves. We consider linear combinations over larger fields, specifically, $GF(2^n)$ for $n$ that divides the length of the bit sequence. Indeed, this means that we partition the bits to blocks of length $n$ and treat each block as the representation of a field element. Various properties of the resulting field element can then be tested. We show that the latter $GF(2^n)$-linear tests are at least as powerful as the $GF(2)$-linear tests. This holds even for a very limited final test of the resulting field element (e.g., checking only the first bit). This is shown constructively in the sense that we show for each linear combination over $GF(2)$, an explicit linear combination over $GF(2^n)$ whose first bit (for instance) has the same bias. One corollary of the above is that the generator producing a random geometric series over $GF(2^n)$, namely $(a,b) mapsto (a^i cdot b)_{i=0}^ell$, is $frac{ell}{2^n}$-biased.

The Maximum Communication Complexity of Multi-party Pointer Jumping., by Joshua Brody

We study the one-way number-on-the-forhead (NOF) communication complexity of the k-layer pointer jumping problem. Strong lower bounds for this problem would have important implications in circuit complexity. All of our results apply to myopic protocols (where players see only one layer ahead, but can still see arbitrarily far behind them.) Furthermore, our results apply to the maximum communication complexity, where a protocol is charged for the maximum communication sent by a single player rather than the total communication sent by all players. Our main result is a lower bound of n/2 bits for deterministic protocols, independent of the number of players. We also provide a matching upper bound, as well as an Omega(n/(k log n)) lower bound for randomised protocols, improving on the bounds of Chakrabarti. In the non-Boolean version of the problem, we give a lower bound of n(log^{(k-1)} n)(1-o(1)) bits, essentially matching the upper bound from Damm, Junka, and Sgall.

Bounded Independence Fools Halfspaces, by Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco Servedio, Emanuele Viola

We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error eps for k = O(log^2(1/eps)/eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing that k = Omega(1/(eps^2 cdot log(1/eps))). Using standard constructions of k-wise independent distributions, we obtain the first explicit pseudorandom generators G: {-1,1}^s --> {-1,1}^n that fool halfspaces. Specifically, we fool halfspaces with error eps and seed length s = k log n = O(log n cdot log^2(1/eps) /eps^2). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Computational Complexity 2007)

A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences, by Joshua Brody, Amit Chakrabarti

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 $n/2 + sqrt n$) or small (i.e., at most $n/2 - sqrt n$); they do not care if it is neither large nor small. This $Theta(sqrt n)$ gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an $Omega(n)$ lower bound on this problem was known only in the one-way setting. We prove an $Omega(n)$ lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that $epsilon$-approximately counting the number of distinct elements in a data stream requires $Omega(1/epsilon^2)$ space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight $n - Theta(sqrt{n}log n)$ lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an $Omega(n)$ lower bound on the one-way randomized communication complexity.

On the asymptotic Nullstellensatz and Polynomial Calculus proof complexity, by Soren Riis

We show that the asymptotic complexity of uniformly generated (expressible in First-Order (FO) logic) propositional tautologies for the Nullstellensatz proof system (NS) as well as for Polynomial Calculus, (PC) has four distinct types of asymptotic behavior over fields of finite characteristic. More precisely, based on some highly non-trivial work by Krajicek, we show that for each prime $p$ there exists a function $l(n) in Omega(log(n))$ for NS and $l(n) in Omega(log(log(n))$ for PC, such that the propositional translation of any FO formula (that fails in all finite models), has degree proof complexity over fields of characteristic $p$, that behave in $4$ mutually distinct ways: (i) The degree complexity is bound by a constant. (ii) The degree complexity is at least $l(n)$ for all values of $n$. (iii) The degree complexity is at least $l(n)$ except in a finite number of regular subsequences of inifinite size, where the degree is constant. (iv) The degree complexity fluctuates between constant and $l(n)$ (and possibly beyond) in a very particular way. We leave it as an open question whether the classification remains valid for $l(n) in n^{Omega(1)}$ or even for $l(n) in Omega(n)$. Finally, we show that for any non-empty proper subset $A subseteq { (i), (ii), (iii), (iv)}$ the decision problem of whether a given input FO formula $psi$ has type belonging to $A$ - is undecidable.

Limits to List Decoding Random Codes, by Atri Rudra

It has been known since [Zyablov and Pinsker 1982] that a random $q$-ary code of rate $1-H_q(rho)-eps$ (where $00$ and $H_q(cdot)$ is the $q$-ary entropy function) with high probability is a $(rho,1/eps)$-list decodable code. (That is, every Hamming ball of radius at most $rho n$ has at most $1/eps$ codewords in it.) In this paper we prove the ``converse" result. In particular, we prove that for emph{every} $0

Balanced Hashing, Color Coding and Approximate Counting, by Noga Alon, Shai Gutner

Color Coding is an algorithmic technique for deciding efficiently if a given input graph contains a path of a given length (or another small subgraph of constant tree-width). Applications of the method in computational biology motivate the study of similar algorithms for counting the number of copies of a given subgraph. While it is unlikely that exact counting of this type can be performed efficiently, as the problem is $#W[1]$-complete even for paths, approximate counting is possible, and leads to the investigation of an intriguing variant of families of perfect hash functions. A family of functions from $[n]$ to $[k]$ is an $(epsilon,k)$-balanced family of hash functions, if there exists a positive $T$ so that for every $K subset [n]$ of size $|K|=k$, the number of functions in the family that are one-to-one on $K$ is between $(1-epsilon)T$ and $(1+epsilon)T$. The family is perfectly $k$-balanced if it is $(0,k)$-balanced. We show that every such perfectly $k$-balanced family is of size at least $c(k) n^{lfloor k/2 rfloor}$, and that for every $epsilon>frac{1}{poly(k)}$ there are explicit constructions of $(epsilon,k)$-balanced families of hash functions from $[n]$ to $[k]$ of size $e^{(1+o(1))k} log n$. This is tight up to the $o(1)$-term in the exponent, and supplies deterministic polynomial time algorithms for approximately counting the number of paths or cycles of a specified length $k$ (or copies of any graph $H$ with $k$ vertices and bounded tree-width) in a given input graph of size $n$, up to relative error $epsilon$, for all $k leq O(log n)$.

Poly-logarithmic independence fools AC0 circuits, by Mark Braverman

We prove that poly-sized AC0 circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log^2 n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has later given a much simpler proof for Bazzi's theorem.

Checking Equality of Matroid Linear Representations and the Cycle Matching Problem, by T.C. Vijayaraghavan

Given linear representations M_1 and M_2 of matroids over a field F, we consider the problem (denoted by ECLR), of checking if M_1 and M_2 represent the same matroid. We show that when F=Z_2, ECLR{Z_2} is complete for $parityL$. Let M_1,M_2in Q ^{mtimes n} be two matroid linear representations given as input. Then any set of indexes, columns corresponding to which are linearly dependent in one representation but are linearly independent in another is a witness that M_1 and M_2 represent different matroids over Q. We show that the decision and the search version of this problem are polynomial time equivalent. We consider the CYCLEMATCHING problem of checking if for a pair of undirected graphs G_1=(V_1,E_1) and G_2=(V_2,E_2) given as input with |V_1|=|V_2|=n, whether any set of vertices having indexes in Xsubseteq {1,...,n} form a cycle in G_1 if and only if the corresponding set of vertices form a cycle in G_2. We show that CYCLEMATCHING is complete for L. Also the problem of counting the number of Xsubseteq {1,...,n}$ such that vertices with indexes in X form a cycle in one of the input graphs but not in the other is shown to be $sharpP$-complete.

Min-Rank Conjecture for Log-Depth Circuits, by Stasys Jukna, Georg Schnitger

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate of which can only depend on variables corresponding to *-entries in the i-th row of A. We conjecture that no such system can have more than 2^{n-ccdot mk(A)} solutions, where c>0 is an absolute constant and mr(A) is the smallest rank over GF(2) of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x --> Mx. The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.

Tensor Products of Weakly Smooth Codes are Robust, by Eli Ben-Sasson, Michael Viderman, Michael Viderman

We continue the study of {em robust} tensor codes and expand the class of base codes that can be used as a starting point for the construction of locally testable codes via robust two-wise tensor products. In particular, we show that all unique-neighbor expander codes and all locally correctable codes, when tensored with any other good-distance code, are robust and hence can be used to construct locally testable codes. Previous works by required stronger expansion properties to obtain locally testable codes. Our proofs follow by defining the notion of {em weakly smooth} codes that generalize the {em smooth} codes of I.Dinur et al. We show that weakly smooth codes are sufficient for constructing robust tensor codes. Using the weaker definition, we are able to expand the family of base codes to include the aforementioned ones.

On basing ZK != BPP on the hardness of PAC learning, by David Xiao

Learning is a central task in computer science, and there are various formalisms for capturing the notion. One important model studied in computational learning theory is the PAC model of Valiant (CACM 1984). On the other hand, in cryptography the notion of ``learning nothing'' is often modelled by the simulation paradigm: in an interactive protocol, a party learns nothing if it can produce a transcript of the protocol by itself that is indistinguishable from what it gets by interacting with other parties. The most famous example of this paradigm is zero knowledge proofs, introduced by Goldwasser, Micali, and Rackoff (SICOMP 1989). Applebaum, Barak, and Xiao (FOCS 2008) established a connection between these two different notions of learning by observing that if there exist non-trivial languages with zero-knowledge proofs (ie $ZK neq BPP$), then no polynomial-time algorithm can PAC learn polynomial-size circuits. In this paper, we consider the reverse implication: is it true that if learning is hard then zero-knowledge proofs exist for non-trivial languages? We rule out two classes of techniques for proving this statement: 1. Relativizing techniques: there exists an oracle $calO$ relative to which learning polynomial-size circuits is hard and yet $ZK^calO = BPP^calO$. 2. Black-box techniques: if there is a (semi-)black-box proof that uses the hardness of PAC learning polynomial-size circuits to construct a zero knowledge proof for some language $L$, then in fact $L in AM cap coAM$. Together, these results rule out all known techniques for proving that hardness of learning implies $ZK neq BPP$, including partially non-black-box techniques such as those of Barak (FOCS 2001). In addition, our technique relies on a new kind of separating oracle that may be of independent interest.

Bit-Probe Lower Bounds for Succinct Data Structures, by Emanuele Viola

We prove lower bounds on the redundancy necessary to represent a set $S$ of objects using a number of bits close to the information-theoretic minimum $log_2 |S|$, while answering various queries by probing few bits. Our main results are: begin{itemize} item To represent $n$ ternary values $t in zot^n$ in terms of $u$ bits $b in zo^u$ while accessing a single value $t_i in zot$ by probing $q$ bits of $b$, one needs $$u geq (log_2 3)n + n/2^{O(q)}.$$ This matches an exciting representation by P{v a}tra{c s}cu (FOCS 2008) where $u leq (log_2 3)n + n/2^{q^{Omega(1)}}$. We also note that results on logarithmic forms imply the lower bound $u geq (log_2 3)n + n/log^{O(1)} n$ if we access $t_i$ by probing one cell of $log n$ bits. item To represent sets of size $n/3$ from a universe of $n$ elements in terms of $u$ bits $b in zo^u$ while answering membership queries by probing $q$ bits of $b$, one needs $$u geq log_2 binom{n}{n/3} + n/2^{O(q)} - log n.$$ end{itemize} Both results above hold even if the probe locations are determined adaptively. Ours are the first lower bounds for these fundamental problems; we obtain them drawing on ideas used in lower bounds for locally decodable codes.

Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers, by Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan

We extend the ``method of multiplicities'' to get the following results, of interest in combinatorics and randomness extraction. begin{enumerate} item We show that every Kakeya set in $F_q^n$, the $n$-dimensional vector space over the finite field on $q$ elements, must be of size at least $q^n/2^n$. This bound is tight to within a $2 + o(1)$ factor for every $n$ as $q to infty$. item We give improved ``randomness mergers'', i.e., seeded functions that take as input $k$ (possibly correlated) random variables in ${0,1}^N$ and a short random seed and output a single random variable in ${0,1}^N$ that is statistically close to having entropy $(1-delta) cdot N$ when one of the $k$ input variables is distributed uniformly. The seed we require is only $(1/delta)cdot log k$-bits long, which significantly improves upon previous construction of mergers. end{enumerate} The ``method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset {em with high multiplicity}. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes {em with high multiplicity} outside the set. This novelty leads to significantly tighter analyses. To get the extended method of multiplicities we provide a number of basic technical results about multiplicity of zeroes of polynomials that may be of general use. For instance, we strengthen the Schwartz-Zippel lemma to show that the expected multiplicity of zeroes of a non-zero degree $d$ polynomial at a random point in $S^n$, for any finite subset $S$ of the underlying field, is at most $d/|S|$ (a fact that does not seem to have been noticed in the CS literature before).

Comments on ECCC Report TR06-133: The Resolution Width Problem is EXPTIME-Complete, by Alex Hertel, Alasdair Urquhart

We discovered a serious error in one of our previous submissions to ECCC and wish to make sure that this mistake is publicly known. The main argument of the report TR06-133 is in error. The paper claims to prove the result of the title by reduction from the (Exists,k)-pebble game, shown to be EXPTIME-complete by Kolaitis and Panttaja. This note shows that the principal lemma is incorrect by providing a simple counter-example.

Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution, by Eli Ben-Sasson, Jakob Nordström

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6-CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Omega(n/log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n/log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the black-white pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordstrom 2006, Nordstrom and Hastad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard black-white pebbling price.

Artin automorphisms, Cyclotomic function fields, and Folded list-decodable codes, by Venkatesan Guruswami

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 in Galois extensions. Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible $M in F_q[T]$. The Reed-Solomon case corresponds to the simplest such extension (corresponding to the case $M=T$). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree one places that can be used for encoding. Our methods shed new light on algebraic codes and their list decoding, and lead to new codes achieving list decoding capacity. Quantitatively, these codes provide list decoding (and list recovery/soft decoding) guarantees similar to folded Reed-Solomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no brute-force search) binary concatenated codes for list decoding up to the Zyablov radius.