Pseudo-random graphs and bit probe schemes with one-sided error

We study probabilistic bit-probe schemes for the membership problem. Given a set A of at most n elements from the universe of size m we organize such a structure that queries of type "Is x in A?" can be answered very quickly. H.Buhrman, P.B.Miltersen, J.Radhakrishnan, and S.Venkatesh proposed a bit-probe scheme based on expanders. Their scheme needs space of $O(n\log m)$ bits, and requires to read only one randomly chosen bit from the memory to answer a query. The answer is correct with high probability with two-sided errors. In this paper we show that for the same problem there exists a bit-probe scheme with one-sided error that needs space of O(n\log^2 m+\poly(\log m)) bits. The difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh is that we consider a bit-probe scheme with an auxiliary word. This means that in our scheme the memory is split into two parts of different size: the main storage of $O(n\log^2 m)$ bits and a short word of $\log^{O(1)}m$ bits that is pre-computed once for the stored set A and `cached'. To answer a query "Is x in A?" we allow to read the whole cached word and only one bit from the main storage. For some reasonable values of parameters our space bound is better than what can be achieved by any scheme without cached data.Comment: 19 page

Bounded Independence Fools Degree-2 Threshold Functions

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem 8.1 to m^4, not m^

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Affine Extractors and AC0-Parity

We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem

Satisfiability and Derandomization for Small Polynomial Threshold Circuits

A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth. - Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only. - Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1/c^d} wires such that C has at most 2^{n^{1-1/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity. - Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1/epsilon)

Algorithms and Lower Bounds in Circuit Complexity

Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits

Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas

We give an explicit pseudorandom generator (PRG) for read-once AC^0, i.e., constant-depth read-once formulas over the basis {wedge, vee, neg} with unbounded fan-in. The seed length of our PRG is O~(log(n/epsilon)). Previously, PRGs with near-optimal seed length were known only for the depth-2 case [Gopalan et al., 2012]. For a constant depth d > 2, the best prior PRG is a recent construction by Forbes and Kelley with seed length O~(log^2 n + log n log(1/epsilon)) for the more general model of constant-width read-once branching programs with arbitrary variable order [Michael A. Forbes and Zander Kelley, 2018]. Looking beyond read-once AC^0, we also show that our PRG fools read-once AC^0[oplus] with seed length O~(t + log(n/epsilon)), where t is the number of parity gates in the formula. Our construction follows Ajtai and Wigderson\u27s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989]. We assume by recursion that we already have a PRG for depth-d AC^0 formulas. To fool depth-(d + 1) AC^0 formulas, we use the given PRG, combined with a small-bias distribution and almost k-wise independence, to sample a pseudorandom restriction. The analysis of Forbes and Kelley [Michael A. Forbes and Zander Kelley, 2018] shows that our restriction approximately preserves the expectation of the formula. The crux of our work is showing that after poly(log log n) independent applications of our pseudorandom restriction, the formula simplifies in the sense that every gate other than the output has only polylog n remaining children. Finally, as the last step, we use a recent PRG by Meka, Reingold, and Tal [Meka et al., 2019] to fool this simpler formula

Pseudorandom Generators for Low Sensitivity Functions

A Boolean function is said to have maximal sensitivity s if s is the largest number of Hamming neighbors of a point which differ from it in function value. We initiate the study of pseudorandom generators fooling low-sensitivity functions as an intermediate step towards settling the sensitivity conjecture. We construct a pseudorandom generator with seed-length 2^{O(s^{1/2})} log(n) that fools Boolean functions on n variables with maximal sensitivity at most s. Prior to our work, the (implicitly) best pseudorandom generators for this class of functions required seed-length 2^{O(s)} log(n)