Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width

A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$,$[n]=\{1,2,...,n\}$, and$\chi_I(x)=\prod_{i\in I}x_i$and$\hat{f}(I)$are non-zero reals. The degree of$f$is$\max \{|I|:\ I\in {\cal F}\}$and the width of$f$is the minimum integer$\rho$such that every$i\in [n]$appears in at most$\rho$sets in$\cal F$. For$i\in [n]$, let$\mathbf{x}_i$be a random variable taking values 1 or -1 uniformly and independently from all other variables$\mathbf{x}_j$,$j\neq i.$Let$\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The$p$-norm of$f$is$||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$for any$p\ge 1$. It is well-known that$||f||_q\ge ||f||_p$whenever$q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on$f$: if$f$is of degree$d$and$q> p>1$then$ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$This inequality is called the Hypercontractive Inequality. We show that one can replace$d$by$\rho$in the Hypercontractive Inequality for each$q> p\ge 2$as follows:$ ||f||_q\le ((2r)!\rho^{r-1})^{1/(2r)}||f||_p,$where$r=\lceil q/2\rceil$. For the case$q=4$and$p=2$, which is important in many applications, we prove a stronger inequality:$ ||f||_4\le (2\rho+1)^{1/4}||f||_2.

Approximability and proof complexity

This work is concerned with the proof-complexity of certifying that optimization problems do \emph{not} have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any $n$-variable degree-$d$ proof can be found in time $n^{O(d)}$. Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "$d/2$-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-$d$ SOS proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al.\ can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 ($> .878$) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?Comment: 34 page

Parameterized Constraint Satisfaction Problems: a Survey

We consider constraint satisfaction problems parameterized above or below guaranteed values. One example is MaxSat parameterized above m/2: given a CNF formula F with m clauses, decide whether there is a truth assignment that satisfies at least m/2 + k clauses, where k is the parameter. Among other problems we deal with are MaxLin2-AA (given a system of linear equations over F_2 in which each equation has a positive integral weight, decide whether there is an assignment to the variables that satisfies equations of total weight at least W/2+k, where W is the total weight of all equations), Max-r-Lin2-AA (the same as MaxLin2-AA, but each equation has at most r variables, where r is a constant) and Max-r-Sat-AA (given a CNF formula F with m clauses in which each clause has at most r literals, decide whether there is a truth assignment satisfying at least sum_{i=1}^m (1-2^{r_i})+k clauses, where k is the parameter, r_i is the number of literals in clause i, and r is a constant). We also consider Max-r-CSP-AA, a natural generalization of both Max-r-Lin2-AA and Max-r-Sat-AA, order (or, permutation) constraint satisfaction problems parameterized above the average value and some other problems related to MaxSat. We discuss results, both polynomial kernels and parameterized algorithms, obtained for the problems mainly in the last few years as well as some open questions

Using and saving randomness

Randomness is ubiquitous and exceedingly useful in computer science. For example, in sparse recovery, randomized algorithms are more efficient and robust than their deterministic counterparts. At the same time, because random sources from the real world are often biased and defective with limited entropy, high-quality randomness is a precious resource. This motivates the studies of pseudorandomness and randomness extraction. In this thesis, we explore the role of randomness in these areas. Our research contributions broadly fall into two categories: learning structured signals and constructing pseudorandom objects. Learning a structured signal. One common task in audio signal processing is to compress an interval of observation through finding the dominating k frequencies in its Fourier transform. We study the problem of learning a Fourier-sparse signal from noisy samples, where [0, T] is the observation interval and the frequencies can be “off-grid”. Previous methods for this problem required the gap between frequencies to be above 1/T, which is necessary to robustly identify individual frequencies. We show that this gap is not necessary to recover the signal as a whole: for arbitrary k-Fourier-sparse signals under ℓ₂ bounded noise, we provide a learning algorithm with a constant factor growth of the noise and sample complexity polynomial in k and logarithmic in the bandwidth and signal-to-noise ratio. In addition to this, we introduce a general method to avoid a condition number depending on the signal family F and the distribution D of measurement in the sample vi complexity. In particular, for any linear family F with dimension d and any distribution D over the domain of F, we show that this method provides a robust learning algorithm with O(d log d) samples. Furthermore, we improve the sample complexity to O(d) via spectral sparsification (optimal up to a constant factor), which provides the best known result for a range of linear families such as low degree multivariate polynomials. Next, we generalize this result to an active learning setting, where we get a large number of unlabeled points from an unknown distribution and choose a small subset to label. We design a learning algorithm optimizing both the number of unlabeled points and the number of labels. Pseudorandomness. Next, we study hash families, which have simple forms in theory and efficient implementations in practice. The size of a hash family is crucial for many applications such as derandomization. In this thesis, we study the upper bound on the size of hash families to fulfill their applications in various problems. We first investigate the number of hash functions to constitute a randomness extractor, which is equivalent to the degree of the extractor. We present a general probabilistic method that reduces the degree of any given strong extractor to almost optimal, at least when outputting few bits. For various almost universal hash families including Toeplitz matrices, Linear Congruential Hash, and Multiplicative Universal Hash, this approach significantly improves the upper bound on the degree of strong extractors in these hash families. Then we consider explicit hash families and multiple-choice schemes in the classical problems of placing balls into bins. We construct explicit hash families of almost-polynomial size that derandomizes two classical multiple-choice schemes, which match the maximum loads of a perfectly random hash function.Computer Science

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics

Efficient deterministic approximate counting for low-degree polynomial threshold functions

We give a deterministic algorithm for approximately counting satisfying assignments of a degree-$d$ polynomial threshold function (PTF). Given a degree-$d$ input polynomial $p(x_1,\dots,x_n)$ over $R^n$ and a parameter $\epsilon> 0$, our algorithm approximates $\Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0]$ to within an additive $\pm \epsilon$ in time $O_{d,\epsilon}(1)\cdot \mathop{poly}(n^d)$. (Any sort of efficient multiplicative approximation is impossible even for randomized algorithms assuming $NP\not=RP$.) Note that the running time of our algorithm (as a function of $n^d$, the number of coefficients of a degree-$d$ PTF) is a \emph{fixed} polynomial. The fastest previous algorithm for this problem (due to Kane), based on constructions of unconditional pseudorandom generators for degree-$d$ PTFs, runs in time $n^{O_{d,c}(1) \cdot \epsilon^{-c}}$ for all $c > 0$. The key novel contributions of this work are: A new multivariate central limit theorem, proved using tools from Malliavin calculus and Stein's Method. This new CLT shows that any collection of Gaussian polynomials with small eigenvalues must have a joint distribution which is very close to a multidimensional Gaussian distribution. A new decomposition of low-degree multilinear polynomials over Gaussian inputs. Roughly speaking we show that (up to some small error) any such polynomial can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues. We use these new ingredients to give a deterministic algorithm for a Gaussian-space version of the approximate counting problem, and then employ standard techniques for working with low-degree PTFs (invariance principles and regularity lemmas) to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian version

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