arXiv subjects

Shachar Lovett

Publications and source records attributed to Shachar Lovett.

At least 73 records · Page 4Linked to original sources

arXiv2015

On the Beck-Fiala Conjecture for Random Set Systems

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,Σ)$, where each element $x \in X$ lies in $t$ randomly selected sets of $Σ$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We show that when $|Σ| \ge |X|$ the hereditary discrepancy of $(X,Σ)$ is with high probability $O(\sqrt{t \log t})$; and when $|X| \gg |Σ|^t$ the hereditary discrepancy of $(X,Σ)$ is with high probability $O(1)$. The first bound combines the Lov{á}sz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.

math.CO↗

arXiv2015

Large Supports are required for Well-Supported Nash Equilibria

We prove that for any constant $k$ and any $ε<1$, there exist bimatrix win-lose games for which every $ε$-WSNE requires supports of cardinality greater than $k$. To do this, we provide a graph-theoretic characterization of win-lose games that possess $ε$-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight to construct win-lose games that do not satisfy the requirements of the characterization. These constructions disprove graph theoretic conjectures of Daskalakis, Mehta and Papadimitriou, and Myers.

cs.GT↗

arXiv2015

On the structure of the spectrum of small sets

Let $G$ be a finite abelian group and $A$ a subset of $G$. The spectrum of $A$ is the set of its large Fourier coefficients. Known combinatorial results on the structure of spectrum, such as Chang's theorem, become trivial in the regime $|A| = |G|^α$ whenever $α\le c$, where $c \ge 1/2$ is some absolute constant. On the other hand, there are statistical results, which apply only to a noticeable fraction of the elements, which give nontrivial bounds even to much smaller sets. One such theorem (due to Bourgain) goes as follows. For a noticeable fraction of pairs $γ_1,γ_2 $ in the spectrum, $γ_1+γ_2$ belongs to the spectrum of the same set with a smaller threshold. Here we show that this result can be made combinatorial by restricting to a large subset. That is, we show that for any set $A$ there exists a large subset $A'$, such that the sumset of the spectrum of $A'$ has bounded size. Our results apply to sets of size $|A| = |G|^α$ for any constant $α>0$, and even in some sub-constant regime.

math.CO↗

arXiv2014

Nonclassical polynomials as a barrier to polynomial lower bounds

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.

cs.CC↗

arXiv2014

List decoding Reed-Muller codes over small fields

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like Reed-Solomon or Reed-Muller codes. Fix a finite field $\mathbb{F}$. The Reed-Muller code $\mathrm{RM}_{\mathbb{F}}(n,d)$ is defined by $n$-variate degree-$d$ polynomials over $\mathbb{F}$. In this work, we study the list decoding radius of Reed-Muller codes over a constant prime field $\mathbb{F}=\mathbb{F}_p$, constant degree $d$ and large $n$. We show that the list decoding radius is equal to the minimal distance of the code. That is, if we denote by $δ(d)$ the normalized minimal distance of $\mathrm{RM}_{\mathbb{F}}(n,d)$, then the number of codewords in any ball of radius $δ(d)-\varepsilon$ is bounded by $c=c(p,d,\varepsilon)$ independent of $n$. This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008], who among other results proved it in the special case of $\mathbb{F}=\mathbb{F}_2$; and extends the work of Gopalan [FOCS 2010] who proved the conjecture in the case of $d=2$. We also analyse the number of codewords in balls of radius exceeding the minimal distance of the code. For $e \leq d$, we show that the number of codewords of $\mathrm{RM}_{\mathbb{F}}(n,d)$ in a ball of radius $δ(e) - \varepsilon$ is bounded by $\exp(c \cdot n^{d-e})$, where $c=c(p,d,\varepsilon)$ is independent of $n$. The dependence on $n$ is tight. This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who proved similar bounds over $\mathbb{F}_2$. The proof relies on several new ingredients: an extension of the Frieze-Kannan weak regularity to general function spaces, higher-order Fourier analysis, and an extension of the Schwartz-Zippel lemma to compositions of polynomials.

cs.CC↗

arXiv2014

Group representations that resist random sampling

We show that there exists a family of groups $G_n$ and nontrivial irreducible representations $ρ_n$ such that, for any constant $t$, the average of $ρ_n$ over $t$ uniformly random elements $g_1, \ldots, g_t \in G_n$ has operator norm $1$ with probability approaching 1 as $n \rightarrow \infty$. More quantitatively, we show that there exist families of finite groups for which $Ω(\log \log |G|)$ random elements are required to bound the norm of a typical representation below $1$. This settles a conjecture of A. Wigderson.

math.CO↗

arXiv2014

An Improved Lower Bound for Arithmetic Regularity

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{é}di regularity lemma in graph theory. It shows that for any abelian group $G$ and any bounded function $f:G \to [0,1]$, there exists a subgroup $H \le G$ of bounded index such that, when restricted to most cosets of $H$, the function $f$ is pseudorandom in the sense that all its nontrivial Fourier coefficients are small. Quantitatively, if one wishes to obtain that for $1-ε$ fraction of the cosets, the nontrivial Fourier coefficients are bounded by $ε$, then Green shows that $|G/H|$ is bounded by a tower of twos of height $1/ε^3$. He also gives an example showing that a tower of height $Ω(\log 1/ε)$ is necessary. Here, we give an improved example, showing that a tower of height $Ω(1/ε)$ is necessary.

math.CO↗

arXiv2014

General systems of linear forms: equidistribution and true complexity

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate the average, it suffices to know the joint distribution of the polynomials applied to the linear forms. We prove a near-equidistribution theorem that describes these distributions for the group $\mathbb{F}_p^n$ when $p$ is a fixed prime. This fundamental fact is equivalent to a strong near-orthogonality statement regarding the higher-order characters, and was previously known only under various extra assumptions about the linear forms. As an application of our near-equidistribution theorem, we settle a conjecture of Gowers and Wolf on the true complexity of systems of linear forms for the group $\mathbb{F}_p^n$.

math.NT↗

arXiv2014

The Freiman--Ruzsa Theorem over Finite Fields

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion.

math.CO↗

arXiv2014

Recent advances on the log-rank conjecture in communication complexity

The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix, up to polynomial factors. Despite much research, we still know very little about this conjecture. Recently, there has been renewed interest in this conjecture and its relations to other fundamental problems in complexity theory. This survey describes some of the recent progress, and hints at potential directions for future research.

cs.CC↗

arXiv2014

0-1 Integer Linear Programming with a Linear Number of Constraints

We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time $2^{(1-\text{poly}(1/c))n}$ where n is the number of variables and cn is the number of constraints. The key idea for the algorithm is a reduction to the Vector Domination problem and a new algorithm for that subproblem.

cs.CC↗

arXiv2013

Communication is bounded by root of rank

We prove that any total boolean function of rank $r$ can be computed by a deterministic communication protocol of complexity $O(\sqrt{r} \cdot \log(r))$. Equivalently, any graph whose adjacency matrix has rank $r$ has chromatic number at most $2^{O(\sqrt{r} \cdot \log(r))}$. This gives a nearly quadratic improvement in the dependence on the rank over previous results.

cs.CC↗

arXiv2013

Correlation Testing for Affine Invariant Properties on $\mathbb{F}_p^n$ in the High Error Regime

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n \rightarrow \mathbb{F}_p$ with polynomials of degree at most $d \le p$ is non-negligible, while making only a constant number of queries to the function. This is an instance of {\em correlation testing}. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analog of {\em proximity oblivious testing}, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree $d$ polynomials for some fixed $d$. We stress that our result holds also for non-linear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdös, Lovász and Spencer to the context of additive number theory.

cs.CC↗

arXiv2013

Nontrivial t-Designs over Finite Fields Exist for All t

A $t$-$(n,k,λ)$ design over $\F_q$ is a collection of $k$-dimensional subspaces of $\F_q^n$, called blocks, such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly $λ$ blocks. Such $t$-designs over $\F_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,λ)$ designs over $\F_q$ are currently known to exist only for $t \leq 3$. Herein, we prove that simple (meaning, without repeated blocks) nontrivial $t$-$(n,k,λ)$ designs over $\F_q$ exist for all $t$ and $q$, provided that $k > 12t$ and $n$ is sufficiently large. This may be regarded as a $q$-analog of the celebrated Teirlinck theorem for combinatorial designs.

math.CO↗

arXiv2013

Estimating the distance from testable affine-invariant properties

Let $\cal{P}$ be an affine invariant property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\cal{P}$ with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over $\mathbb{F}_2$. Our test is simple: take a restriction of $f$ to a constant dimensional affine subspace, and measure its distance from $\cal{P}$. We show that by choosing the dimension large enough, this approximates with high probability the global distance of $f$ from $\cP$. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].

cs.CC↗

arXiv2013

New Lower Bounds for Matching Vector Codes

A Matching Vector (MV) family modulo $m$ is a pair of ordered lists $U=(u_1,...,u_t)$ and $V=(v_1,...,v_t)$ where $u_i,v_j \in \mathbb{Z}_m^n$ with the following inner product pattern: for any $i$, $< u_i,v_i>=0$, and for any $i \ne j$, $< u_i,v_j> \ne 0$. A MV family is called $q$-restricted if inner products $< u_i,v_j>$ take at most $q$ different values. Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs). There, $q$-restricted MV families are used to construct LDCs with $q$ queries, and there is special interest in the regime where $q$ is constant. When $m$ is a prime it is known that such constructions yield codes with exponential block length. However, for composite $m$ the behaviour is dramatically different. A recent work by Efremenko [STOC 2009] (based on an approach initiated by Yekhanin [JACM 2008]) gives the first sub-exponential LDC with constant queries. It is based on a construction of a MV family of super-polynomial size by Grolmusz [Combinatorica 2000] modulo composite $m$. In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families. When $q$ is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length. When the modulus $m$ is constant (as it is in the construction of Efremenko) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over $\mathbb{Z}_m$.

cs.CC↗

arXiv2013

Every locally characterized affine-invariant property is testable

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.

cs.CC↗

arXiv2012

Constructive Discrepancy Minimization by Walking on The Edges

Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of n sets in a universe of size n, there always exists a coloring which achieves discrepancy 6\sqrt{n}. The original proof of Spencer was existential in nature, and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal (FOCS 2010) gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call "Edge-Walk". Our algorithm and its analysis use only basic linear algebra and is "truly" constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

cs.DS↗