Pub Date : 2023-08-23DOI: 10.48550/arXiv.2308.12451
Zander Kelley, Shachar Lovett, R. Meka
We study the power of randomness in the Number-on-Forehead (NOF) model in communication complexity. We construct an explicit 3-player function $f:[N]^3 to {0,1}$, such that: (i) there exist a randomized NOF protocol computing it that sends a constant number of bits; but (ii) any deterministic or nondeterministic NOF protocol computing it requires sending about $(log N)^{1/3}$ many bits. This exponentially improves upon the previously best-known such separation. At the core of our proof is an extension of a recent result of the first and third authors on sets of integers without 3-term arithmetic progressions into a non-arithmetic setting.
{"title":"Explicit separations between randomized and deterministic Number-on-Forehead communication","authors":"Zander Kelley, Shachar Lovett, R. Meka","doi":"10.48550/arXiv.2308.12451","DOIUrl":"https://doi.org/10.48550/arXiv.2308.12451","url":null,"abstract":"We study the power of randomness in the Number-on-Forehead (NOF) model in communication complexity. We construct an explicit 3-player function $f:[N]^3 to {0,1}$, such that: (i) there exist a randomized NOF protocol computing it that sends a constant number of bits; but (ii) any deterministic or nondeterministic NOF protocol computing it requires sending about $(log N)^{1/3}$ many bits. This exponentially improves upon the previously best-known such separation. At the core of our proof is an extension of a recent result of the first and third authors on sets of integers without 3-term arithmetic progressions into a non-arithmetic setting.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82281485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-18DOI: 10.48550/arXiv.2308.09582
Yotam Dikstein, Irit Dinur
Given a family $X$ of subsets of $[n]$ and an ensemble of local functions ${f_s:stoSigma; | ; sin X}$, an agreement test is a randomized property tester that is supposed to test whether there is some global function $G:[n]toSigma$ such that $f_s=G|_s$ for many sets $s$. A"classical"small-soundness agreement theorem is a list-decoding $(LD)$ statement, saying that [tag{$LD$} Agree({f_s})>varepsilon quad Longrightarrow quad exists G^1,dots, G^ell,quad P_s[f_soverset{0.99}{approx}G^i|_s]geq poly(varepsilon),;i=1,dots,ell. ] Such a statement is motivated by PCP questions and has been shown in the case where $X=binom{[n]}k$, or where $X$ is a collection of low dimensional subspaces of a vector space. In this work we study small the case of on high dimensional expanders $X$. It has been an open challenge to analyze their small soundness behavior. Surprisingly, the small soundness behavior turns out to be governed by the topological covers of $X$.We show that: 1. If $X$ has no connected covers, then $(LD)$ holds, provided that $X$ satisfies an additional expansion property. 2. If $X$ has a connected cover, then $(LD)$ necessarily fails. 3. If $X$ has a connected cover (and assuming the additional expansion property), we replace the $(LD)$ by a weaker statement we call lift-decoding: [ tag{$LFD$} Agree({f_s})>varepsilon Longrightarrow quad existstext{ cover }rho:Ytwoheadrightarrow X,text{ and }G:Y(0)toSigma,text{ such that }] [P_{{tilde stwoheadrightarrow s}}[f_s overset{0.99}{approx} G|_{tilde s}] geq poly(varepsilon),] where ${tilde stwoheadrightarrow s}$ means that $rho(tilde s)=s$. The additional expansion property is cosystolic expansion of a complex derived from $X$ holds for the spherical building and for quotients of the Bruhat-Tits building.
给定一个$[n]$的子集族$X$和一个局部函数集合${f_s:stoSigma; | ; sin X}$，一致性测试是一个随机属性测试器，用来测试是否存在一些全局函数$G:[n]toSigma$，使得$f_s=G|_s$对于许多集合$s$。一个“经典的”小完备性一致定理是一个列表解码$(LD)$语句，它说[tag{$LD$} Agree({f_s})>varepsilon quad Longrightarrow quad exists G^1,dots, G^ell,quad P_s[f_soverset{0.99}{approx}G^i|_s]geq poly(varepsilon),;i=1,dots,ell. ]这样的语句是由PCP问题激发的，并且在$X=binom{[n]}k$或$X$是向量空间的低维子空间集合的情况下得到了证明。在这项工作中，我们研究了小的情况下，高维膨胀$X$。分析它们的小健康行为一直是一个公开的挑战。令人惊讶的是，小稳健性行为被证明是由$X$的拓扑覆盖所控制的。如果$X$没有连接的盖，则$(LD)$成立，前提是$X$满足一个附加的展开属性。2. 如果$X$有一个连接的封面，那么$(LD)$必然失败。3.如果$X$有一个连接的覆盖(并假设有额外的扩展属性)，我们用一个更弱的语句来替换$(LD)$，我们称之为提升解码:[ tag{$LFD$} Agree({f_s})>varepsilon Longrightarrow quad existstext{ cover }rho:Ytwoheadrightarrow X,text{ and }G:Y(0)toSigma,text{ such that }][P_{{tilde stwoheadrightarrow s}}[f_s overset{0.99}{approx} G|_{tilde s}] geq poly(varepsilon),]，其中${tilde stwoheadrightarrow s}$表示$rho(tilde s)=s$。附加的膨胀特性是由$X$衍生的综合体的收缩膨胀，适用于球形建筑和Bruhat-Tits建筑的商数。
{"title":"Agreement theorems for high dimensional expanders in the small soundness regime: the role of covers","authors":"Yotam Dikstein, Irit Dinur","doi":"10.48550/arXiv.2308.09582","DOIUrl":"https://doi.org/10.48550/arXiv.2308.09582","url":null,"abstract":"Given a family $X$ of subsets of $[n]$ and an ensemble of local functions ${f_s:stoSigma; | ; sin X}$, an agreement test is a randomized property tester that is supposed to test whether there is some global function $G:[n]toSigma$ such that $f_s=G|_s$ for many sets $s$. A\"classical\"small-soundness agreement theorem is a list-decoding $(LD)$ statement, saying that [tag{$LD$} Agree({f_s})>varepsilon quad Longrightarrow quad exists G^1,dots, G^ell,quad P_s[f_soverset{0.99}{approx}G^i|_s]geq poly(varepsilon),;i=1,dots,ell. ] Such a statement is motivated by PCP questions and has been shown in the case where $X=binom{[n]}k$, or where $X$ is a collection of low dimensional subspaces of a vector space. In this work we study small the case of on high dimensional expanders $X$. It has been an open challenge to analyze their small soundness behavior. Surprisingly, the small soundness behavior turns out to be governed by the topological covers of $X$.We show that: 1. If $X$ has no connected covers, then $(LD)$ holds, provided that $X$ satisfies an additional expansion property. 2. If $X$ has a connected cover, then $(LD)$ necessarily fails. 3. If $X$ has a connected cover (and assuming the additional expansion property), we replace the $(LD)$ by a weaker statement we call lift-decoding: [ tag{$LFD$} Agree({f_s})>varepsilon Longrightarrow quad existstext{ cover }rho:Ytwoheadrightarrow X,text{ and }G:Y(0)toSigma,text{ such that }] [P_{{tilde stwoheadrightarrow s}}[f_s overset{0.99}{approx} G|_{tilde s}] geq poly(varepsilon),] where ${tilde stwoheadrightarrow s}$ means that $rho(tilde s)=s$. The additional expansion property is cosystolic expansion of a complex derived from $X$ holds for the spherical building and for quotients of the Bruhat-Tits building.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82905594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-14DOI: 10.4230/LIPIcs.SAT.2022.5
Agnes Schleitzer, Olaf Beyersdorff
To date, we know only a few handcrafted quantified Boolean formulas (QBFs) that are hard for central QBF resolution systems such as Q-Res and QU-Res, and only one specific QBF family to separate Q-Res and QU-Res. Here we provide a general method to construct hard formulas for Q-Res and QU-Res. The construction uses simple propositional formulas (e.g. minimally unsatisfiable formulas) in combination with easy QBF gadgets (Σb2 formulas without constant winning strategies). This leads to a host of new hard formulas, including new classes of hard random QBFs. We further present generic constructions for formulas separating Q-Res and QU-Res, and for separating Q-Res and LD-Q-Res.
{"title":"Classes of Hard Formulas for QBF Resolution","authors":"Agnes Schleitzer, Olaf Beyersdorff","doi":"10.4230/LIPIcs.SAT.2022.5","DOIUrl":"https://doi.org/10.4230/LIPIcs.SAT.2022.5","url":null,"abstract":"To date, we know only a few handcrafted quantified Boolean formulas (QBFs) that are hard for central QBF resolution systems such as Q-Res and QU-Res, and only one specific QBF family to separate Q-Res and QU-Res.\u0000Here we provide a general method to construct hard formulas for Q-Res and QU-Res. The construction uses simple propositional formulas (e.g. minimally unsatisfiable formulas) in combination with easy QBF gadgets (Σb2 formulas without constant winning strategies). This leads to a host of new hard formulas, including new classes of hard random QBFs.\u0000We further present generic constructions for formulas separating Q-Res and QU-Res, and for separating Q-Res and LD-Q-Res.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"25 1","pages":"5:1-5:18"},"PeriodicalIF":0.0,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87520665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-12DOI: 10.48550/arXiv.2308.06600
Amey Bhangale, Subhash Khot, Dor Minzer
For a prime $p$, a restricted arithmetic progression in $mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from ${0,1,2}^n$. What is the size of the largest $Asubseteq mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such set is at most $frac{C}{(logloglog n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/log^{*} n)$, which follows from the density Hales-Jewett theorem.
{"title":"Effective Bounds for Restricted 3-Arithmetic Progressions in Fpn","authors":"Amey Bhangale, Subhash Khot, Dor Minzer","doi":"10.48550/arXiv.2308.06600","DOIUrl":"https://doi.org/10.48550/arXiv.2308.06600","url":null,"abstract":"For a prime $p$, a restricted arithmetic progression in $mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from ${0,1,2}^n$. What is the size of the largest $Asubseteq mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such set is at most $frac{C}{(logloglog n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/log^{*} n)$, which follows from the density Hales-Jewett theorem.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85854810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-09DOI: 10.48550/arXiv.2308.05079
Franccois Le Gall, Yupan Liu, Qisheng Wang
Driven by exploring the power of quantum computation with a limited number of qubits, we present a novel complete characterization for space-bounded quantum computation, which encompasses settings with one-sided error (unitary coRQL) and two-sided error (BQL), approached from a quantum state testing perspective: - The first family of natural complete problems for unitary coRQL, i.e., space-bounded quantum state certification for trace distance and Hilbert-Schmidt distance; - A new family of natural complete problems for BQL, i.e., space-bounded quantum state testing for trace distance, Hilbert-Schmidt distance, and quantum entropy difference. In the space-bounded quantum state testing problem, we consider two logarithmic-qubit quantum circuits (devices) denoted as $Q_0$ and $Q_1$, which prepare quantum states $rho_0$ and $rho_1$, respectively, with access to their ``source code''. Our goal is to decide whether $rho_0$ is $epsilon_1$-close to or $epsilon_2$-far from $rho_1$ with respect to a specified distance-like measure. Interestingly, unlike time-bounded state testing problems, which exhibit computational hardness depending on the chosen distance-like measure (either QSZK-complete or BQP-complete), our results reveal that the space-bounded state testing problems, considering all three measures, are computationally as easy as preparing quantum states. Our results primarily build upon a space-efficient variant of the quantum singular value transformation (QSVT) introduced by Gily'en, Su, Low, and Wiebe (STOC 2019), which is of independent interest. Our technique provides a unified approach for designing space-bounded quantum algorithms. Specifically, we show that implementing QSVT for any bounded polynomial that approximates a piecewise-smooth function incurs only a constant overhead in terms of the space required for special forms of the projected unitary encoding.
{"title":"Space-bounded quantum state testing via space-efficient quantum singular value transformation","authors":"Franccois Le Gall, Yupan Liu, Qisheng Wang","doi":"10.48550/arXiv.2308.05079","DOIUrl":"https://doi.org/10.48550/arXiv.2308.05079","url":null,"abstract":"Driven by exploring the power of quantum computation with a limited number of qubits, we present a novel complete characterization for space-bounded quantum computation, which encompasses settings with one-sided error (unitary coRQL) and two-sided error (BQL), approached from a quantum state testing perspective: - The first family of natural complete problems for unitary coRQL, i.e., space-bounded quantum state certification for trace distance and Hilbert-Schmidt distance; - A new family of natural complete problems for BQL, i.e., space-bounded quantum state testing for trace distance, Hilbert-Schmidt distance, and quantum entropy difference. In the space-bounded quantum state testing problem, we consider two logarithmic-qubit quantum circuits (devices) denoted as $Q_0$ and $Q_1$, which prepare quantum states $rho_0$ and $rho_1$, respectively, with access to their ``source code''. Our goal is to decide whether $rho_0$ is $epsilon_1$-close to or $epsilon_2$-far from $rho_1$ with respect to a specified distance-like measure. Interestingly, unlike time-bounded state testing problems, which exhibit computational hardness depending on the chosen distance-like measure (either QSZK-complete or BQP-complete), our results reveal that the space-bounded state testing problems, considering all three measures, are computationally as easy as preparing quantum states. Our results primarily build upon a space-efficient variant of the quantum singular value transformation (QSVT) introduced by Gily'en, Su, Low, and Wiebe (STOC 2019), which is of independent interest. Our technique provides a unified approach for designing space-bounded quantum algorithms. Specifically, we show that implementing QSVT for any bounded polynomial that approximates a piecewise-smooth function incurs only a constant overhead in terms of the space required for special forms of the projected unitary encoding.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81502683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.48550/arXiv.2308.04599
Abhranil Chatterjee, Mrinal Kumar, Ben lee Volk
We show that for every homogeneous polynomial of degree $d$, if it has determinantal complexity at most $s$, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most $O(d^5s)$. Moreover, we show that for $textit{most}$ homogeneous polynomials, the width of the resulting homogeneous ABP is just $s-1$ and the size is at most $O(ds)$. Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree, and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor. While determinantal complexity and ABP complexity are classically known to be polynomially equivalent, the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.
{"title":"Determinants vs. Algebraic Branching Programs","authors":"Abhranil Chatterjee, Mrinal Kumar, Ben lee Volk","doi":"10.48550/arXiv.2308.04599","DOIUrl":"https://doi.org/10.48550/arXiv.2308.04599","url":null,"abstract":"We show that for every homogeneous polynomial of degree $d$, if it has determinantal complexity at most $s$, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most $O(d^5s)$. Moreover, we show that for $textit{most}$ homogeneous polynomials, the width of the resulting homogeneous ABP is just $s-1$ and the size is at most $O(ds)$. Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree, and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor. While determinantal complexity and ABP complexity are classically known to be polynomially equivalent, the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"238 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77003870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-30DOI: 10.48550/arXiv.2307.16248
Amey Bhangale, Subhash Khot, Dor Minzer
We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $Sigma,Gamma$ and $Phi$ are alphabets of constant size, and $mu$ is a pairwise connected distribution over $SigmatimesGammatimesPhi$ with no $(mathbb{Z},+)$ embeddings in which the probability of each atom is $Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $fcolon Sigma^ntomathbb{C}$, $gcolon Gamma^ntomathbb{C}$, $hcolon Phi^ntomathbb{C}$ satisfying [ left|mathbb{E}_{(x,y,z)sim mu^{otimes n}}big[f(x)g(y)h(z)big]right|geq varepsilon ] must arise from an Abelian group associated with the distribution $mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $tilde{f}(x) = chi(sigma(x_1),ldots,sigma(x_n)) L (x)$, where $sigmacolon Sigma to H$ is some map, $chiin hat{H}^{otimes n}$ is a character, and $Lcolon Sigma^ntomathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.
{"title":"On Approximability of Satisfiable k-CSPs: IV","authors":"Amey Bhangale, Subhash Khot, Dor Minzer","doi":"10.48550/arXiv.2307.16248","DOIUrl":"https://doi.org/10.48550/arXiv.2307.16248","url":null,"abstract":"We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $Sigma,Gamma$ and $Phi$ are alphabets of constant size, and $mu$ is a pairwise connected distribution over $SigmatimesGammatimesPhi$ with no $(mathbb{Z},+)$ embeddings in which the probability of each atom is $Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $fcolon Sigma^ntomathbb{C}$, $gcolon Gamma^ntomathbb{C}$, $hcolon Phi^ntomathbb{C}$ satisfying [ left|mathbb{E}_{(x,y,z)sim mu^{otimes n}}big[f(x)g(y)h(z)big]right|geq varepsilon ] must arise from an Abelian group associated with the distribution $mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $tilde{f}(x) = chi(sigma(x_1),ldots,sigma(x_n)) L (x)$, where $sigmacolon Sigma to H$ is some map, $chiin hat{H}^{otimes n}$ is a character, and $Lcolon Sigma^ntomathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90614029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-11DOI: 10.48550/arXiv.2305.06821
B. P. Cavalar, I. Oliveira
We establish new separations between the power of monotone and general (non-monotone) Boolean circuits: - For every $k geq 1$, there is a monotone function in ${sf AC^0}$ that requires monotone circuits of depth $Omega(log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai and Gurevich (1987). In addition, our separation holds for a monotone graph property, which was unknown even in the context of ${sf AC^0}$ versus ${sf mAC^0}$. - For every $k geq 1$, there is a monotone function in ${sf AC^0}[oplus]$ that requires monotone circuits of size $exp(Omega(log^k n))$. This makes progress towards a question posed by Grigni and Sipser (1992). These results show that constant-depth circuits can be more efficient than monotone circuits when computing monotone functions. In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an ${sf AC^0}$ circuit of size $s$ and depth $d$ can be computed by a monotone circuit of size $2^{n - n/O(log s)^{d-1}}$. We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in ${sf AC^0}$ admits a polynomial size monotone circuit, then ${sf NC^2}$ is not contained in ${sf NC^1}$ . Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems established by G"o"os et al. (2019) via lifting techniques. Adapting results of Schaefer (1978) and Allender et al. (2009), we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.
{"title":"Constant-depth circuits vs. monotone circuits","authors":"B. P. Cavalar, I. Oliveira","doi":"10.48550/arXiv.2305.06821","DOIUrl":"https://doi.org/10.48550/arXiv.2305.06821","url":null,"abstract":"We establish new separations between the power of monotone and general (non-monotone) Boolean circuits: - For every $k geq 1$, there is a monotone function in ${sf AC^0}$ that requires monotone circuits of depth $Omega(log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai and Gurevich (1987). In addition, our separation holds for a monotone graph property, which was unknown even in the context of ${sf AC^0}$ versus ${sf mAC^0}$. - For every $k geq 1$, there is a monotone function in ${sf AC^0}[oplus]$ that requires monotone circuits of size $exp(Omega(log^k n))$. This makes progress towards a question posed by Grigni and Sipser (1992). These results show that constant-depth circuits can be more efficient than monotone circuits when computing monotone functions. In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an ${sf AC^0}$ circuit of size $s$ and depth $d$ can be computed by a monotone circuit of size $2^{n - n/O(log s)^{d-1}}$. We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in ${sf AC^0}$ admits a polynomial size monotone circuit, then ${sf NC^2}$ is not contained in ${sf NC^1}$ . Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems established by G\"o\"os et al. (2019) via lifting techniques. Adapting results of Schaefer (1978) and Allender et al. (2009), we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"124 1","pages":"29:1-29:37"},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77528104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-07DOI: 10.48550/arXiv.2305.04363
Yuval Filmus, Itai Leigh, Artur Riazanov, Dmitry Sokolov
A circuit $mathcal{C}$ samples a distribution $mathbf{X}$ with an error $epsilon$ if the statistical distance between the output of $mathcal{C}$ on the uniform input and $mathbf{X}$ is $epsilon$. We study the hardness of sampling a uniform distribution over the set of $n$-bit strings of Hamming weight $k$ denoted by $mathbf{U}^n_k$ for _decision forests_, i.e. every output bit is computed as a decision tree of the inputs. For every $k$ there is an $O(log n)$-depth decision forest sampling $mathbf{U}^n_k$ with an inverse-polynomial error [Viola 2012, Czumaj 2015]. We show that for every $epsilon>0$ there exists $tau$ such that for decision depth $tau log (n/k) / log log (n/k)$, the error for sampling $mathbf{U}_k^n$ is at least $1-epsilon$. Our result is based on the recent robust sunflower lemma [Alweiss, Lovett, Wu, Zhang 2021, Rao 2019]. Our second result is about matching a set of $n$-bit strings with the image of a $d$-_local_ circuit, i.e. such that each output bit depends on at most $d$ input bits. We study the set of all $n$-bit strings whose Hamming weight is at least $n/2$. We improve the previously known locality lower bound from $Omega(log^* n)$ [Beyersdorff, Datta, Krebs, Mahajan, Scharfenberger-Fabian, Sreenivasaiah, Thomas and Vollmer, 2013] to $Omega(sqrt{log n})$, leaving only a quartic gap from the best upper bound of $O(log^2 n)$.
{"title":"Sampling and Certifying Symmetric Functions","authors":"Yuval Filmus, Itai Leigh, Artur Riazanov, Dmitry Sokolov","doi":"10.48550/arXiv.2305.04363","DOIUrl":"https://doi.org/10.48550/arXiv.2305.04363","url":null,"abstract":"A circuit $mathcal{C}$ samples a distribution $mathbf{X}$ with an error $epsilon$ if the statistical distance between the output of $mathcal{C}$ on the uniform input and $mathbf{X}$ is $epsilon$. We study the hardness of sampling a uniform distribution over the set of $n$-bit strings of Hamming weight $k$ denoted by $mathbf{U}^n_k$ for _decision forests_, i.e. every output bit is computed as a decision tree of the inputs. For every $k$ there is an $O(log n)$-depth decision forest sampling $mathbf{U}^n_k$ with an inverse-polynomial error [Viola 2012, Czumaj 2015]. We show that for every $epsilon>0$ there exists $tau$ such that for decision depth $tau log (n/k) / log log (n/k)$, the error for sampling $mathbf{U}_k^n$ is at least $1-epsilon$. Our result is based on the recent robust sunflower lemma [Alweiss, Lovett, Wu, Zhang 2021, Rao 2019]. Our second result is about matching a set of $n$-bit strings with the image of a $d$-_local_ circuit, i.e. such that each output bit depends on at most $d$ input bits. We study the set of all $n$-bit strings whose Hamming weight is at least $n/2$. We improve the previously known locality lower bound from $Omega(log^* n)$ [Beyersdorff, Datta, Krebs, Mahajan, Scharfenberger-Fabian, Sreenivasaiah, Thomas and Vollmer, 2013] to $Omega(sqrt{log n})$, leaving only a quartic gap from the best upper bound of $O(log^2 n)$.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"29 1","pages":"36:1-36:21"},"PeriodicalIF":0.0,"publicationDate":"2023-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89254439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-04DOI: 10.48550/arXiv.2305.02512
Louis Golowich
In this paper, we present a new construction of simplicial complexes of subpolynomial degree with arbitrarily good local spectral expansion. Previously, the only known high-dimensional expanders (HDXs) with arbitrarily good expansion and less than polynomial degree were based on one of two constructions, namely Ramanujan complexes and coset complexes. In contrast, our construction is a Cayley complex over the group $mathbb{F}_2^k$, with Cayley generating set given by a Grassmannian HDX. Our construction is in part motivated by a coding-theoretic interpretation of Grassmannian HDXs that we present, which provides a formal connection between Grassmannian HDXs, simplicial HDXs, and LDPC codes. We apply this interpretation to prove a general characterization of the 1-homology groups over $mathbb{F}_2$ of Cayley simplicial complexes over $mathbb{F}_2^k$. Using this result, we construct simplicial complexes on $N$ vertices with arbitrarily good local expansion for which the dimension of the 1-homology group grows as $Omega(log^2N)$. No prior constructions in the literature have been shown to achieve as large a 1-homology group.
{"title":"From Grassmannian to Simplicial High-Dimensional Expanders","authors":"Louis Golowich","doi":"10.48550/arXiv.2305.02512","DOIUrl":"https://doi.org/10.48550/arXiv.2305.02512","url":null,"abstract":"In this paper, we present a new construction of simplicial complexes of subpolynomial degree with arbitrarily good local spectral expansion. Previously, the only known high-dimensional expanders (HDXs) with arbitrarily good expansion and less than polynomial degree were based on one of two constructions, namely Ramanujan complexes and coset complexes. In contrast, our construction is a Cayley complex over the group $mathbb{F}_2^k$, with Cayley generating set given by a Grassmannian HDX. Our construction is in part motivated by a coding-theoretic interpretation of Grassmannian HDXs that we present, which provides a formal connection between Grassmannian HDXs, simplicial HDXs, and LDPC codes. We apply this interpretation to prove a general characterization of the 1-homology groups over $mathbb{F}_2$ of Cayley simplicial complexes over $mathbb{F}_2^k$. Using this result, we construct simplicial complexes on $N$ vertices with arbitrarily good local expansion for which the dimension of the 1-homology group grows as $Omega(log^2N)$. No prior constructions in the literature have been shown to achieve as large a 1-homology group.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76469897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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