arXiv - CS - Computational Complexity最新文献_第8页

From Proof Complexity to Circuit Complexity via Interactive Protocols 通过交互式协议从证明复杂性到电路复杂性

arXiv - CS - Computational Complexity Pub Date : 2024-05-03 DOI: arxiv-2405.02232

Noel Arteche, Erfan Khaniki, Ján Pich, Rahul Santhanam

{"title":"From Proof Complexity to Circuit Complexity via Interactive Protocols","authors":"Noel Arteche, Erfan Khaniki, Ján Pich, Rahul Santhanam","doi":"arxiv-2405.02232","DOIUrl":"https://doi.org/arxiv-2405.02232","url":null,"abstract":"Folklore in complexity theory suspects that circuit lower bounds against\u0000$mathbf{NC}^1$ or $mathbf{P}/operatorname{poly}$, currently out of reach,\u0000are a necessary step towards proving strong proof complexity lower bounds for\u0000systems like Frege or Extended Frege. Establishing such a connection formally,\u0000however, is already daunting, as it would imply the breakthrough separation\u0000$mathbf{NEXP} notsubseteq mathbf{P}/operatorname{poly}$, as recently\u0000observed by Pich and Santhanam (2023). We show such a connection conditionally for the Implicit Extended Frege proof\u0000system ($mathsf{iEF}$) introduced by Kraj'iv{c}ek (The Journal of Symbolic\u0000Logic, 2004), capable of formalizing most of contemporary complexity theory. In\u0000particular, we show that if $mathsf{iEF}$ proves efficiently the standard\u0000derandomization assumption that a concrete Boolean function is hard on average\u0000for subexponential-size circuits, then any superpolynomial lower bound on the\u0000length of $mathsf{iEF}$ proofs implies $#mathbf{P} notsubseteq\u0000mathbf{FP}/operatorname{poly}$ (which would in turn imply, for example,\u0000$mathbf{PSPACE} notsubseteq mathbf{P}/operatorname{poly}$). Our proof\u0000exploits the formalization inside $mathsf{iEF}$ of the soundness of the\u0000sum-check protocol of Lund, Fortnow, Karloff, and Nisan (Journal of the ACM,\u00001992). This has consequences for the self-provability of circuit upper bounds\u0000in $mathsf{iEF}$. Interestingly, further improving our result seems to require\u0000progress in constructing interactive proof systems with more efficient provers.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883818","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}

引用次数: 0

Maximizing Network Phylogenetic Diversity 最大化网络系统发育多样性

arXiv - CS - Computational Complexity Pub Date : 2024-05-02 DOI: arxiv-2405.01091

Leo van Iersel, Mark Jones, Jannik Schestag, Celine Scornavacca, Mathias Weller

引用次数: 0

A Smoothed Analysis of the Space Complexity of Computing a Chaotic Sequence 计算混沌序列空间复杂性的平滑分析

arXiv - CS - Computational Complexity Pub Date : 2024-05-01 DOI: arxiv-2405.00327

Naoaki Okada, Shuji Kijima

引用次数: 0

An Oracle with no $mathrm{UP}$-Complete Sets, but $mathrm{NP}=mathrm{PSPACE}$ 没有 $mathrm{UP}$ 完整集合，但有 $mathrm{NP}=mathrm{PSPACE}$ 的 Oracle

arXiv - CS - Computational Complexity Pub Date : 2024-04-29 DOI: arxiv-2404.19104

David Dingel, Fabian Egidy, Christian Glaßer

引用次数: 0

Limits of Sequential Local Algorithms on the Random $k$-XORSAT Problem 随机 $k$-XORSAT 问题上序列局部算法的极限

arXiv - CS - Computational Complexity Pub Date : 2024-04-27 DOI: arxiv-2404.17775

Kingsley Yung

{"title":"Limits of Sequential Local Algorithms on the Random $k$-XORSAT Problem","authors":"Kingsley Yung","doi":"arxiv-2404.17775","DOIUrl":"https://doi.org/arxiv-2404.17775","url":null,"abstract":"The random $k$-XORSAT problem is a random constraint satisfaction problem of\u0000$n$ Boolean variables and $m=rn$ clauses, which a random instance can be\u0000expressed as a $Gmathbb{F}(2)$ linear system of the form $Ax=b$, where $A$ is\u0000a random $m times n$ matrix with $k$ ones per row, and $b$ is a random vector.\u0000It is known that there exist two distinct thresholds $r_{core}(k) < r_{sat}(k)$\u0000such that as $n rightarrow infty$ for $r < r_{sat}(k)$ the random instance\u0000has solutions with high probability, while for $r_{core} < r < r_{sat}(k)$ the\u0000solution space shatters into an exponential number of clusters. Sequential\u0000local algorithms are a natural class of algorithms which assign values to\u0000variables one by one iteratively. In each iteration, the algorithm runs some\u0000heuristics, called local rules, to decide the value assigned, based on the\u0000local neighborhood of the selected variables under the factor graph\u0000representation of the instance. We prove that for any $r > r_{core}(k)$ the sequential local algorithms with\u0000certain local rules fail to solve the random $k$-XORSAT with high probability.\u0000They include (1) the algorithm using the Unit Clause Propagation as local rule\u0000for $k ge 9$, and (2) the algorithms using any local rule that can calculate\u0000the exact marginal probabilities of variables in instances with factor graphs\u0000that are trees, for $kge 13$. The well-known Belief Propagation and Survey\u0000Propagation are included in (2). Meanwhile, the best known linear-time\u0000algorithm succeeds with high probability for $r < r_{core}(k)$. Our results\u0000support the intuition that $r_{core}(k)$ is the sharp threshold for the\u0000existence of a linear-time algorithm for random $k$-XORSAT.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140826977","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}

引用次数: 0

Maximizing Minimum Cycle Bases Intersection 最大化最小循环基数交叉点

arXiv - CS - Computational Complexity Pub Date : 2024-04-26 DOI: arxiv-2404.17223

Dimitri WatelSAMOVAR, ENSIIE, Marc-Antoine WeisserGALaC, Dominique BarthUVSQ, DAVID, Ylène AboulfathUVSQ, DAVID, Thierry MautorUVSQ, DAVID

引用次数: 0

A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results 非交换秩的多变量到双变量还原及相关结果

arXiv - CS - Computational Complexity Pub Date : 2024-04-25 DOI: arxiv-2404.16382

Vikraman Arvind, Pushkar S Joglekar

引用次数: 0

A Brief Note on a Recent Claim About NP-Hard Problems and BQP 关于 NP 难问题和 BQP 的最新说法的简要说明

arXiv - CS - Computational Complexity Pub Date : 2024-04-25 DOI: arxiv-2406.08495

Michael C. Chavrimootoo

引用次数: 0

Clique Is Hard on Average for Sherali-Adams with Bounded Coefficients 对于具有有界系数的谢拉利-亚当斯算法来说，克利克平均是困难的

arXiv - CS - Computational Complexity Pub Date : 2024-04-25 DOI: arxiv-2404.16722

Susanna F. de Rezende, Aaron Potechin, Kilian Risse

引用次数: 0

A nearly-$4log n$ depth lower bound for formulas with restriction on top 上有限制的公式的近$4/log n$ 深度下限

arXiv - CS - Computational Complexity Pub Date : 2024-04-24 DOI: arxiv-2404.15613

Hao Wu

{"title":"A nearly-$4log n$ depth lower bound for formulas with restriction on top","authors":"Hao Wu","doi":"arxiv-2404.15613","DOIUrl":"https://doi.org/arxiv-2404.15613","url":null,"abstract":"One of the major open problems in complexity theory is to demonstrate an\u0000explicit function which requires super logarithmic depth, a.k.a, the\u0000$mathbf{P}$ versus $mathbf{NC^1}$ problem. The current best depth lower bound\u0000is $(3-o(1))cdot log n$, and it is widely open how to prove a super-$3log n$\u0000depth lower bound. Recently Mihajlin and Sofronova (CCC'22) show if considering\u0000formulas with restriction on top, we can break the $3log n$ barrier. Formally,\u0000they prove there exist two functions $f:{0,1}^n rightarrow\u0000{0,1},g:{0,1}^n rightarrow {0,1}^n$, such that for any constant\u0000$0<alpha<0.4$ and constant $0<epsilon<alpha/2$, their XOR composition\u0000$f(g(x)oplus y)$ is not computable by an AND of $2^{(alpha-epsilon)n}$\u0000formulas of size at most $2^{(1-alpha/2-epsilon)n}$. This implies a modified\u0000version of Andreev function is not computable by any circuit of depth\u0000$(3.2-epsilon)log n$ with the restriction that top $0.4-epsilon$ layers only\u0000consist of AND gates for any small constant $epsilon>0$. They ask whether the\u0000parameter $alpha$ can be push up to nearly $1$ thus implying a nearly-$3.5log\u0000n$ depth lower bound. In this paper, we provide a stronger answer to their question. We show there\u0000exist two functions $f:{0,1}^n rightarrow {0,1},g:{0,1}^n rightarrow\u0000{0,1}^n$, such that for any constant $0<alpha<2-o(1)$, their XOR composition\u0000$f(g(x)oplus y)$ is not computable by an AND of $2^{alpha n}$ formulas of\u0000size at most $2^{(1-alpha/2-o(1))n}$. This implies a $(4-o(1))log n$ depth\u0000lower bound with the restriction that top $2-o(1)$ layers only consist of AND\u0000gates. We prove it by observing that one crucial component in Mihajlin and\u0000Sofronova's work, called the well-mixed set of functions, can be significantly\u0000simplified thus improved. Then with this observation and a more careful\u0000analysis, we obtain these nearly tight results.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801326","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}

引用次数: 0