arXiv - CS - Computational Complexity最新文献_第4页

Limitations of Affine Integer Relaxations for Solving Constraint Satisfaction Problems 解决约束满足问题的仿射整数松弛的局限性

arXiv - CS - Computational Complexity Pub Date : 2024-07-12 DOI: arxiv-2407.09097

Moritz Lichter, Benedikt Pago

引用次数: 0

Circuits and Backdoors: Five Shades of the SETH 电路与后门SETH 的五种色调

arXiv - CS - Computational Complexity Pub Date : 2024-07-12 DOI: arxiv-2407.09683

Michael Lampis

{"title":"Circuits and Backdoors: Five Shades of the SETH","authors":"Michael Lampis","doi":"arxiv-2407.09683","DOIUrl":"https://doi.org/arxiv-2407.09683","url":null,"abstract":"The SETH is a hypothesis of fundamental importance to (fine-grained)\u0000parameterized complexity theory and many important tight lower bounds are based\u0000on it. This situation is somewhat problematic, because the validity of the SETH\u0000is not universally believed and because in some senses the SETH seems to be\u0000\"too strong\" a hypothesis for the considered lower bounds. Motivated by this,\u0000we consider a number of reasonable weakenings of the SETH that render it more\u0000plausible, with sources ranging from circuit complexity, to backdoors for\u0000SAT-solving, to graph width parameters, to weighted satisfiability problems.\u0000Despite the diversity of the different formulations, we are able to uncover\u0000several non-obvious connections using tools from classical complexity theory.\u0000This leads us to a hierarchy of five main equivalence classes of hypotheses,\u0000with some of the highlights being the following: We show that beating brute force search for SAT parameterized by a modulator\u0000to a graph of bounded pathwidth, or bounded treewidth, or logarithmic\u0000tree-depth, is actually the same question, and is in fact equivalent to beating\u0000brute force for circuits of depth $epsilon n$; we show that beating brute\u0000force search for a strong 2-SAT backdoor is equivalent to beating brute force\u0000search for a modulator to logarithmic pathwidth; we show that beting brute\u0000force search for a strong Horn backdoor is equivalent to beating brute force\u0000search for arbitrary circuit SAT.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719612","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

Approximate Degree Composition for Recursive Functions 递归函数的近似度合成

arXiv - CS - Computational Complexity Pub Date : 2024-07-11 DOI: arxiv-2407.08385

Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh

{"title":"Approximate Degree Composition for Recursive Functions","authors":"Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh","doi":"arxiv-2407.08385","DOIUrl":"https://doi.org/arxiv-2407.08385","url":null,"abstract":"Determining the approximate degree composition for Boolean functions remains\u0000a significant unsolved problem in Boolean function complexity. In recent\u0000decades, researchers have concentrated on proving that approximate degree\u0000composes for special types of inner and outer functions. An important and\u0000extensively studied class of functions are the recursive functions,\u0000i.e.~functions obtained by composing a base function with itself a number of\u0000times. Let $h^d$ denote the standard $d$-fold composition of the base function\u0000$h$. The main result of this work is to show that the approximate degree composes\u0000if either of the following conditions holds: begin{itemize} item The outer function $f:{0,1}^nto {0,1}$ is a recursive function of\u0000the form $h^d$, with $h$ being any base function and $d= Omega(loglog n)$. item The inner function is a recursive function of the form $h^d$, with $h$\u0000being any constant arity base function (other than AND and OR) and $d=\u0000Omega(loglog n)$, where $n$ is the arity of the outer function. end{itemize} In terms of proof techniques, we first observe that the lower bound for\u0000composition can be obtained by introducing majority in between the inner and\u0000the outer functions. We then show that majority can be emph{efficiently\u0000eliminated} if the inner or outer function is a recursive function.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611391","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

Coordinating "7 Billion Humans" is hard 协调 "70 亿人类 "很困难

arXiv - CS - Computational Complexity Pub Date : 2024-07-09 DOI: arxiv-2407.07246

Alessandro Panconesi, Pietro Maria Posta, Mirko Giacchini

引用次数: 0

Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension 学习低内在维度概念的平滑分析法

arXiv - CS - Computational Complexity Pub Date : 2024-07-01 DOI: arxiv-2407.00966

Gautam Chandrasekaran, Adam Klivans, Vasilis Kontonis, Raghu Meka, Konstantinos Stavropoulos

{"title":"Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension","authors":"Gautam Chandrasekaran, Adam Klivans, Vasilis Kontonis, Raghu Meka, Konstantinos Stavropoulos","doi":"arxiv-2407.00966","DOIUrl":"https://doi.org/arxiv-2407.00966","url":null,"abstract":"In traditional models of supervised learning, the goal of a learner -- given\u0000examples from an arbitrary joint distribution on $mathbb{R}^d times {pm\u00001}$ -- is to output a hypothesis that is competitive (to within $epsilon$) of\u0000the best fitting concept from some class. In order to escape strong hardness\u0000results for learning even simple concept classes, we introduce a\u0000smoothed-analysis framework that requires a learner to compete only with the\u0000best classifier that is robust to small random Gaussian perturbation. This subtle change allows us to give a wide array of learning results for any\u0000concept that (1) depends on a low-dimensional subspace (aka multi-index model)\u0000and (2) has a bounded Gaussian surface area. This class includes functions of\u0000halfspaces and (low-dimensional) convex sets, cases that are only known to be\u0000learnable in non-smoothed settings with respect to highly structured\u0000distributions such as Gaussians. Surprisingly, our analysis also yields new results for traditional\u0000non-smoothed frameworks such as learning with margin. In particular, we obtain\u0000the first algorithm for agnostically learning intersections of $k$-halfspaces\u0000in time $k^{poly(frac{log k}{epsilon gamma}) }$ where $gamma$ is the\u0000margin parameter. Before our work, the best-known runtime was exponential in\u0000$k$ (Arriaga and Vempala, 1999).","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141518329","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

An XOR Lemma for Deterministic Communication Complexity 确定性通信复杂性的 XOR 定理

arXiv - CS - Computational Complexity Pub Date : 2024-07-01 DOI: arxiv-2407.01802

Siddharth Iyer, Anup Rao

引用次数: 0

On the approximability of graph visibility problems 论图形可见性问题的近似性

arXiv - CS - Computational Complexity Pub Date : 2024-06-29 DOI: arxiv-2407.00409

Davide Bilò, Alessia Di Fonso, Gabriele Di Stefano, Stefano Leucci

{"title":"On the approximability of graph visibility problems","authors":"Davide Bilò, Alessia Di Fonso, Gabriele Di Stefano, Stefano Leucci","doi":"arxiv-2407.00409","DOIUrl":"https://doi.org/arxiv-2407.00409","url":null,"abstract":"Visibility problems have been investigated for a long time under different\u0000assumptions as they pose challenging combinatorial problems and are connected\u0000to robot navigation problems. The mutual-visibility problem in a graph $G$ of\u0000$n$ vertices asks to find the largest set of vertices $Xsubseteq V(G)$, also\u0000called $mu$-set, such that for any two vertices $u,vin X$, there is a\u0000shortest $u,v$-path $P$ where all internal vertices of $P$ are not in $X$. This\u0000means that $u$ and $v$ are visible w.r.t. $X$. Variations of this problem are\u0000known as total, outer, and dual mutual-visibility problems, depending on the\u0000visibility property of vertices inside and/or outside $X$. The\u0000mutual-visibility problem and all its variations are known to be\u0000$mathsf{NP}$-complete on graphs of diameter $4$. In this paper, we design a polynomial-time algorithm that finds a $mu$-set\u0000with size $Omegaleft( sqrt{n/ overline{D}} right)$, where $overline D$ is\u0000the average distance between any two vertices of $G$. Moreover, we show\u0000inapproximability results for all visibility problems on graphs of diameter $2$\u0000and strengthen the inapproximability ratios for graphs of diameter $3$ or\u0000larger. More precisely, for graphs of diameter at least $3$ and for every\u0000constant $varepsilon > 0$, we show that mutual-visibility and dual\u0000mutual-visibility problems are not approximable within a factor of\u0000$n^{1/3-varepsilon}$, while outer and total mutual-visibility problems are not\u0000approximable within a factor of $n^{1/2 - varepsilon}$, unless\u0000$mathsf{P}=mathsf{NP}$. Furthermore we study the relationship between the mutual-visibility number\u0000and the general position number in which no three distinct vertices $u,v,w$ of\u0000$X$ belong to any shortest path of $G$.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507159","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

Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs 距离与传递性：驯服时态图中可达性的新参数

arXiv - CS - Computational Complexity Pub Date : 2024-06-27 DOI: arxiv-2406.19514

Arnaud Casteigts, Nils Morawietz, Petra Wolf

{"title":"Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs","authors":"Arnaud Casteigts, Nils Morawietz, Petra Wolf","doi":"arxiv-2406.19514","DOIUrl":"https://doi.org/arxiv-2406.19514","url":null,"abstract":"A temporal graph is a graph whose edges only appear at certain points in\u0000time. Reachability in these graphs is defined in terms of paths that traverse\u0000the edges in chronological order (temporal paths). This form of reachability is\u0000neither symmetric nor transitive, the latter having important consequences on\u0000the computational complexity of even basic questions, such as computing\u0000temporal connected components. In this paper, we introduce several parameters\u0000that capture how far a temporal graph $mathcal{G}$ is from being transitive,\u0000namely, emph{vertex-deletion distance to transitivity} and\u0000emph{arc-modification distance to transitivity}, both being applied to the\u0000reachability graph of $mathcal{G}$. We illustrate the impact of these\u0000parameters on the temporal connected component problem, obtaining several\u0000tractability results in terms of fixed-parameter tractability and polynomial\u0000kernels. Significantly, these results are obtained without restrictions of the\u0000underlying graph, the snapshots, or the lifetime of the input graph. As such,\u0000our results isolate the impact of non-transitivity and confirm the key role\u0000that it plays in the hardness of temporal graph problems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141518330","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

On Fourier analysis of sparse Boolean functions over certain Abelian groups 论某些阿贝尔群上稀疏布尔函数的傅立叶分析

arXiv - CS - Computational Complexity Pub Date : 2024-06-26 DOI: arxiv-2406.18700

Sourav Chakraborty, Swarnalipa Datta, Pranjal Dutta, Arijit Ghosh, Swagato Sanyal

{"title":"On Fourier analysis of sparse Boolean functions over certain Abelian groups","authors":"Sourav Chakraborty, Swarnalipa Datta, Pranjal Dutta, Arijit Ghosh, Swagato Sanyal","doi":"arxiv-2406.18700","DOIUrl":"https://doi.org/arxiv-2406.18700","url":null,"abstract":"Given an Abelian group G, a Boolean-valued function f: G -> {-1,+1}, is said\u0000to be s-sparse, if it has at most s-many non-zero Fourier coefficients over the\u0000domain G. In a seminal paper, Gopalan et al. proved \"Granularity\" for Fourier\u0000coefficients of Boolean valued functions over Z_2^n, that have found many\u0000diverse applications in theoretical computer science and combinatorics. They\u0000also studied structural results for Boolean functions over Z_2^n which are\u0000approximately Fourier-sparse. In this work, we obtain structural results for\u0000approximately Fourier-sparse Boolean valued functions over Abelian groups G of\u0000the form,G:= Z_{p_1}^{n_1} times ... times Z_{p_t}^{n_t}, for distinct primes\u0000p_i. We also obtain a lower bound of the form 1/(m^{2}s)^ceiling(phi(m)/2), on\u0000the absolute value of the smallest non-zero Fourier coefficient of an s-sparse\u0000function, where m=p_1 ... p_t, and phi(m)=(p_1-1) ... (p_t-1). We carefully\u0000apply probabilistic techniques from Gopalan et al., to obtain our structural\u0000results, and use some non-trivial results from algebraic number theory to get\u0000the lower bound. We construct a family of at most s-sparse Boolean functions over Z_p^n, where\u0000p > 2, for arbitrarily large enough s, where the minimum non-zero Fourier\u0000coefficient is 1/omega(n). The \"Granularity\" result of Gopalan et al. implies\u0000that the absolute values of non-zero Fourier coefficients of any s-sparse\u0000Boolean valued function over Z_2^n are 1/O(s). So, our result shows that one\u0000cannot expect such a lower bound for general Abelian groups. Using our new structural results on the Fourier coefficients of sparse\u0000functions, we design an efficient testing algorithm for Fourier-sparse Boolean\u0000functions, thata requires poly((ms)^phi(m),1/epsilon)-many queries. Further, we\u0000prove an Omega(sqrt{s}) lower bound on the query complexity of any adaptive\u0000sparsity testing algorithm.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507161","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

Consistent Query Answering over SHACL Constraints SHACL 约束条件下的一致性查询应答

arXiv - CS - Computational Complexity Pub Date : 2024-06-24 DOI: arxiv-2406.16653

Shqiponja Ahmetaj, Timo Camillo Merkl, Reinhard Pichler

引用次数: 0