arXiv - CS - Computational Complexity最新文献

arXiv - CS - Computational Complexity Pub Date : 2024-09-16 DOI: arxiv-2409.10464

Alek Westover, Edward Yu, Kai Zheng

{"title":"New Direct Sum Tests","authors":"Alek Westover, Edward Yu, Kai Zheng","doi":"arxiv-2409.10464","DOIUrl":"https://doi.org/arxiv-2409.10464","url":null,"abstract":"A function $f:[n]^{d} to mathbb{F}_2$ is a defn{direct sum} if there are\u0000functions $L_i:[n]to mathbb{F}_2$ such that ${f(x) = sum_{i}L_i(x_i)}$. In\u0000this work we give multiple results related to the property testing of direct\u0000sums. Our first result concerns a test proposed by Dinur and Golubev in 2019. We\u0000call their test the Diamond test and show that it is indeed a direct sum\u0000tester. More specifically, we show that if a function $f$ is $epsilon$-far\u0000from being a direct sum function, then the Diamond test rejects $f$ with\u0000probability at least $Omega_{n,epsilon}(1)$. Even in the case of $n = 2$, the\u0000Diamond test is, to the best of our knowledge, novel and yields a new tester\u0000for the classic property of affinity. Apart from the Diamond test, we also analyze a broad family of direct sum\u0000tests, which at a high level, run an arbitrary affinity test on the restriction\u0000of $f$ to a random hypercube inside of $[n]^d$. This family of tests includes\u0000the direct sum test analyzed in cite{di19}, but does not include the Diamond\u0000test. As an application of our result, we obtain a direct sum test which works\u0000in the online adversary model of cite{KRV}. Finally, we also discuss a Fourier analytic interpretation of the diamond\u0000tester in the $n=2$ case, as well as prove local correction results for direct\u0000sum as conjectured by Dinur and Golubev.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142264887","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

Complexity and algorithms for Swap median and relation to other consensus problems 交换中值的复杂性和算法以及与其他共识问题的关系

arXiv - CS - Computational Complexity Pub Date : 2024-09-15 DOI: arxiv-2409.09734

Luís Cunha, Thiago Lopes, Arnaud Mary

{"title":"Complexity and algorithms for Swap median and relation to other consensus problems","authors":"Luís Cunha, Thiago Lopes, Arnaud Mary","doi":"arxiv-2409.09734","DOIUrl":"https://doi.org/arxiv-2409.09734","url":null,"abstract":"Genome rearrangements are events in which large blocks of DNA exchange pieces\u0000during evolution. The analysis of such events is a tool for understanding\u0000evolutionary genomics, based on finding the minimum number of rearrangements to\u0000transform one genome into another. In a general scenario, more than two genomes\u0000are considered and we have new challenges. The {sc Median} problem consists in\u0000finding, given three permutations and a distance metric, a permutation $s$ that\u0000minimizes the sum of the distances between $s$ and each input. We study the\u0000{sc median} problem over emph{swap} distances in permutations, for which the\u0000computational complexity has been open for almost 20 years (Eriksen,\u0000emph{Theor. Compt. Sci.}, 2007). We consider this problem through some\u0000branches. We associate median solutions and interval convex sets, where the\u0000concept of graph convexity inspires the following investigation: Does a median\u0000permutation belong to every shortest path between one of the pairs of input\u0000permutations? We are able to partially answer this question, and as a\u0000by-product we solve a long open problem by proving that the {sc Swap Median}\u0000problem is NP-hard. Furthermore, using a similar approach, we show that the\u0000{sc Closest} problem, which seeks to minimize the maximum distance between the\u0000solution and the input permutations, is NP-hard even considering three input\u0000permutations. This gives a sharp dichotomy into the P vs. NP-hard approaches,\u0000since considering two input permutations the problem is easily solvable and\u0000considering any number of input permutations it is known to be NP-hard since\u00002007 (Popov, emph{Theor. Compt. Sci.}, 2007). In addition, we show that {sc\u0000Swap Median} and {sc Swap Closest} are APX-hard problems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142264891","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

Journalists, Emotions, and the Introduction of Generative AI Chatbots: A Large-Scale Analysis of Tweets Before and After the Launch of ChatGPT 记者、情感与生成式人工智能聊天机器人的引入：对 ChatGPT 推出前后推文的大规模分析

arXiv - CS - Computational Complexity Pub Date : 2024-09-13 DOI: arxiv-2409.08761

Seth C. Lewis, David M. Markowitz, Jon Benedik Bunquin

引用次数: 0

Almost-catalytic Computation 几乎催化的计算

arXiv - CS - Computational Complexity Pub Date : 2024-09-11 DOI: arxiv-2409.07208

Sagar Bisoyi, Krishnamoorthy Dinesh, Bhabya Deep Rai, Jayalal Sarma

{"title":"Almost-catalytic Computation","authors":"Sagar Bisoyi, Krishnamoorthy Dinesh, Bhabya Deep Rai, Jayalal Sarma","doi":"arxiv-2409.07208","DOIUrl":"https://doi.org/arxiv-2409.07208","url":null,"abstract":"Designing algorithms for space bounded models with restoration requirements\u0000on the space used by the algorithm is an important challenge posed about the\u0000catalytic computation model introduced by Buhrman et al. (2014). Motivated by\u0000the scenarios where we do not need to restore unless is useful, we define\u0000$ACL(A)$ to be the class of languages that can be accepted by almost-catalytic\u0000Turing machines with respect to $A$ (which we call the catalytic set), that\u0000uses at most $clog n$ work space and $n^c$ catalytic space. We show that if there are almost-catalytic algorithms for a problem with\u0000catalytic set as $A subseteq Sigma^*$ and its complement respectively, then\u0000the problem can be solved by a ZPP algorithm. Using this, we derive that to\u0000design catalytic algorithms, it suffices to design almost-catalytic algorithms\u0000where the catalytic set is the set of strings of odd weight ($PARITY$). Towards\u0000this, we consider two complexity measures of the set $A$ which are maximized\u0000for $PARITY$ - random projection complexity (${cal R}(A)$) and the subcube\u0000partition complexity (${cal P}(A)$). By making use of error-correcting codes, we show that for all $k ge 1$,\u0000there is a language $A_k subseteq Sigma^*$ such that $DSPACE(n^k) subseteq\u0000ACL(A_k)$ where for every $m ge 1$, $mathcal{R}(A_k cap {0,1}^m) ge\u0000frac{m}{4}$ and $mathcal{P}(A_k cap {0,1}^m)=2^{m/4}$. This contrasts the\u0000catalytic machine model where it is unclear if it can accept all languages in\u0000$DSPACE(log^{1+epsilon} n)$ for any $epsilon > 0$. Improving the partition complexity of the catalytic set $A$ further, we show\u0000that for all $k ge 1$, there is a $A_k subseteq {0,1}^*$ such that\u0000$mathsf{DSPACE}(log^k n) subseteq ACL(A_k)$ where for every $m ge 1$,\u0000$mathcal{R}(A_k cap {0,1}^m) ge frac{m}{4}$ and $mathcal{P}(A_k cap\u0000{0,1}^m)=2^{m/4+Omega(log m)}$.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185868","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

Fast Simulation of Cellular Automata by Self-Composition 通过自组合快速模拟细胞自动机

arXiv - CS - Computational Complexity Pub Date : 2024-09-11 DOI: arxiv-2409.07065

Joseph Natal, Oleksiy Al-saadi

引用次数: 0

Fully Characterizing Lossy Catalytic Computation 全面描述有损催化计算

arXiv - CS - Computational Complexity Pub Date : 2024-09-08 DOI: arxiv-2409.05046

Marten Folkertsma, Ian Mertz, Florian Speelman, Quinten Tupker

{"title":"Fully Characterizing Lossy Catalytic Computation","authors":"Marten Folkertsma, Ian Mertz, Florian Speelman, Quinten Tupker","doi":"arxiv-2409.05046","DOIUrl":"https://doi.org/arxiv-2409.05046","url":null,"abstract":"A catalytic machine is a model of computation where a traditional\u0000space-bounded machine is augmented with an additional, significantly larger,\u0000\"catalytic\" tape, which, while being available as a work tape, has the caveat\u0000of being initialized with an arbitrary string, which must be preserved at the\u0000end of the computation. Despite this restriction, catalytic machines have been\u0000shown to have surprising additional power; a logspace machine with a polynomial\u0000length catalytic tape, known as catalytic logspace ($CL$), can compute problems\u0000which are believed to be impossible for $L$. A fundamental question of the model is whether the catalytic condition, of\u0000leaving the catalytic tape in its exact original configuration, is robust to\u0000minor deviations. This study was initialized by Gupta et al. (2024), who\u0000defined lossy catalytic logspace ($LCL[e]$) as a variant of $CL$ where we allow\u0000up to $e$ errors when resetting the catalytic tape. They showed that $LCL[e] =\u0000CL$ for any $e = O(1)$, which remains the frontier of our understanding. In this work we completely characterize lossy catalytic space\u0000($LCSPACE[s,c,e]$) in terms of ordinary catalytic space ($CSPACE[s,c]$). We\u0000show that $$LCSPACE[s,c,e] = CSPACE[Theta(s + e log c), Theta(c)]$$ In other\u0000words, allowing $e$ errors on a catalytic tape of length $c$ is equivalent, up\u0000to a constant stretch, to an equivalent errorless catalytic machine with an\u0000additional $e log c$ bits of ordinary working memory. As a consequence, we show that for any $e$, $LCL[e] = CL$ implies $SPACE[e\u0000log n] subseteq ZPP$, thus giving a barrier to any improvement beyond\u0000$LCL[O(1)] = CL$. We also show equivalent results for non-deterministic and\u0000randomized catalytic space.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185870","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

A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity 量子鸽洞原理和通信复杂性的两种半无限松弛

arXiv - CS - Computational Complexity Pub Date : 2024-09-06 DOI: arxiv-2409.04592

Pavel Dvořák, Bruno Loff, Suhail Sherif

引用次数: 0

Two-Sided Lossless Expanders in the Unbalanced Setting 非平衡设置中的双面无损扩展器

arXiv - CS - Computational Complexity Pub Date : 2024-09-06 DOI: arxiv-2409.04549

Eshan Chattopadhyay, Mohit Gurumukhani, Noam Ringach, Yunya Zhao

引用次数: 0

Query complexity lower bounds for local list-decoding and hard-core predicates (even for small rate and huge lists) 本地列表解码和硬核谓词的查询复杂度下限（即使对于小速率和大列表也是如此）

arXiv - CS - Computational Complexity Pub Date : 2024-09-03 DOI: arxiv-2409.01708

Noga Ron-Zewi, Ronen Shaltiel, Nithin Varma

{"title":"Query complexity lower bounds for local list-decoding and hard-core predicates (even for small rate and huge lists)","authors":"Noga Ron-Zewi, Ronen Shaltiel, Nithin Varma","doi":"arxiv-2409.01708","DOIUrl":"https://doi.org/arxiv-2409.01708","url":null,"abstract":"A binary code Enc$:{0,1}^k to {0,1}^n$ is $(0.5-epsilon,L)$-list\u0000decodable if for all $w in {0,1}^n$, the set List$(w)$ of all messages $m\u0000in {0,1}^k$ such that the relative Hamming distance between Enc$(m)$ and $w$\u0000is at most $0.5 -epsilon$, has size at most $L$. Informally, a $q$-query local\u0000list-decoder for Enc is a randomized procedure Dec$:[k]times [L] to {0,1}$\u0000that when given oracle access to a string $w$, makes at most $q$ oracle calls,\u0000and for every message $m in text{List}(w)$, with high probability, there\u0000exists $j in [L]$ such that for every $i in [k]$, with high probability,\u0000Dec$^w(i,j)=m_i$. We prove lower bounds on $q$, that apply even if $L$ is huge (say\u0000$L=2^{k^{0.9}}$) and the rate of Enc is small (meaning that $n ge 2^{k}$): 1. For $epsilon geq 1/k^{nu}$ for some universal constant $0< nu < 1$, we\u0000prove a lower bound of $q=Omega(frac{log(1/delta)}{epsilon^2})$, where\u0000$delta$ is the error probability of the local list-decoder. This bound is\u0000tight as there is a matching upper bound by Goldreich and Levin (STOC 1989) of\u0000$q=O(frac{log(1/delta)}{epsilon^2})$ for the Hadamard code (which has\u0000$n=2^k$). This bound extends an earlier work of Grinberg, Shaltiel and Viola\u0000(FOCS 2018) which only works if $n le 2^{k^{gamma}}$ for some universal\u0000constant $0<gamma <1$, and the number of coins tossed by Dec is small (and\u0000therefore does not apply to the Hadamard code, or other codes with low rate). 2. For smaller $epsilon$, we prove a lower bound of roughly $q =\u0000Omega(frac{1}{sqrt{epsilon}})$. To the best of our knowledge, this is the\u0000first lower bound on the number of queries of local list-decoders that gives $q\u0000ge k$ for small $epsilon$. We also prove black-box limitations for improving some of the parameters of\u0000the Goldreich-Levin hard-core predicate construction.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185873","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

Partial and weighted matrix multiplication 部分矩阵乘法和加权矩阵乘法

arXiv - CS - Computational Complexity Pub Date : 2024-08-28 DOI: arxiv-2408.15728

Péter Vrana

引用次数: 0