arXiv - CS - Computational Complexity最新文献_第9页

arXiv - CS - Computational Complexity Pub Date : 2024-04-24 DOI: arxiv-2404.16562

J. Andres Montoya

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Complexity of Planar Graph Orientation Consistency, Promise-Inference, and Uniqueness, with Applications to Minesweeper Variants 平面图方向一致性、承诺推理和唯一性的复杂性，以及在扫雷变体中的应用

arXiv - CS - Computational Complexity Pub Date : 2024-04-22 DOI: arxiv-2404.14519

MIT Hardness Group, Della Hendrickson, Andy Tockman

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Tetris with Few Piece Types 只有几种棋子的俄罗斯方块

arXiv - CS - Computational Complexity Pub Date : 2024-04-16 DOI: arxiv-2404.10712

MIT Hardness Group, Erik D. Demaine, Holden Hall, Jeffery Li

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PSPACE-Hard 2D Super Mario Games: Thirteen Doors PSPACE-高难度 2D 超级马里奥游戏：十三道门

arXiv - CS - Computational Complexity Pub Date : 2024-04-16 DOI: arxiv-2404.10380

MIT Hardness Group, Hayashi Ani, Erik D. Demaine, Holden Hall, Matias Korman

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On the complexity of some cycle convexity parameters 论某些周期凸性参数的复杂性

arXiv - CS - Computational Complexity Pub Date : 2024-04-14 DOI: arxiv-2404.09236

Carlos V. G. C. Lima, Thiago Marcilon, Pedro Paulo de Medeiros

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Almost Optimal Time Lower Bound for Approximating Parameterized Clique, CSP, and More, under ETH 近似参数化 Clique、CSP 等的近似最优时间下限，ETH 下

arXiv - CS - Computational Complexity Pub Date : 2024-04-13 DOI: arxiv-2404.08870

Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, Kewen Wu

{"title":"Almost Optimal Time Lower Bound for Approximating Parameterized Clique, CSP, and More, under ETH","authors":"Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, Kewen Wu","doi":"arxiv-2404.08870","DOIUrl":"https://doi.org/arxiv-2404.08870","url":null,"abstract":"The Parameterized Inapproximability Hypothesis (PIH), which is an analog of\u0000the PCP theorem in parameterized complexity, asserts that, there is a constant\u0000$varepsilon> 0$ such that for any computable function\u0000$f:mathbb{N}tomathbb{N}$, no $f(k)cdot n^{O(1)}$-time algorithm can, on\u0000input a $k$-variable CSP instance with domain size $n$, find an assignment\u0000satisfying $1-varepsilon$ fraction of the constraints. A recent work by\u0000Guruswami, Lin, Ren, Sun, and Wu (STOC'24) established PIH under the\u0000Exponential Time Hypothesis (ETH). In this work, we improve the quantitative aspects of PIH and prove (under\u0000ETH) that approximating sparse parameterized CSPs within a constant factor\u0000requires $n^{k^{1-o(1)}}$ time. This immediately implies that, assuming ETH,\u0000finding a $(k/2)$-clique in an $n$-vertex graph with a $k$-clique requires\u0000$n^{k^{1-o(1)}}$ time. We also prove almost optimal time lower bounds for\u0000approximating $k$-ExactCover and Max $k$-Coverage. Our proof follows the blueprint of the previous work to identify a\u0000\"vector-structured\" ETH-hard CSP whose satisfiability can be checked via an\u0000appropriate form of \"parallel\" PCP. Using further ideas in the reduction, we\u0000guarantee additional structures for constraints in the CSP. We then leverage\u0000this to design a parallel PCP of almost linear size based on Reed-Muller codes\u0000and derandomized low degree testing.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140602041","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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Resolution Over Linear Equations: Combinatorial Games for Tree-like Size and Space 线性方程的解析：树状大小和空间的组合游戏

arXiv - CS - Computational Complexity Pub Date : 2024-04-12 DOI: arxiv-2404.08370

Svyatoslav Gryaznov, Sergei Ovcharov, Artur Riazanov

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Lifting with Inner Functions of Polynomial Discrepancy 用多项式差异内函数提升

arXiv - CS - Computational Complexity Pub Date : 2024-04-11 DOI: arxiv-2404.07606

Yahel Manor, Or Meir

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Optimal Communication Complexity of Chained Index 链式索引的最佳通信复杂度

arXiv - CS - Computational Complexity Pub Date : 2024-04-10 DOI: arxiv-2404.07026

Janani Sundaresan

{"title":"Optimal Communication Complexity of Chained Index","authors":"Janani Sundaresan","doi":"arxiv-2404.07026","DOIUrl":"https://doi.org/arxiv-2404.07026","url":null,"abstract":"We study the CHAIN communication problem introduced by Cormode et al. [ICALP\u00002019]. It is a generalization of the well-studied INDEX problem. For $kgeq 1$,\u0000in CHAIN$_{n,k}$, there are $k$ instances of INDEX, all with the same answer.\u0000They are shared between $k+1$ players as follows. Player 1 has the first string\u0000$X^1 in {0,1}^n$, player 2 has the first index $sigma^1 in [n]$ and the\u0000second string $X^2 in {0,1}^n$, player 3 has the second index $sigma^2 in\u0000[n]$ along with the third string $X^3 in {0,1}^n$, and so on. Player $k+1$\u0000has the last index $sigma^k in [n]$. The communication is one way from each\u0000player to the next, starting from player 1 to player 2, then from player 2 to\u0000player 3 and so on. Player $k+1$, after receiving the message from player $k$,\u0000has to output a single bit which is the answer to all $k$ instances of INDEX. It was proved that the CHAIN$_{n,k}$ problem requires $Omega(n/k^2)$\u0000communication by Cormode et al., and they used it to prove streaming lower\u0000bounds for approximation of maximum independent sets. Subsequently, it was used\u0000by Feldman et al. [STOC 2020] to prove lower bounds for streaming submodular\u0000maximization. However, these works do not get optimal bounds on the\u0000communication complexity of CHAIN$_{n,k}$, and in fact, it was conjectured by\u0000Cormode et al. that $Omega(n)$ bits are necessary, for any $k$. As our main result, we prove the optimal lower bound of $Omega(n)$ for\u0000CHAIN$_{n,k}$. This settles the open conjecture of Cormode et al. in the\u0000affirmative. The key technique is to use information theoretic tools to analyze\u0000protocols over the Jensen-Shannon divergence measure, as opposed to total\u0000variation distance. As a corollary, we get an improved lower bound for\u0000approximation of maximum independent set in vertex arrival streams through a\u0000reduction from CHAIN directly.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583040","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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Superpolynomial Lower Bounds for Smooth 3-LCCs and Sharp Bounds for Designs 光滑 3-LCC 的超多项式下界和设计的锐界

arXiv - CS - Computational Complexity Pub Date : 2024-04-09 DOI: arxiv-2404.06513

Pravesh K. Kothari, Peter Manohar

{"title":"Superpolynomial Lower Bounds for Smooth 3-LCCs and Sharp Bounds for Designs","authors":"Pravesh K. Kothari, Peter Manohar","doi":"arxiv-2404.06513","DOIUrl":"https://doi.org/arxiv-2404.06513","url":null,"abstract":"We give improved lower bounds for binary $3$-query locally correctable codes\u0000(3-LCCs) $C colon {0,1}^k rightarrow {0,1}^n$. Specifically, we prove: (1) If $C$ is a linear design 3-LCC, then $n geq 2^{(1 - o(1))sqrt{k} }$. A\u0000design 3-LCC has the additional property that the correcting sets for every\u0000codeword bit form a perfect matching and every pair of codeword bits is queried\u0000an equal number of times across all matchings. Our bound is tight up to a\u0000factor $sqrt{8}$ in the exponent of $2$, as the best construction of binary\u0000$3$-LCCs (obtained by taking Reed-Muller codes on $mathbb{F}_4$ and applying a\u0000natural projection map) is a design $3$-LCC with $n leq 2^{sqrt{8 k}}$. Up to\u0000a $sqrt{8}$ factor, this resolves the Hamada conjecture on the maximum\u0000$mathbb{F}_2$-codimension of a $4$-design. (2) If $C$ is a smooth, non-linear $3$-LCC with near-perfect completeness,\u0000then, $n geq k^{Omega(log k)}$. (3) If $C$ is a smooth, non-linear $3$-LCC with completeness $1 -\u0000varepsilon$, then $n geq tilde{Omega}(k^{frac{1}{2varepsilon}})$. In\u0000particular, when $varepsilon$ is a small constant, this implies a lower bound\u0000for general non-linear LCCs that beats the prior best $n geq\u0000tilde{Omega}(k^3)$ lower bound of [AGKM23] by a polynomial factor. Our design LCC lower bound is obtained via a fine-grained analysis of the\u0000Kikuchi matrix method applied to a variant of the matrix used in [KM23]. Our\u0000lower bounds for non-linear codes are obtained by designing a from-scratch\u0000reduction from nonlinear $3$-LCCs to a system of \"chain polynomial equations\":\u0000polynomial equations with similar structure to the long chain derivations that\u0000arise in the lower bounds for linear $3$-LCCs [KM23].","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583028","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