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A Critique of Du's "A Polynomial-Time Algorithm for 3-SAT 对杜维明《3-SAT 的多项式时间算法》的批判
arXiv - CS - Computational Complexity Pub Date : 2024-04-05 DOI: arxiv-2404.04395
Yumeng He, Matan Kotler-Berkowitz, Harry Liuson, Zeyu Nie
{"title":"A Critique of Du's \"A Polynomial-Time Algorithm for 3-SAT","authors":"Yumeng He, Matan Kotler-Berkowitz, Harry Liuson, Zeyu Nie","doi":"arxiv-2404.04395","DOIUrl":"https://doi.org/arxiv-2404.04395","url":null,"abstract":"In this paper, we examine the claims made by the paper \"A polynomial-time\u0000algorithm for 3-SAT\" by Lizhi Du. The paper claims to provide a polynomial-time\u0000algorithm for solving the NP-complete problem 3-SAT. In examining the paper's\u0000argument, we find a flaw in one of the main sections of its algorithm. We argue\u0000that this flaw causes the paper's algorithm to incorrectly decide that an\u0000infinite family of satisfiable 3-CNF boolean formulas are not satisfiable.\u0000Therefore, the paper does not establish that P = NP.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583148","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
Random Circuit Sampling: Fourier Expansion and Statistics 随机电路采样：傅立叶展开和统计
arXiv - CS - Computational Complexity Pub Date : 2024-04-01 DOI: arxiv-2404.00935
Gil Kalai, Yosef Rinott, Tomer Shoham
{"title":"Random Circuit Sampling: Fourier Expansion and Statistics","authors":"Gil Kalai, Yosef Rinott, Tomer Shoham","doi":"arxiv-2404.00935","DOIUrl":"https://doi.org/arxiv-2404.00935","url":null,"abstract":"Considerable effort in experimental quantum computing is devoted to noisy\u0000intermediate scale quantum computers (NISQ computers). Understanding the effect\u0000of noise is important for various aspects of this endeavor including notable\u0000claims for achieving quantum supremacy and attempts to demonstrate quantum\u0000error correcting codes. In this paper we use Fourier methods combined with\u0000statistical analysis to study the effect of noise. In particular, we use\u0000Fourier analysis to refine the linear cross-entropy fidelity estimator. We use\u0000both analytical methods and simulations to study the effect of readout and gate\u0000errors, and we use our analysis to study the samples of Google's 2019 quantum\u0000supremacy experiment.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583144","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
Two prover perfect zero knowledge for MIP* MIP* 的两个完美零知识证明者
arXiv - CS - Computational Complexity Pub Date : 2024-04-01 DOI: arxiv-2404.00926
Kieran Mastel, William Slofstra
{"title":"Two prover perfect zero knowledge for MIP*","authors":"Kieran Mastel, William Slofstra","doi":"arxiv-2404.00926","DOIUrl":"https://doi.org/arxiv-2404.00926","url":null,"abstract":"The recent MIP*=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows\u0000that the complexity class MIP* of multiprover proof systems with entangled\u0000provers contains all recursively enumerable languages. Prior work of Grilo,\u0000Slofstra, and Yuen [FOCS '19] further shows (via a technique called simulatable\u0000codes) that every language in MIP* has a perfect zero knowledge (PZK) MIP*\u0000protocol. The MIP*=RE theorem uses two-prover one-round proof systems, and\u0000hence such systems are complete for MIP*. However, the construction in Grilo,\u0000Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect\u0000zero knowledge with two provers via simulatable codes. This leads to a natural\u0000question: are there two-prover PZK-MIP* protocols for all of MIP*? In this paper, we show that every language in MIP* has a two-prover one-round\u0000PZK-MIP* protocol, answering the question in the affirmative. For the proof, we\u0000use a new method based on a key consequence of the MIP*=RE theorem, which is\u0000that every MIP* protocol can be turned into a family of boolean constraint\u0000system (BCS) nonlocal games. This makes it possible to work with MIP* protocols\u0000as boolean constraint systems, and in particular allows us to use a variant of\u0000a construction due to Dwork, Feige, Kilian, Naor, and Safra [Crypto '92] which\u0000gives a classical MIP protocol for 3SAT with perfect zero knowledge. To show\u0000quantum soundness of this classical construction, we develop a toolkit for\u0000analyzing quantum soundness of reductions between BCS games, which we expect to\u0000be useful more broadly. This toolkit also applies to commuting operator\u0000strategies, and our argument shows that every language with a commuting\u0000operator BCS protocol has a two prover PZK commuting operator protocol.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583130","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 simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operations CSP 二分猜想和 XY 对称运算的简化证明
arXiv - CS - Computational Complexity Pub Date : 2024-04-01 DOI: arxiv-2404.01080
Dmitriy Zhuk
{"title":"A simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operations","authors":"Dmitriy Zhuk","doi":"arxiv-2404.01080","DOIUrl":"https://doi.org/arxiv-2404.01080","url":null,"abstract":"We develop a new theory of strong subalgebras and linear congruences that are\u0000defined globally. Using this theory we provide a new proof of the correctness\u0000of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a\u0000new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the\u0000new theory we prove that composing a weak near-unanimity operation of an odd\u0000arity $n$ we can derive an $n$-ary operation that is symmetric on all\u0000two-element sets. Thus, CSP over a constraint language $Gamma$ on a finite\u0000domain is tractable if and only if there exist infinitely many polymorphisms of\u0000$Gamma$ that are symmetric on all two-element sets.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583149","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 the Complexity of Minimizing Energy Consumption of Partitioning DAG Tasks 论分区 DAG 任务能耗最小化的复杂性
arXiv - CS - Computational Complexity Pub Date : 2024-04-01 DOI: arxiv-2404.01022
Wei Liu, Jian-Jia Chen, Yongjie Yang
{"title":"On the Complexity of Minimizing Energy Consumption of Partitioning DAG Tasks","authors":"Wei Liu, Jian-Jia Chen, Yongjie Yang","doi":"arxiv-2404.01022","DOIUrl":"https://doi.org/arxiv-2404.01022","url":null,"abstract":"We study a graph partition problem where we are given a directed acyclic\u0000graph (DAG) whose vertices and arcs can be respectively regarded as tasks and\u0000dependencies among tasks. The objective of the problem is to minimize the total\u0000energy consumed for completing these tasks by assigning the tasks to k\u0000heterogeneous machines. We first show that the problem is NP-hard. Then, we\u0000present polynomial-time algorithms for two special cases where there are only\u0000two machines and where the input DAG is a directed path. Finally, we study a\u0000natural variant where there are only two machines with one of them being\u0000capable of executing a limited number of tasks. We show that this special case\u0000remains computationally hard.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583134","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
No Complete Problem for Constant-Cost Randomized Communication 恒定成本随机通信没有完整问题
arXiv - CS - Computational Complexity Pub Date : 2024-03-31 DOI: arxiv-2404.00812
Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, Pooya Hatami
{"title":"No Complete Problem for Constant-Cost Randomized Communication","authors":"Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, Pooya Hatami","doi":"arxiv-2404.00812","DOIUrl":"https://doi.org/arxiv-2404.00812","url":null,"abstract":"We prove that the class of communication problems with public-coin randomized\u0000constant-cost protocols, called $BPP^0$, does not contain a complete problem.\u0000In other words, there is no randomized constant-cost problem $Q in BPP^0$,\u0000such that all other problems $P in BPP^0$ can be computed by a constant-cost\u0000deterministic protocol with access to an oracle for $Q$. We also show that the\u0000$k$-Hamming Distance problems form an infinite hierarchy within $BPP^0$.\u0000Previously, it was known only that Equality is not complete for $BPP^0$. We\u0000introduce a new technique, using Ramsey theory, that can prove lower bounds\u0000against arbitrary oracles in $BPP^0$, and more generally, we show that\u0000$k$-Hamming Distance matrices cannot be expressed as a Boolean combination of\u0000any constant number of matrices which forbid large Greater-Than subproblems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583143","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
Random Reed-Solomon Codes are List Recoverable with Optimal List Size 随机里德-所罗门码可通过最佳列表大小进行列表恢复
arXiv - CS - Computational Complexity Pub Date : 2024-03-30 DOI: arxiv-2404.00206
Dean Doron, S. Venkitesh
{"title":"Random Reed-Solomon Codes are List Recoverable with Optimal List Size","authors":"Dean Doron, S. Venkitesh","doi":"arxiv-2404.00206","DOIUrl":"https://doi.org/arxiv-2404.00206","url":null,"abstract":"We prove that Reed-Solomon (RS) codes with random evaluation points are list\u0000recoverable up to capacity with optimal output list size, for any input list\u0000size. Namely, given an input list size $ell$, a designated rate $R$, and any\u0000$varepsilon > 0$, we show that a random RS code is list recoverable from\u0000$1-R-varepsilon$ fraction of errors with output list size $L =\u0000O(ell/varepsilon)$, for field size $q=exp(ell,1/varepsilon) cdot n^2$. In\u0000particular, this shows that random RS codes are list recoverable beyond the\u0000``list recovery Johnson bound''. Such a result was not even known for arbitrary\u0000random linear codes. Our technique follows and extends the recent line of work\u0000on list decoding of random RS codes, specifically the works of Brakensiek,\u0000Gopi, and Makam (STOC 2023), and of Guo and Zhang (FOCS 2023).","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583035","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
Computational Complexity of the Recoverable Robust Shortest Path Problem with Discrete Recourse 具有离散追索权的可恢复稳健最短路径问题的计算复杂性
arXiv - CS - Computational Complexity Pub Date : 2024-03-29 DOI: arxiv-2403.20000
Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński
{"title":"Computational Complexity of the Recoverable Robust Shortest Path Problem with Discrete Recourse","authors":"Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński","doi":"arxiv-2403.20000","DOIUrl":"https://doi.org/arxiv-2403.20000","url":null,"abstract":"In this paper the recoverable robust shortest path problem is investigated.\u0000Discrete budgeted interval uncertainty representation is used to model\u0000uncertain second-stage arc costs. The known complexity results for this problem\u0000are strengthened. It is shown that it is Sigma_3^p-hard for the arc exclusion\u0000and the arc symmetric difference neighborhoods. Furthermore, it is also proven\u0000that the inner adversarial problem for these neighborhoods is Pi_2^p-hard.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583137","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
Local Correction of Linear Functions over the Boolean Cube 布尔立方上线性函数的局部修正
arXiv - CS - Computational Complexity Pub Date : 2024-03-29 DOI: arxiv-2403.20305
Prashanth Amireddy, Amik Raj Behera, Manaswi Paraashar, Srikanth Srinivasan, Madhu Sudan
{"title":"Local Correction of Linear Functions over the Boolean Cube","authors":"Prashanth Amireddy, Amik Raj Behera, Manaswi Paraashar, Srikanth Srinivasan, Madhu Sudan","doi":"arxiv-2403.20305","DOIUrl":"https://doi.org/arxiv-2403.20305","url":null,"abstract":"We consider the task of locally correcting, and locally list-correcting,\u0000multivariate linear functions over the domain ${0,1}^n$ over arbitrary fields\u0000and more generally Abelian groups. Such functions form error-correcting codes\u0000of relative distance $1/2$ and we give local-correction algorithms correcting\u0000up to nearly $1/4$-fraction errors making $widetilde{mathcal{O}}(log n)$\u0000queries. This query complexity is optimal up to $mathrm{poly}(loglog n)$\u0000factors. We also give local list-correcting algorithms correcting $(1/2 -\u0000varepsilon)$-fraction errors with $widetilde{mathcal{O}}_{varepsilon}(log\u0000n)$ queries. These results may be viewed as natural generalizations of the classical work\u0000of Goldreich and Levin whose work addresses the special case where the\u0000underlying group is $mathbb{Z}_2$. By extending to the case where the\u0000underlying group is, say, the reals, we give the first non-trivial locally\u0000correctable codes (LCCs) over the reals (with query complexity being sublinear\u0000in the dimension (also known as message length)). The central challenge in constructing the local corrector is constructing\u0000``nearly balanced vectors'' over ${-1,1}^n$ that span $1^n$ -- we show how to\u0000construct $mathcal{O}(log n)$ vectors that do so, with entries in each vector\u0000summing to $pm1$. The challenge to the local-list-correction algorithms, given\u0000the local corrector, is principally combinatorial, i.e., in proving that the\u0000number of linear functions within any Hamming ball of radius\u0000$(1/2-varepsilon)$ is $mathcal{O}_{varepsilon}(1)$. Getting this general\u0000result covering every Abelian group requires integrating a variety of known\u0000methods with some new combinatorial ingredients analyzing the structural\u0000properties of codewords that lie within small Hamming balls.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583043","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
Pseudoentanglement Ain't Cheap 伪量子纠缠并不便宜
arXiv - CS - Computational Complexity Pub Date : 2024-03-29 DOI: arxiv-2404.00126
Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang
{"title":"Pseudoentanglement Ain't Cheap","authors":"Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang","doi":"arxiv-2404.00126","DOIUrl":"https://doi.org/arxiv-2404.00126","url":null,"abstract":"We show that any pseudoentangled state ensemble with a gap of $t$ bits of\u0000entropy requires $Omega(t)$ non-Clifford gates to prepare. This bound is tight\u0000up to polylogarithmic factors if linear-time quantum-secure pseudorandom\u0000functions exist. Our result follows from a polynomial-time algorithm to\u0000estimate the entanglement entropy of a quantum state across any cut of qubits.\u0000When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli\u0000operators, our algorithm produces an estimate that is within an additive factor\u0000of $frac{t}{2}$ bits of the true entanglement entropy.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583042","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
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