Pub Date : 2017-11-03DOI: 10.1007/s00037-020-00197-5
E. Hemaspaandra, L. Hemaspaandra, Holger Spakowski, O. Watanabe
{"title":"The Robustness of LWPP and WPP, with an Application to Graph Reconstruction","authors":"E. Hemaspaandra, L. Hemaspaandra, Holger Spakowski, O. Watanabe","doi":"10.1007/s00037-020-00197-5","DOIUrl":"https://doi.org/10.1007/s00037-020-00197-5","url":null,"abstract":"","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":"29 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00037-020-00197-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45818080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-13DOI: 10.1007/s00037-017-0162-2
L. A. Goldberg, Heng Guo
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Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size nO(logn)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$n^{O(log n)}$$end{document} and O(polylog(n)) depth. Moreover, the depth of the circuit can be reduced to O(log2n) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix, when the given matrix is of the form A0+A1x1+⋯+Amxmdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$A_{0} + A_{1 }x_{1} + cdots + A_{m} x_{m}$$end{document}, for an arbitrary matrix A0 and rank-1 matrices A1,A2,⋯,Amdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$A_{1}, A_{2}, dots, A_{m}$$end{document}. This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.
给定同一地面集上的两个拟阵，拟阵交集问题要求找到一个最大大小的公共独立集。在线性拟阵的情况下，该问题有一个随机并行算法，但没有确定性算法。我们给出了该算法的几乎完全去随机化，这意味着线性拟阵相交问题是在准NC中。也就是说，它具有准多项式大小nO（logn）documentclass[12pt]｛minimum｝usepackage｛amsmath｝usepackage｛wasysym｝usecpackage｛｛amsfonts｝ucepackage｛s amssymb｝use package{amsbsy｝ usepackage{mathrsfs｝userpackage{upgek｝setlength｛doddsidemargin｝｛-69pt｝ begin｛document｝$n^｛O（logn）｝$end｛document｝和O（polylog（n）））深度的统一电路。此外，在零特征场的情况下，电路的深度可以减小到O（log2n）。这推广了二部分完全匹配问题的类似结果。我们的主要技术贡献是对两个拟阵的公共基族的孤立引理进行去随机化。针对Edmonds问题的一个特殊情况，即符号矩阵的奇异性检验，我们利用我们的隔离结果给出了一个拟多项式时间黑盒算法，当给定矩阵的形式为A0+A1x1+…+Amxmdocumentclass[12pt]｛minimal｝usepackage｛amsmath｝usepackage｛wasysym｝ usepackage{amsfonts}usepackage{amssymb｝usepackaging｛amsbsy｝ usepackage{mathrsfs｝usepackage｛upgek｝setlength｛doddsedmargin｝｛-69pt｝ begin｛document｝$A_｛0｝+A｛1｝x_｛任意矩阵A0和秩1矩阵A1、A2、A3，……，Amdocumentclass[12pt]｛minimum｝usepackage｛amsmath｝ usepackage{wasysym｝ use package｛｛amsfonts｝ usapackage{amssymb｝ userpackage{amsbsy｝use package{mathrsfs｝ user package{upgeek｝setlength｛oddsedmargin｝｛-69pt｝ begin｛document｝$A_｛1｝，A_｛。这也可以看作是对应行列式多项式的黑盒多项式恒等式测试算法。该结果的另一个结果是最大秩矩阵完成问题的确定性解。最后，我们用我们的结果找到了准NC中线性拟阵并集的一个确定性表示。
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Pub Date : 2017-03-22DOI: 10.1007/s00037-018-0175-5
Mika Göös, T. Pitassi, Thomas Watson
{"title":"Query-to-Communication Lifting for PNP","authors":"Mika Göös, T. Pitassi, Thomas Watson","doi":"10.1007/s00037-018-0175-5","DOIUrl":"https://doi.org/10.1007/s00037-018-0175-5","url":null,"abstract":"","PeriodicalId":51005,"journal":{"name":"Computational Complexity","volume":"28 1","pages":"113-144"},"PeriodicalIF":1.4,"publicationDate":"2017-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00037-018-0175-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46709610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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