Completeness for First-order Properties on Sparse Structures with Algorithmic Applications
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Revision2OfTR16-053.pdf
Description
Accepted version
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968.7 KB
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b31be9522f7b0edc34d3ac63d2bafb85
Author(s)
Gao, Jiawei
Impagliazzo, Russell
Kolokolova, Antonina
Williams, Ryan
Date Issued
2019
Journal
ACM Transactions on Algorithms
Publisher
Association for Computing Machinery (ACM)
Version
Author's final manuscript
Abstract
© 2018 Copyright held by the owner/author(s). Properties definable in first-order logic are algorithmically interesting for both theoretical and pragmatic reasons. Many of the most studied algorithmic problems, such as Hitting Set and Orthogonal Vectors, are first-order, and the first-order properties naturally arise as relational database queries. A relatively straightforward algorithm for evaluating a propertywith k + 1 quantifiers takes timeO(mk ) and, assuming the Strong Exponential Time Hypothesis (SETH), some such properties require O(mk-ϵ ) time for any ϵ > 0. (Here, m represents the size of the input structure, i.e., the number of tuples in all relations.) We give algorithms for every first-order property that improves this upper bound to mk /2Θ( √ log n) , i.e., an improvement by a factor more than any poly-log, but less than the polynomial required to refute SETH. Moreover,we showthat further improvement is equivalent to improving algorithms for sparse instances of the well-studied Orthogonal Vectors problem. Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of References [3, 16]. While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3196275
