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Completeness for First-order Properties on Sparse Structures with Algorithmic Applications

Authors:

Russell Impagliazzo,

Antonina Kolokolova,

Ryan WilliamsAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 15, Issue 2

Article No.: 23, Pages 1 - 35

https://doi.org/10.1145/3196275

Published: 18 December 2018 Publication History

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Abstract

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 property with k+1 quantifiers takes time O(m^k) and, assuming the Strong Exponential Time Hypothesis (SETH), some such properties require O(m^k−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 m^k/2^{Θ (√ 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 show that 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.

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Cited By

Huang ZLong YSaranurak TWang BSaha BServedio R(2023)Tight Conditional Lower Bounds for Vertex Connectivity ProblemsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585223(1384-1395)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585223
Papp PAnegg GYzelman AAgrawal KShun J(2023)Partitioning Hypergraphs is Hard: Models, Inapproximability, and ApplicationsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591087(415-425)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591087
Akmal SChen LJin CRaj MWilliams R(2023)Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained ComplexityAlgorithmica10.1007/s00453-023-01102-685:8(2395-2426)Online publication date: 17-Feb-2023
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Index Terms

Completeness for First-order Properties on Sparse Structures with Algorithmic Applications
1. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 15, Issue 2

Special Issue on Soda'17 and Regular Papers

April 2019

407 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3292530

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2018 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Publication History

Published: 18 December 2018

Accepted: 01 March 2018

Revised: 01 March 2018

Received: 01 February 2017

Published in TALG Volume 15, Issue 2

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Cited By

Huang ZLong YSaranurak TWang BSaha BServedio R(2023)Tight Conditional Lower Bounds for Vertex Connectivity ProblemsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585223(1384-1395)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585223
Papp PAnegg GYzelman AAgrawal KShun J(2023)Partitioning Hypergraphs is Hard: Models, Inapproximability, and ApplicationsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591087(415-425)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591087
Akmal SChen LJin CRaj MWilliams R(2023)Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained ComplexityAlgorithmica10.1007/s00453-023-01102-685:8(2395-2426)Online publication date: 17-Feb-2023
https://dl.acm.org/doi/10.1007/s00453-023-01102-6
Dell HLapinskas JMeeks K(2022)Approximately Counting and Sampling Small Witnesses Using a Colorful Decision OracleSIAM Journal on Computing10.1137/19M130604X51:4(849-899)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1137/19M130604X
Abboud ABringmann KHermelin DShabtay D(2022)Scheduling lower bounds via AND subset sumJournal of Computer and System Sciences10.1016/j.jcss.2022.01.005127:C(29-40)Online publication date: 1-Aug-2022
https://dl.acm.org/doi/10.1016/j.jcss.2022.01.005
Bläsius TFriedrich TLischeid JMeeks KSchirneck M(2022)Efficiently enumerating hitting sets of hypergraphs arising in data profilingJournal of Computer and System Sciences10.1016/j.jcss.2021.10.002124:C(192-213)Online publication date: 1-Mar-2022
https://dl.acm.org/doi/10.1016/j.jcss.2021.10.002
Nederlof JWęgrzycki KKhuller SVassilevska Williams V(2021)Improving Schroeppel and Shamir’s algorithm for subset sum via orthogonal vectorsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451024(1670-1683)Online publication date: 15-Jun-2021
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Birnick JBläsius TFriedrich TNaumann FPapenbrock TSchirneck M(2020)Hitting set enumeration with partial information for unique column combination discoveryProceedings of the VLDB Endowment10.14778/3407790.340782413:12(2270-2283)Online publication date: 14-Sep-2020
https://dl.acm.org/doi/10.14778/3407790.3407824

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