License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.20
URN: urn:nbn:de:0030-drops-140894
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14089/
Bafna, Mitali ;
Vyas, Nikhil
Optimal Fine-Grained Hardness of Approximation of Linear Equations
Abstract
The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system (A,b), for A ∈ ℝ^{n×n} and b ∈ ℝⁿ, we wish to find a vector x ∈ ℝⁿ such that Ax = b. The current best algorithms for solving dense linear systems reduce the problem to matrix multiplication, and run in time O(n^ω). We consider the problem of finding ε-approximate solutions to linear systems with respect to the L₂-norm, that is, given a satisfiable linear system (A ∈ ℝ^{n×n}, b ∈ ℝⁿ), find an x ∈ ℝⁿ such that ||Ax - b||₂ ≤ ε||b||₂. Our main result is a fine-grained reduction from computing the rank of a matrix to finding ε-approximate solutions to linear systems. In particular, if the best known Õ(n^ω) time algorithm for computing the rank of n × O(n) matrices is optimal (which we conjecture is true), then finding an ε-approximate solution to a dense linear system also requires Ω̃(n^ω) time, even for ε as large as (1 - 1/poly(n)). We also prove (under some modified conjectures for the rank-finding problem) optimal hardness of approximation for sparse linear systems, linear systems over positive semidefinite matrices and well-conditioned linear systems. At the heart of our results is a novel reduction from the rank problem to a decision version of the approximate linear systems problem. This reduction preserves properties such as matrix sparsity and bit complexity.
BibTeX - Entry
@InProceedings{bafna_et_al:LIPIcs.ICALP.2021.20,
author = {Bafna, Mitali and Vyas, Nikhil},
title = {{Optimal Fine-Grained Hardness of Approximation of Linear Equations}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {20:1--20:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-195-5},
ISSN = {1868-8969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14089},
URN = {urn:nbn:de:0030-drops-140894},
doi = {10.4230/LIPIcs.ICALP.2021.20},
annote = {Keywords: Linear Equations, Fine-Grained Complexity, Hardness of Approximation}
}
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Keywords: |
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Linear Equations, Fine-Grained Complexity, Hardness of Approximation |
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Collection: |
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48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.07.2021 |
