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 εapproximate solutions to linear systems with respect to the L₂norm, that is, given a satisfiable linear system (A ∈ ℝ^{n×n}, b ∈ ℝⁿ), find an x ∈ ℝⁿ such that Ax  b₂ ≤ εb₂. Our main result is a finegrained reduction from computing the rank of a matrix to finding εapproximate 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 εapproximate 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 rankfinding problem) optimal hardness of approximation for sparse linear systems, linear systems over positive semidefinite matrices and wellconditioned 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 FineGrained Hardness of Approximation of Linear Equations}},
booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages = {20:120:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771955},
ISSN = {18688969},
year = {2021},
volume = {198},
editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14089},
URN = {urn:nbn:de:0030drops140894},
doi = {10.4230/LIPIcs.ICALP.2021.20},
annote = {Keywords: Linear Equations, FineGrained Complexity, Hardness of Approximation}
}
Keywords: 

Linear Equations, FineGrained Complexity, Hardness of Approximation 
Collection: 

48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) 
Issue Date: 

2021 
Date of publication: 

02.07.2021 