### Abstract

In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be "non-trivial". Here is an informal statement of our result: it is NP-hard to distinguish whether there is a non-trivial assignment that satisfies 1-δ fraction of the equations or every non-trivial assignment fails to satisfy a constant fraction of the equations with a "margin" of Ω(√δ). We develop linearity and dictatorship testing procedures for functions f: R ^{n} -> R over a Gaussian space, which could be of independent interest. We believe that studying the complexity of linear equations over reals, apart from being a natural pursuit, can lead to progress on the Unique Games Conjecture.

Original language | English (US) |
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Title of host publication | STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing |

Pages | 413-419 |

Number of pages | 7 |

DOIs | |

State | Published - 2011 |

Event | 43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States Duration: Jun 6 2011 → Jun 8 2011 |

### Other

Other | 43rd ACM Symposium on Theory of Computing, STOC'11 |
---|---|

Country | United States |

City | San Jose, CA |

Period | 6/6/11 → 6/8/11 |

### Fingerprint

### Keywords

- hardness of approximation
- linear equations
- PCP

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing*(pp. 413-419) https://doi.org/10.1145/1993636.1993692

**NP-hardness of approximately solving linear equations over reals.** / Khot, Subhash; Moshkovitz, Dana.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing.*pp. 413-419, 43rd ACM Symposium on Theory of Computing, STOC'11, San Jose, CA, United States, 6/6/11. https://doi.org/10.1145/1993636.1993692

}

TY - GEN

T1 - NP-hardness of approximately solving linear equations over reals

AU - Khot, Subhash

AU - Moshkovitz, Dana

PY - 2011

Y1 - 2011

N2 - In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be "non-trivial". Here is an informal statement of our result: it is NP-hard to distinguish whether there is a non-trivial assignment that satisfies 1-δ fraction of the equations or every non-trivial assignment fails to satisfy a constant fraction of the equations with a "margin" of Ω(√δ). We develop linearity and dictatorship testing procedures for functions f: R n -> R over a Gaussian space, which could be of independent interest. We believe that studying the complexity of linear equations over reals, apart from being a natural pursuit, can lead to progress on the Unique Games Conjecture.

AB - In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be "non-trivial". Here is an informal statement of our result: it is NP-hard to distinguish whether there is a non-trivial assignment that satisfies 1-δ fraction of the equations or every non-trivial assignment fails to satisfy a constant fraction of the equations with a "margin" of Ω(√δ). We develop linearity and dictatorship testing procedures for functions f: R n -> R over a Gaussian space, which could be of independent interest. We believe that studying the complexity of linear equations over reals, apart from being a natural pursuit, can lead to progress on the Unique Games Conjecture.

KW - hardness of approximation

KW - linear equations

KW - PCP

UR - http://www.scopus.com/inward/record.url?scp=79959709044&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959709044&partnerID=8YFLogxK

U2 - 10.1145/1993636.1993692

DO - 10.1145/1993636.1993692

M3 - Conference contribution

AN - SCOPUS:79959709044

SN - 9781450306911

SP - 413

EP - 419

BT - STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing

ER -