NP-hardness of approximately solving linear equations over reals

Subhash Khot, Dana Moshkovitz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationSTOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing
Pages413-419
Number of pages7
DOIs
StatePublished - 2011
Event43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States
Duration: Jun 6 2011Jun 8 2011

Other

Other43rd ACM Symposium on Theory of Computing, STOC'11
CountryUnited States
CitySan Jose, CA
Period6/6/116/8/11

Fingerprint

Linear equations
Hardness
Testing

Keywords

  • hardness of approximation
  • linear equations
  • PCP

ASJC Scopus subject areas

  • Software

Cite this

Khot, S., & Moshkovitz, D. (2011). NP-hardness of approximately solving linear equations over reals. In 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.

STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing. 2011. p. 413-419.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Khot, S & Moshkovitz, D 2011, NP-hardness of approximately solving linear equations over reals. in 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
Khot S, Moshkovitz D. NP-hardness of approximately solving linear equations over reals. In STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing. 2011. p. 413-419 https://doi.org/10.1145/1993636.1993692
Khot, Subhash ; Moshkovitz, Dana. / NP-hardness of approximately solving linear equations over reals. STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing. 2011. pp. 413-419
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