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 - PCP

KW - hardness of approximation

KW - linear equations

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

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 413

EP - 419

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

T2 - 43rd ACM Symposium on Theory of Computing, STOC'11

Y2 - 6 June 2011 through 8 June 2011

ER -