Tight hardness of the non-commutative Grothendieck problem. (English) Zbl 1387.68122

Briët, Jop; Regev, Oded; Saket, Rishi

Theory Comput. 13, Paper No. 15, 24 p. (2017).

Summary: We prove that for any \(\varepsilon > 0\) it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor \(1/2 + \varepsilon\), which matches the approximation ratio of the algorithm of A. Naor et al. [Theory Comput. 10, Paper No. 11, 257–295 (2014; Zbl 1302.68323)]. Our proof uses an embedding of \(\ell_2\) into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight NP-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture [S. Khot and A. Naor, Mathematika 55, No. 1–2, 129–165 (2009; Zbl 1195.68114)].

MSC:

68Q17	Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
32A70	Functional analysis techniques applied to functions of several complex variables
68W25	Approximation algorithms
90C22	Semidefinite programming

Keywords:

hardness of approximation; semidefinite programming; Grothendieck inequality

Citations:

Zbl 1302.68323; Zbl 1195.68114