Tight hardness of the non-commutative Grothendieck problem

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 Naor, Regev, and Vidick (STOC'13). 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

Towards a better approximation for sparsest cut?

We give a new $(1+\epsilon)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-$r$ Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time $2^{O(r)} \mathrm{poly}(n)$. We also show similar approximation algorithms in graphs with genus $g$ with an analogous local expansion condition. This is the first algorithm we know of that achieves $(1+\epsilon)$-approximation on such general family of graphs

Tight hardness of the non-commutative Grothendieck problem

We prove that for any ε > 0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1=2+ε, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC’13). Our proof uses an embedding of ℓ2 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 (Khot and Naor, Mathematika 2009)