Inapproximability of Matrix \(\boldsymbol{p \rightarrow q}\) Norms

Abstract

We study the problem of computing the norm of a matrix , defined as . This problem generalizes the spectral norm of a matrix and the Grothendieck problem ( , ) and has been widely studied in various regimes. When , the problem exhibits a dichotomy: constant factor approximation algorithms are known if , and the problem is hard to approximate within almost polynomial factors when . The regime when , known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with and was studied by Barak et al. [Proceedings of the 44th Annual ACM Symposium on Theory of Computing, 2012, pp. 307–326], who gave subexponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the exponential time hypothesis. However, no NP-hardness of approximation is known for these problems for any . We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any with , is hard to approximate within assuming . En route to the above result, we also prove almost tight results for the case when with .