Approximating Matrix p-norms

We consider the problem of computing the q 7→ p norm of a matrix A, which is defined for p, q ≥ 1, as ‖A‖q 7→p = max x 6=~0 ‖Ax‖p ‖x‖q . This is in general a non-convex optimization problem, and is a natural generalization of the well-studied question of computing singular values (this corresponds to p = q = 2). Different settings of parameters give rise to a variety of known interesting problems (such as the Grothendieck problem when p = 1 and q = ∞). However, very little is understood about the approximability of the problem for different values of p, q. Our first result is an efficient algorithm for computing the q 7→ p norm of matrices with non-negative entries, when q ≥ p ≥ 1. The algorithm we analyze is based on a natural fixed point iteration, which can be seen as an analog of power iteration for computing eigenvalues. We then present an application of our techniques to the problem of constructing a scheme for oblivious routing in the lp norm. This makes constructive a recent existential result of Englert and Räcke [ER09] on O(log n) competitive oblivious routing schemes (which they make constructive only for p = 2). On the other hand, when we do not have any restrictions on the entries (such as non-negativity), we prove that the problem is NP-hard to approximate to any constant factor, for 2 < p ≤ q and p ≤ q < 2 (these are precisely the ranges of p, q with p ≤ q where constant factor approximations are not known). In this range, our techniques also show that if NP / ∈ DTIME(n), the problem cannot be approximated to a factor 2 1−ε , for any constant ε > 0. Center for Computational Intractability, and Department of Computer Science, Princeton University. Supported by NSF CCF 0832797. Email: [email protected] Center for Computational Intractability, and Department of Computer Science, Princeton University. Supported by NSF CCF 0832797. Email: [email protected]