Approximating Matrix p-norms

We consider the problem of computing the q bar right arrow p norm of a matrix A, which is defined for p, q >= 1, as parallel to A parallel to(qbar right arrowp) = max (x not equal 0 ->) parallel to Ax parallel to(p)/parallel to x parallel to(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 = infinity). 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 bar right arrow 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 l(p) norm. This makes constructive a recent existential result of Englert and Racke [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 is not an element of DTIME(n(polylog(n))), the problem cannot be approximated to a factor 2((log n)1-epsilon), for any constant epsilon > 0.