Some optimal inapproximability results
Journal of the ACM (JACM)
Approximating the cut-norm via Grothendieck's inequality
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Hardness of Approximating the Shortest Vector Problem in Lattices
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
The UGC hardness threshold of the ℓp Grothendieck problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Optimal hierarchical decompositions for congestion minimization in networks
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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We consider the problem of computing the q ↦ p norm of a matrix A, which is defined for p, q ≥ 1, as [EQUATION] 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 ↦ 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 p, q with p ≤ q where constant factor approximations are not known). In this range, our techniques also show that if NP ∉ DTIME(npolylog(n)), the problem cannot be approximated to a factor 2(log n)1-ε, for any constant ε 0.