The UGC hardness threshold of the ep Grothendieck problem

Abstract

For p ≥ 2 we consider the problem of, given an n × n matrix A = (aij) whose diagonal entries vanish, approximating in polynomial time the number {display equation} (where optimization is taken over real numbers). When p = 2 this is simply the problem of computing the maximum eigenvalue of A, while for p = ∞ (actually it suffices to take p a log n) it is the Grothendieck problem on the complete graph, which was shown to have a O(log n) approximation algorithm in [27, 26, 15], and was used in [15] to design the best known algorithm for the problem of computing the maximum correlation in Correlation Clustering. Thus the problem of approximating Optp (A) interpolates between the spectral (p = 2) case and the Correlation Clustering (p = ∞) case. From a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard-wall potential well. We design a polynomial time algorithm which, given p ≥ 2 and an n x n matrix A = (aij) with zeros on the diagonal, computes Optp (A) up to a factor p/e + 30 log p. On the other hand, assuming the unique games conjecture (UGC) we show that it is NP-hard to approximate (1.2) up to a factor smaller than p/e + 1/4. Hence as p → ∞ the UGC-hardness threshold for computing Optp (A) is exactly p/e (1 + o(1)).