The UGC hardness threshold of the ℓp Grothendieck problem

For <i>p</i> ≥ 2 we consider the problem of, given an <i>n</i> × <i>n</i> matrix <i>A</i> = (<i>a_ij</i>) whose diagonal entries vanish, approximating in polynomial time the number {display equation} (where optimization is taken over real numbers). When <i>p</i> = 2 this is simply the problem of computing the maximum eigenvalue of <i>A</i>, while for <i>p</i> = ∞ (actually it suffices to take <i>p</i> ≈ log <i>n</i>) it is the Grothendieck problem on the complete graph, which was shown to have a <i>O</i>(log <i>n</i>) 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 Opt_<i>p</i> (<i>A</i>) interpolates between the spectral (<i>p</i> = 2) case and the Correlation Clustering (<i>p</i> = ∞) 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 <i>p</i> ≥ 2 and an <i>n x n</i> matrix <i>A</i> = (<i>a_ij</i>) with zeros on the diagonal, computes Opt_<i>p</i> (<i>A</i>) up to a factor <i>p/e</i> + 30 log <i>p.</i> 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 <i>p/e</i> + 1/4. Hence as <i>p</i> → ∞ the UGC-hardness threshold for computing Opt_<i>p</i> (<i>A</i>) is exactly <i>p/e</i> (1 + <i>o</i>(1)).