The Positive Semidefinite Grothendieck Problem with Rank Constraint

Abstract

Given a positive integer n and a positive semidefinite matrix A = (Aij) ∈ R m×m , the positive semidefinite Grothendieck problem with rank-n-constraint (SDPn) iswhere x1, . . . , xm ∈ S n−1 .In this paper we design a randomized polynomial-time approximation algorithm for SDPn achieving an approximation ratio of γ(n) = 2 n Γ ((n + 1)/2) Γ (n/2) 2 = 1 − Θ(1/n).We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial-time algorithm which approximates SDPn with a ratio greater than γ(n). We improve the approximation ratio of the best known polynomial-time algorithm for SDP1 from 2/π to 2/(πγ(m)) = 2/π + Θ(1/m), and we show a tighter approximation ratio for SDPn when A is the Laplacian matrix of a graph with nonnegative edge weights.

A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement

“…After the appearance of this work in preprint form, it was shown in [BOFV10b] that for positive semidefinite matrices M, like the ones considered here, the lower bound on K G (∞ → m) given in Theorem 1 is in fact optimal. Note that for general matrices, trivially, we have K G (n → m) ≤ K G (n).…”

Spectral bounds for the independence ratio and the chromatic number of an operator

“…Indeed, let T be a compact transitivity group of G. Then a = 1 V and K = R T (K) form a feasible solution of (8) …”

“…Set a = 1 V , so that D a = I, and K = λ −1 Z. Then we have that K respects G and that D a + K 0, so a and K are a feasible solution of (8).…”

“…To see the reverse inequality when G has a compact transitivity group T , let a and K be a feasible solution of (8). Notice a = 1 V and K = R T (K) are also feasible for (8), and ((…”

“…Computing the ϑ-number of a graph is a basic subroutine in the design of many approximation algorithms, see e.g. Karger, Motwani, and Sudan [20], Kleinberg, Goemans [21], Alon, Makarychev, Makarychev, and Naor [2], and Briët, Oliveira, and Vallentin [8]. Bachoc, Nebe, Oliveira, and Vallentin [6] generalized the ϑ-number from finite graphs to infinite graphs whose vertex sets are compact metric spaces.…”