The positive semidefinite Grothendieck problem with rank constraint

TL;DR

This paper introduces a polynomial-time approximation algorithm for the positive semidefinite Grothendieck problem with rank constraints, achieving near-optimal ratios and improving previous bounds, with implications under the unique games conjecture.

Contribution

The paper presents a new approximation algorithm for SDP_n with a proven optimal ratio under the unique games conjecture and improves existing approximation bounds.

Findings

Achieves an approximation ratio of rac{2}{n}(rac{\u0393((n+1)/2)}{\u0393(n/2)})^2 for SDP_n.

Under the unique games conjecture, the approximation ratio is proven optimal.

Improves the approximation ratio for SDP_1 from 2/pi to 2/(pi 0c1; c1(m)).

Abstract

Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint (SDP_n) is maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation ratio of \gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 - \Theta(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 SDP_n with a ratio greater than \gamma(n). We improve the approximation ratio of the best known polynomial time algorithm for SDP_1 from 2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter approximation ratio for SDP_n when A is the Laplacian matrix of a graph with nonnegative edge weights.