THEORY OF COMPUTING, Volume 10 (4), 2014, pp. 77–105 www.theoryofcomputing.org
Grothendieck Inequalities for Semidefinite Programs with Rank Constraint
Jop Briët∗ Fernando Mário de Oliveira Filho† Frank Vallentin‡
Received May 4, 2012; Revised January 21, 2014; Published May 31, 2014
Abstract: Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constraint is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give two applications: approximating ground states in the n-vector model in statistical mechanics and XOR games in quantum information theory.
ACM Classification: G.1.6, G.2.2 AMS Classification: 68W25, 90C22 Key words and phrases: randomized rounding, Grothendieck inequality, n-vector model, XOR games
1 Introduction