An Approximation Algorithm for the Bipartite Grothendieck Problem

This problem is NP-hard, so there is no known algorithm to determine the solution in polynomial time. Instead we seek to compute an approximate solution in polynomial time. One strategy for accomplishing this, first pioneered by Goemans & Williamson in Ref. [3], is to employ a semidefinite relaxation. This technique consists of three steps: first construct a related problem that is a semidefinite program (and hence can be solved in polynomial time), then solve this problem, and finally develop a polynomial time procedure to round this solution back into a valid input to the original problem that achieves some fraction of the optimal objective value.