Separating MAX 2-AND, MAX DI-CUT and MAX CUT

In 2008, Raghavendra’s paper “Optimal algorithms and inapproximability results for every CSP?” showed that assuming the Unique Games Conjecture (or at least that unique games is hard), for any CSP on a finite set of predicates, the optimal worst-case polynomial time approximation algorithm is to use a standard semidefinite program and then apply a rounding algorithm. However, since the set of potential rounding functions is extremely large, Raghavendra’s result does not tell us what the approximation ratio is for a given CSP. In fact, there are still several basic questions which are still open. For example, the question of whether there is a 7/8 approximation ratio for MAX SAT (where the clauses can have any length) is still open.