In recent years, an algorithm based on Lasserre semidefinite programming relaxation has been proposed towards efficiently solving the Unique Games problem. If the algorithm works, it would yield a disproval of the Unique Games Conjecture. On the other hand, constructing an integrality gap instance for the relaxation would indicate that the algorithm does not work.

In the speaker's opinion, the existence of integrality gap for the Lasserre relaxation is the most pressing question in approximability. The Lasserre relaxation has a dual view as a Sum-of-Squares proof system and the integrality gap for the former corresponds to a degree lower bound for the latter.

The talk proposes a candidate Lasserre integrality gap construction for the Unique Games problem. Specifically, a concrete instance of the Unique Games problem is constructed. The instance is shown to have a good SDP solution. The authors believe that the instance does not have a good integral solution, but are unable to prove so.

The construction is based on a suggestion in [K Moshkovitz, STOC'11] wherein the authors study the complexity of approximately solving a system of linear equations over reals and suggest it as an avenue towards a (positive) resolution of the Unique Games Conjecture.

Joint work with Dana Moshkovitz.