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The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure forconstructing a sequence of increasingly tight semidefinite relaxations. It isknown that the hierarchy converges to the 0/1 polytope in n levels and capturesthe convex relaxations used in the best available approximation algorithms fora wide variety of optimization problems.  In this paper we characterize the set of 0/1 integer linear problems andunconstrained 0/1 polynomial optimization problems that can still have anintegrality gap at level n-1. These problems are the hardest for the Lasserrehierarchy in this sense.

On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy