  The Min-sum single machine scheduling problem (denoted 1||sum f_j)
generalizes a large number of sequencing problems. The first constant
approximation guarantees have been obtained only recently and are based on
natural time-indexed LP relaxations strengthened with the so called
Knapsack-Cover inequalities (see Bansal and Pruhs, Cheung and Shmoys and the
recent 4+\epsilon-approximation by Mestre and Verschae). These relaxations have
an integrality gap of 2, since the Min-knapsack problem is a special case. No
APX-hardness result is known and it is still conceivable that there exists a
PTAS. Interestingly, the Lasserre hierarchy relaxation, when the objective
function is incorporated as a constraint, reduces the integrality gap for the
Min-knapsack problem to 1+\epsilon.
  In this paper we study the complexity of the Min-sum single machine
scheduling problem under algorithms from the Lasserre hierarchy. We prove the
first lower bound for this model by showing that the integrality gap is
unbounded at level \Omega(\sqrt{n}) even for a variant of the problem that is
solvable in O(n log n) time by the Moore-Hodgson algorithm, namely Min-number
of tardy jobs. We consider a natural formulation that incorporates the
objective function as a constraint and prove the result by partially
diagonalizing the matrix associated with the relaxation and exploiting this
characterization.
