  We give an optimal upper bound for the maximum-norm distance from a vertex of
a knapsack polyhedron to its nearest feasible lattice point. In a randomised
setting, we show that the upper bound can be significantly improved on average.
As a corollary, we obtain an optimal upper bound for the additive integrality
gap of integer knapsack problems and show that the integrality gap of a
"typical" knapsack problem is drastically smaller than the integrality gap that
occurs in a worst case scenario. We also prove that, in a generic case, the
integer programming gap admits a natural optimal lower bound.
