  We prove that even in average case, the Euclidean Traveling Salesman Problem
exhibits an integrality gap of $(1+\epsilon)$ for $\epsilon>0$ when the
Held-Karp Linear Programming relaxation is augmented by all comb inequalities
of bounded size. This implies that large classes of branch-and-cut algorithms
take exponential time for the Euclidean TSP, even on random inputs.
