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.

翻译：我们证明，即使在平均情况下，当Held-Karp线性规划松弛被所有有界大小的组合不等式增强时，欧几里得旅行商问题仍表现出$(1+\epsilon)$的积分间隙，其中$\epsilon>0$。这意味着，即使在随机输入上，对于欧几里得TSP，大规模的分支切割算法也需要指数时间。