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：我们证明了当Held-Karp线性规划放松被所有大小有界的comb不等式所增强时，在平均情况下，欧几里得旅行商问题仍然存在一个$(1+\epsilon)$的完整度缺口，其中$\epsilon>0$。这意味着即使在随机输入上，大类的分支定界算法也需要指数时间来解决欧几里得旅行商问题。