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.

中文翻译：

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典型的Euclidean TSP实例的梳状不等式

我们证明，即使在一般情况下，当Held-Karp线性规划松弛因有界大小的所有梳形不等式而增加时，欧氏旅行推销员问题在$ \ epsilon> 0 $的情况下仍表现出$（1+ \ epsilon）$的完整性缺口。这意味着即使是在随机输入上，大类的分支剪切算法也需要花费大量时间来处理Euclidean TSP。