The greedy and nearest-neighbor TSP heuristics can both have $\log n$ approximation factors from optimal in worst case, even just for $n$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $d$-Ahlfor's regular metric space (which includes any $d$-manifold like the $d$-cube $[0,1]^d$ in the case $d$ is an integer but also fractals of dimension $d$ when $d$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have \emph{additive} errors from optimal on the order of the \emph{optimal} tour length through \emph{random} points in the same space, for $d>1$.

翻译：贪婪算法和最近邻TSP启发式算法在最坏情况下最优解的近似因子均可达到$\log n$，即使对于欧几里得空间中的$n$个点也是如此。本文指出，这种近似因子仅在最优路径异常短时才会出现。具体而言，对于任意固定的$d$-Ahlfor正则度量空间（当$d$为整数时包含$d$维流形如$d$立方体$[0,1]^d$，当$d$为实数时包含$d$维分形），我们的结果表明：对于$d>1$，当从同一空间中随机选取点时，贪婪算法和最近邻启发式算法相较于最优解的误差为\emph{加性}误差，且其数量级与\emph{最优}路径长度相当。