We obtain hardness of approximation results for the $\ell_p$-Shortest Path problem, a variant of the classic Shortest Path problem with vector costs. For every integer $p \in [2,\infty)$, we show a hardness of $\Omega(p(\log n / \log^2\log n)^{1-1/p})$ for both polynomial- and quasi-polynomial-time approximation algorithms. This nearly matches the approximation factor of $O(p(\log n / \log\log n)^{1-1/p})$ achieved by a quasi-polynomial-time algorithm of Makarychev, Ovsiankin, and Tani (ICALP 2025). No hardness of approximation results were previously known for any $p < \infty$. We also present results for the case where $p$ is a function of $n$. For $p = \infty$, we establish a hardness of $\tilde\Omega(\log^2 n)$, improving upon the previous $\tilde\Omega(\log n)$ hardness result. Our result nearly matches the $O(\log^2 n)$ approximation guarantee of the quasi-polynomial-time algorithm by Li, Xu, and Zhang (ICALP 2025). Finally, we present asymptotic bounds on higher-order Bell numbers, which might be of independent interest.

翻译：我们针对$\ell_p$-最短路径问题（经典最短路径问题的一种带向量成本的变体）获得了近似难度结果。对于每个整数$p \in [2,\infty)$，我们证明了多项式时间和拟多项式时间近似算法均具有$\Omega(p(\log n / \log^2\log n)^{1-1/p})$的近似难度。这一结果几乎匹配了Makarychev、Ovsiankin和Tani（ICALP 2025）提出的拟多项式时间算法所实现的$O(p(\log n / \log\log n)^{1-1/p})$近似因子。此前对于任意$p < \infty$均未有任何近似难度结果被提出。我们还针对$p$为$n$的函数的情形给出了相关结果。对于$p = \infty$的情况，我们建立了$\tilde\Omega(\log^2 n)$的近似难度，改进了先前$\tilde\Omega(\log n)$的难度结果。该结果几乎匹配了Li、Xu和Zhang（ICALP 2025）提出的拟多项式时间算法所具有的$O(\log^2 n)$近似保证。最后，我们给出了高阶贝尔数的渐近界，该结果可能具有独立的研究价值。