Hardness of Approximation for Shortest Path with Vector Costs

TL;DR

This paper establishes new hardness of approximation results for the vector-cost shortest path problem across various p-norms, nearly matching existing algorithms and providing the first such bounds for p<infinity.

Contribution

It introduces the first known hardness of approximation results for the $ ext{l}_p$-Shortest Path problem for all finite p, and refines bounds for the case p=∞, aligning with current algorithms.

Findings

Hardness of $ ilde heta(p(rac{ ext{log} n}{ ext{log}^2 ext{log} n})^{1-1/p})$ for $p eq ext{infinity}$.

Hardness of $ ilde heta( ext{log}^2 n)$ for $p = ext{infinity}$.

Nearly matching approximation algorithms for all considered cases.

Abstract

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.