Hardness of Approximation for Shortest Path with Vector Costs

2025-10-23

AI Summary

This paper studies the approximation complexity of the Vector Cost Shortest Path problem under the ℓₚ norm. For integer p ∈ [2, ∞), it establishes nearly tight inapproximability lower bounds for the first time; for p = ∞, it improves the hardness to Õ(log² n), approaching the optimal upper bound. Technically, the work integrates PCP theory, combinatorial reductions, and asymptotic analysis of higher-order Bell numbers to construct a refined reduction framework. Key contributions are: (1) a lower bound of Ω(p(log n / log² log n)¹⁻¹/ᵖ), nearly matching the best-known algorithmic upper bound O(p(log n / log log n)¹⁻¹/ᵖ); (2) closing a long-standing gap in inapproximability lower bounds for finite p; and (3) substantially strengthening the inapproximability result for p = ∞, achieving a deeper alignment between theoretical hardness and algorithmic performance.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 $Ω(p(log n / log^2log n)^{1-1/p})$ for both polynomial- and quasi-polynomial-time approximation algorithms. This nearly matches the approximation factor of $O(p(log n / loglog n)^{1-1/p})$ 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 $ ildeΩ(log^2 n)$, improving upon the previous $ ildeΩ(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.

Problem

• Hardness of approximation for shortest path with vector costs
• Matching quasi-polynomial-time approximation algorithms for l_p-shortest path
• Improved hardness results for p=∞ case over previous bounds

Innovation

• Hardness bounds for vector shortest path
• Matching quasi-polynomial approximation algorithms
• Improved infinite norm case hardness results