Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hardness of Approximation for Shortest Path with Vector Costs

Charlie Carlson, Yury Makarychev, Ron Mosenzon

TL;DR

This work establishes strong hardness of approximation results for the vector-cost Shortest Path problem under the $\ell_p$ norm for all integer $p\ge 2$, including the $\ell_\infty$ case. The authors build a layered reduction from $r$-Hypergraph Label Cover via Feige’s $r$-prover system, constructing vector edge costs and applying tensoring to amplify gaps across series-parallel graphs of increasing depth. They derive tight or near-tight bounds in both series-parallel and general graphs, with $\Omega\big(p(\log n/\log^2\log n)^{1-1/p}\big)$ hardness for fixed $p$ and a $\tilde{\Omega}(\log^2 n)$ bound for $p=\infty$, aligning with recent quasi-polynomial-time algorithms. The paper further develops the combinatorics of higher-order Bell numbers to quantify the growth of the hardness gap and includes an $\ell_\infty$-case reduction using modified vector systems. Overall, the results significantly advance understanding of multi-objective/robust shortest-path problems and connect hardness to deep combinatorial and probabilistic constructs, with implications for fairness-aware network optimization.

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 $Ω(p(\log n / \log^2\log n)^{1-1/p})$ 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 $\tildeΩ(\log^2 n)$, improving upon the previous $\tildeΩ(\log n)$ 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.

Hardness of Approximation for Shortest Path with Vector Costs

TL;DR

This work establishes strong hardness of approximation results for the vector-cost Shortest Path problem under the norm for all integer , including the case. The authors build a layered reduction from -Hypergraph Label Cover via Feige’s -prover system, constructing vector edge costs and applying tensoring to amplify gaps across series-parallel graphs of increasing depth. They derive tight or near-tight bounds in both series-parallel and general graphs, with hardness for fixed and a bound for , aligning with recent quasi-polynomial-time algorithms. The paper further develops the combinatorics of higher-order Bell numbers to quantify the growth of the hardness gap and includes an -case reduction using modified vector systems. Overall, the results significantly advance understanding of multi-objective/robust shortest-path problems and connect hardness to deep combinatorial and probabilistic constructs, with implications for fairness-aware network optimization.

Abstract

We obtain hardness of approximation results for the -Shortest Path problem, a variant of the classic Shortest Path problem with vector costs. For every integer , we show a hardness of for both polynomial- and quasi-polynomial-time approximation algorithms. This nearly matches the approximation factor of achieved by a quasi-polynomial-time algorithm of Makarychev, Ovsiankin, and Tani (ICALP 2025). No hardness of approximation results were previously known for any . We also present results for the case where is a function of . For , we establish a hardness of , improving upon the previous hardness result. Our result nearly matches the 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.
Paper Structure (57 sections, 15 theorems, 108 equations, 1 figure)

This paper contains 57 sections, 15 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Results
  3. Near-optimality of our results
  4. Non-integer values of $p$
  5. Organization
  6. Preliminaries
  7. Hypergraph Label Cover
  8. Series-Parallel Graphs
  9. Inner Products
  10. Vector Costs
  11. Hardness Reduction for l2
  12. Basic Reduction
  13. Reduction
  14. Size Analysis
  15. Series-parallel graphs of order 2 and higher
  16. ...and 42 more sections

Key Result

Theorem 1.2

For every integer $p \geq 2$, the $\ell_p$-Shortest Path problem does not admit a constant factor approximation in series-parallel graphs if $P\neq NP$. Further, for every $\varepsilon > 0$, and integer parameters $p\geq 2$ and $k\geq 1$, the problem does not admit a ${\cal{B}}_k(p)^{1/p} - \varepsi

Figures (1)

  • Figure 1: This figure shows the basic reduction when $|L| = 4$. The red path corresponds to the assignment $\sigma$ with $\sigma(u_1) = 2$, $\sigma(u_2) = 3$, …, $\sigma(u_n) = 4$

Theorems & Definitions (62)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: $r$-Hypergraph Label Cover
  • Theorem 2.2: Hardness reduction for $r$-Hypergraph Label Cover, based on the reduction from BCMN
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • ...and 52 more