$\ell_p$-Norm Maximization over Zonotopes Is W[1]-Hard

2026-08-16

AI Summary

This study investigates the maximization of the ℓp norm over zonotopes. Leveraging parameterized complexity theory and the Exponential Time Hypothesis, we establish W[1]-hardness and tight time lower bounds, thereby precluding specific efficient exact algorithms. Building on these hardness results, we propose a deterministic (1−ε)-approximation algorithm and prove its optimality. Furthermore, we extend these theoretical findings to the analysis of ReLU neural networks, providing new computational complexity foundations and an approximation framework for network verification and robustness evaluation. This work effectively bridges convex geometric optimization and deep learning safety research by translating fundamental complexity insights into practical tools for assessing neural network reliability.

Abstract

We study $\ell_p$-norm maximization over zonotopes given by rational generators, with input length $L$. For fixed $p=a/b>1$, the exact Turing baseline runs in $n^{O(d)}b^{O(d)}\mathrm{poly}(L)$ time, but fixed-parameter tractability in the ambient dimension $d$ was open [FGHS25]. We prove W[1]-hardness and, under the Exponential Time Hypothesis (ETH), exclude $ρ_p(d)L^{o(d)}$ time, even for $5$-sparse generators, by encoding binary CSP constraints with normalized positive cap generators. We also give a deterministic $(1-\varepsilon)$-approximation with $\varepsilon^{-(d-1)/2}$ dependence and, among algorithms with fixed-degree polynomial dependence on $L$, rule out $(1/\varepsilon)^{o(d)}$ dependence under ETH. Support-function duality transfers the results to positive-output two-layer ReLU networks.

Problem

• ell_p-norm maximization
• zonotopes
• fixed-parameter tractability
• W[1]-hardness

Innovation

• W[1]-hardness
• Exponential Time Hypothesis
• Zonotope
• Approximation algorithm
• ReLU networks