Parameterized Hardness of Zonotope Containment and Neural Network Verification

Neural networks with ReLU activations are a widely used model in machine learning. It is thus important to have a profound understanding of the properties of the functions computed by such networks. Recently, there has been increasing interest in the (parameterized) computational complexity of determining these properties. In this work, we close several gaps and resolve an open problem posed by Froese et al. [COLT '25] regarding the parameterized complexity of various problems related to network verification. In particular, we prove that, for all $\ell\ge 2$, deciding positivity (and thus surjectivity) of a function $f:\mathbb{R}^d\to\mathbb{R}$ computed by an $\ell$-layer ReLU network is W[$\ell-1$]-hard when parameterized by the input dimension $d$. The case $\ell=2$ implies that zonotope non-containment (a problem that is of independent interest in computational geometry, control theory, and robotics) is W[1]-hard with respect to the ambient dimension $d$. Moreover, we show that approximating the maximum within any multiplicative factor and computing the $L_p$-Lipschitz constant for $p\in(0,\infty]$ in $\ell$-layer networks is NP-hard and W[$\ell-1$]-hard with respect to $d$. For $\ell\ge 3$, approximating the $L_p$-Lipschitz constant is NP- and W[$\ell-2$]-hard. We further show that the above problems are NP- and W[$t$]-hard (for all $t\ge 1$) with respect to $\ell$ for constant $d$. Notably, our hardness results imply that the naive enumeration-based methods for these fundamental problems running in $n^{(\ell-1) d}\cdot\operatorname{poly}(N)$ time are all essentially optimal under the Exponential Time Hypothesis.