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 posted by Froese et al. [COLT '25] regarding the parameterized complexity of various problems related to network verification. In particular, we prove that deciding positivity (and thus surjectivity) of a function $f\colon\mathbb{R}^d\to\mathbb{R}$ computed by a 2-layer ReLU network is W[1]-hard when parameterized by $d$. This result also implies that zonotope (non-)containment is W[1]-hard with respect to $d$, a problem that is of independent interest in computational geometry, control theory, and robotics. Moreover, we show that approximating the maximum within any multiplicative factor in 2-layer ReLU networks, computing the $L_p$-Lipschitz constant for $p\in(0,\infty]$ in 2-layer networks, and approximating the $L_p$-Lipschitz constant in 3-layer networks are NP-hard and W[1]-hard with respect to $d$. Notably, our hardness results are the strongest known so far and imply that the naive enumeration-based methods for solving these fundamental problems are all essentially optimal under the Exponential Time Hypothesis.

翻译：具有ReLU激活函数的神经网络是机器学习中广泛使用的模型。因此，深入理解此类网络计算函数的性质至关重要。近年来，人们越来越关注确定这些性质的（参数化）计算复杂性。在本工作中，我们填补了若干空白，并解决了Froese等人[COLT '25]提出的一个关于网络验证相关问题的参数化复杂性的开放性问题。特别地，我们证明：对于由2层ReLU网络计算的函数$f\colon\mathbb{R}^d\to\mathbb{R}$，判定其正性（进而判定满射性）在参数$d$下是W[1]-难的。这一结果也蕴含了zonotope（非）包含问题在参数$d$下是W[1]-难的，而该问题在计算几何、控制理论和机器人学中具有独立意义。此外，我们证明：在2层ReLU网络中近似最大值的任意乘法因子、计算$p\in(0,\infty]$时的$L_p$-Lipschitz常数，以及在3层网络中近似$L_p$-Lipschitz常数，这些问题均是NP-难的且在参数$d$下是W[1]-难的。值得注意的是，我们的难度结果目前是已知最强的，并且意味着，在指数时间假设下，解决这些基本问题的基于朴素枚举的方法本质上是最优的。