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

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 $\rho_p(d)L^{o(d)}$ 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.