W[1] membership of zonotope norm maximization and ICNN Lipschitz problems

Determine whether exact L_p-norm maximization over generator-represented zonotopes in rational Euclidean space, including its equivalent problem of computing the L_p-Lipschitz constant of two-layer ReLU input-convex neural networks, belongs to the parameterized complexity class W[1] for every fixed rational p in (1,∞), thereby completing the parameterized complexity classification.

Background

The paper establishes W[1]-hardness, with respect to the input dimension, for exact L_p-norm maximization over generator-represented zonotopes for every fixed rational p in (1,∞). Under the Exponential Time Hypothesis, it also proves that no algorithm with running time of the form ρ(d)N^{o(d)} exists for any computable function ρ.

Through duality between zonotopes and bias-free two-layer ReLU input-convex neural networks, the same hardness and ETH lower bounds apply to computing their L_p-Lipschitz constants. These results show hardness but do not establish membership in W[1]. Proving such membership would settle the parameterized complexity status of both equivalent problems completely.