Detecting Points in Integer Cones of Polytopes is Double-Exponentially Hard

Let $d$ be a positive integer. For a finite set $X \subseteq \mathbb{R}^d$, we define its integer cone as the set $\mathsf{IntCone}(X) := \{ \sum_{x \in X} λ_x \cdot x \mid λ_x \in \mathbb{Z}_{\geq 0} \} \subseteq \mathbb{R}^d$. Goemans and Rothvoss showed that, given two polytopes $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^d$ with $\mathcal{P}$ being bounded, one can decide whether $\mathsf{IntCone}(\mathcal{P} \cap \mathbb{Z}^d)$ intersects $\mathcal{Q}$ in time $\mathsf{enc}(\mathcal{P})^{2^{\mathcal{O}(d)}} \cdot \mathsf{enc}(\mathcal{Q})^{\mathcal{O}(1)}$ [J. ACM 2020], where $\mathsf{enc}(\cdot)$ denotes the number of bits required to encode a polytope through a system of linear inequalities. This result is the cornerstone of their XP algorithm for BIN PACKING parameterized by the number of different item sizes. We complement their result by providing a conditional lower bound. In particular, we prove that, unless the ETH fails, there is no algorithm which, given a bounded polytope $\mathcal{P} \subseteq \mathbb{R}^d$ and a point $q \in \mathbb{Z}^d$, decides whether $q \in \mathsf{IntCone}(\mathcal{P} \cap \mathbb{Z}^d)$ in time $\mathsf{enc}(\mathcal{P}, q)^{2^{o(d)}}$. Note that this does not rule out the existence of a fixed-parameter tractable algorithm for the problem, but shows that dependence of the running time on the parameter $d$ must be at least doubly-exponential.