Let $d$ be a positive integer. For a finite set $X \subseteq \mathbb{R}^d$, we define its integer cone as the set $\mathsf{IntCone}(X) := \{ \sum_{x \in X} \lambda_x \cdot x \mid \lambda_x \in \mathbb{Z}_{\geq 0} \} \subseteq \mathbb{R}^d$. 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.

翻译：设 $d$ 为正整数。对于有限集合 $X \subseteq \mathbb{R}^d$，我们定义其整数锥为集合 $\mathsf{IntCone}(X) := \{ \sum_{x \in X} \lambda_x \cdot x \mid \lambda_x \in \mathbb{Z}_{\geq 0} \} \subseteq \mathbb{R}^d$。Goemans 和 Rothvoss 证明了，给定两个多胞体 $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^d$（其中 $\mathcal{P}$ 有界），可以在时间 $\mathsf{enc}(\mathcal{P})^{2^{\mathcal{O}(d)}} \cdot \mathsf{enc}(\mathcal{Q})^{\mathcal{O}(1)}$ 内判定 $\mathsf{IntCone}(\mathcal{P} \cap \mathbb{Z}^d)$ 是否与 $\mathcal{Q}$ 相交 [J. ACM 2020]，其中 $\mathsf{enc}(\cdot)$ 表示通过线性不等式系统编码一个多胞体所需的比特数。该结果是以不同物品数量为参数的 BIN PACKING 问题 XP 算法的基础。我们通过提供一个条件性下界来补充他们的结果。具体而言，我们证明，除非 ETH 失效，否则不存在算法能在时间 $\mathsf{enc}(\mathcal{P}, q)^{2^{o(d)}}$ 内，给定有界多胞体 $\mathcal{P} \subseteq \mathbb{R}^d$ 和点 $q \in \mathbb{Z}^d$，判定 $q \in \mathsf{IntCone}(\mathcal{P} \cap \mathbb{Z}^d)$。注意，这并不排除该问题存在固定参数可解算法，但表明运行时间对参数 $d$ 的依赖至少是双指数的。