发表机构
Kiel University（基尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种构造性 FPT 算法，将 Koana-Kumabe 框架推广至完整锥与多面体交集问题，结合 Carathéodory 型整数锥界与活跃支撑集枚举，将运行时间降至 ETH 下最优的双指数参数依赖，并给出显式解压缩算法。
AI 中文摘要
在一篇里程碑式的论文中，Goemans 和 Rothvoss (2020) 为锥与多面体交集问题建立了一个 XP 算法，运行时间为 $\text{enc}(P)^{2^{O(d)}} \cdot \text{enc}(Q)^{O(1)}$，该问题旨在寻找一个向量 $y \in \text{ this http URL }(P \cap \mathbb{Z}^d) \cap Q$ 以及一个稀疏证书 $\lambda \in \mathbb{Z}_{\ge 0}^{P \cap \mathbb{Z}^d}$，其支撑集大小至多为 $2^{2d+1}$，其中 $P \subseteq \mathbb{R}^d$ 是有界有理多面体，$Q \subseteq \mathbb{R}^d$ 是任意有理多面体。对于高多样性装箱问题，这给出了 ${|I|}^{2^{O(d)}}$ 的运行时间，其中 $|I|$ 表示输入的编码长度。最近，Koana 和 Kumabe (2026) 证明了该问题的判定变体在参数化为物品类型数量 $d$ 时是固定参数可处理的 (FPT)，运行时间为 $2^{d^{O(d)}} \cdot {|I|}^{O(1)} = 2^{2^{O(d \log d)}} \cdot {|I|}^{O(1)}$。在这项工作中，我们将 Koana 和 Kumabe 的框架从标准装箱推广到 Goemans 和 Rothvoss 的完整锥与多面体交集问题，直接涵盖了高多样性装箱、锥内点和调度问题。其次，通过将 Carathéodory 型整数锥界 (Eisenbrand 和 Shmonin, 2006) 与活跃支撑集枚举相结合，我们将运行时间降低到：$2^{2^{O(d)}} \cdot (\text{enc}(P) + \text{enc}(Q))^{O(1)}$。在指数时间假设 (ETH) 下，Kowalik、Lassota、Majewski、Pilipczuk 和 Sokołowski (2024) 针对锥内点问题以及 Jansen、Ohnesorge 和 Pirotton (2026) 针对高多样性装箱问题的双指数下界意味着该参数依赖性是渐近最优的。最后，我们提供了一个显式的解压缩算法，该算法在单指数 FPT 时间内提取一个具有稀疏支撑集 $|\text{supp}(\lambda)| \le 2^{2d+1}$ 的解。
英文摘要
In a landmark paper, Goemans and Rothvoss (2020) established an XP algorithm running in time $\text{enc}(P)^{2^{O(d)}} \cdot \text{enc}(Q)^{O(1)}$ for the Cone and Polytope Intersection problem: finding a vector $y \in \operatorname{int{.}cone}(P \cap \mathbb{Z}^d) \cap Q$ together with a sparse certificate $λ\in \mathbb{Z}_{\ge 0}^{P \cap \mathbb{Z}^d}$ supported on at most $2^{2d+1}$ generators, where $P \subseteq \mathbb{R}^d$ is a bounded rational polyhedron and $Q \subseteq \mathbb{R}^d$ is an arbitrary rational polyhedron. For high-multiplicity bin packing, this gives a running time of ${|I|}^{2^{O(d)}}$, where $|I|$ denotes the encoding length of the input. Recently, Koana and Kumabe (2026) proved that the decision variant of this problem is fixed-parameter tractable (FPT) parameterized by the number of item types $d$ with running time $2^{d^{O(d)}} \cdot {|I|}^{O(1)} = 2^{2^{O(d \log d)}} \cdot {|I|}^{O(1)}$. In this work, we generalize the framework of Koana and Kumabe from standard bin packing to the full Cone and Polytope Intersection Problem of Goemans and Rothvoss, directly encompassing high-multiplicity bin packing, point-in-cone, and scheduling. Secondly, by combining Carathéodory-type integer cone bounds (Eisenbrand and Shmonin, 2006) with active support enumeration, we reduce the running time to: $$2^{2^{O(d)}} \cdot (\text{enc}(P) + \text{enc}(Q))^{O(1)}.$$ Under the Exponential Time Hypothesis (ETH), the double-exponential lower bound of Kowalik, Lassota, Majewski, Pilipczuk, and Sokołowski (2024) for point-in-cone and Jansen, Ohnesorge, and Pirotton (2026) for high-multiplicity bin packing implies that this parameter dependence is asymptotically optimal. Finally, we provide an explicit decompression algorithm that extracts a solution with sparse support $|\text{supp}(λ)| \le 2^{2d+1}$ in single-exponential FPT time.
