发表机构
The University of Tokyo; CyberAgent, Inc.（东京大学; CyberAgent公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了高多样性装箱问题在物品类型数$d$上是否为FPT的开放问题，通过构造至多$(d+1)d^d$个变量的整数线性规划并利用配置类凸包的整数分解性质，给出了确定性$O^*(2^{d^{O(d)}})$时间算法。
AI 中文摘要
装箱问题询问一组物品能否装入给定容量的至多给定数量的箱子中。我们考虑高多样性设置，其中有$d$种不同的物品尺寸，物品尺寸和每种尺寸的物品数量都以二进制编码。Goemans和Rothvos（JACM 2020）给出了一个以$d$为参数的XP算法。该问题是否在$d$上具有固定参数可处理性（FPT）一直是一个核心开放问题。我们通过给出一个确定性的$O^*(2^{d^{O(d)}})$时间算法解决了这个问题。我们将装箱问题表述为一个整数线性规划（ILP），其变量数至多为$(d+1)d^d$。一个箱子配置记录一个箱子中每种类型的物品数量。我们根据这些配置的坐标模$d$的余数对其进行划分。对于每个类别，我们使用一个变量表示箱子数量，使用$d$个变量表示物品总数。每个类别的凸包具有整数分解性质，这保证了每个可行的ILP解都对应一个装箱方案。
英文摘要
Bin packing asks whether a collection of items can be packed into at most a given number of bins of a given capacity. We consider the high-multiplicity setting with $d$ distinct item sizes, in which both the item sizes and the number of items of each size are encoded in binary. Goemans and Rothvos (JACM 2020) gave an XP algorithm parameterized by $d$. Whether this problem is fixed-parameter tractable (FPT) in $d$ has remained a central open problem. We resolve this question by giving a deterministic $O^*(2^{d^{O(d)}})$-time algorithm. We formulate bin packing as an integer linear program (ILP) with at most $(d+1)d^d$ variables. A bin configuration records the number of items of each type in one bin. We partition these configurations by their coordinate remainders modulo $d$. For each class, we use one variable for the bin count and $d$ variables for the total item counts. The convex hull of each class has the integer decomposition property, which guarantees that every feasible ILP solution corresponds to a packing.
