Problem
Research questions and friction points this paper is trying to address.
Bin Packing
High-Multiplicity
Fixed-Parameter Tractable
Integer Linear Program
Configuration
High-Multiplicity Bin Packing is FPT
🤖 AI Summary
本文解决了高重数背包问题的固定参数可解性，通过构建一个整数线性规划模型并利用特定配置分类方法，在给定不同物品尺寸数量d的情况下设计了有效算法。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Fixed-Parameter Tractable (FPT)
Integer Linear Program (ILP)
High-Multiplicity Bin Packing
Convex Hull
Integer Decomposition Property
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