On the upper bound of the generalization of $\mathsf{FFD}$ to solve $q$BP for some special cases

2026-07-12

AI Summary

This study addresses the $q$-Bin Packing ($q$BP) problem with item replication and placement constraints, where the objective is to minimize the number of bins used such that each bin contains at most one copy of any item and respects a given capacity limit. Focusing on the generalized First Fit Decreasing (FFD) algorithm, the paper establishes, for the first time, a performance guarantee for $\mathsf{FFD}_q$ in a special case of $q$BP without relying on the "single-item-at-the-end" assumption. By constructing a carefully designed subinstance ${D'}_q$ and leveraging techniques from combinatorial optimization and approximation analysis, the authors prove that $\mathsf{FFD}_q(D_q) \leq \frac{11}{9} \mathsf{OPT}(D_q) + 3q$. This result provides the first effective approximation ratio bound for this variant of the bin packing problem.

Abstract

We consider a variant of the bin packing problem with constraints on the number of copies of each item and their placement in the packing. The input $D_q := DD\ldots$ is defined as $q$ consecutive copies of the multiset $D$, with a fixed bin capacity $S$. Note that, for each item in $D$, there are $q$ copies in $D_q$. The goal is to pack all the items in $D_q$ into the minimum number of bins, such that each bin contains at most one copy of each item and the total size of all items in a bin does not exceed the bin capacity $S$. We call this problem $q$BP. First Fit Decreasing ($\mathsf{FFD}$) is a classical bin packing algorithm: it first orders the items in nonincreasing order, then packs the next item into the first bin where it fits. In the literature, $\mathsf{FFD}$ proofs rely on the assumption that the last bin in the $\mathsf{FFD}$ packing contains only a single item. This assumption does not naturally extend to the $q$BP problem. In this paper, we circumvent this difficulty by analyzing $\mathsf{FFDq(D_q)}$ on a carefully chosen subinstance ${D'}_q \subseteq D_q$ ($q$ consecutive copies of $D$, each copy sorted in non-increasing order) while preserving the same upper bound for the original input $D_q$. We show that the approximation ratio of $\mathsf{FFDq(D_q)}$ for some special cases is \begin{align*} \mathsf{FFDq(D_q)} \leq \frac{11}{9}\mathsf{OPT(D_q)} + 3q \end{align*} where $\mathsf{FFDq}$ and $\mathsf{OPT}$ denote the number of bins used by the $\mathsf{FFD}$ generalization and by an optimal algorithm, respectively.

Problem

• bin packing
• qBP
• First Fit Decreasing
• approximation ratio
• generalization bound

Innovation

• qBP
• First Fit Decreasing
• approximation ratio
• bin packing with copies
• generalization analysis