Bin-packing algorithms | 装箱问题算法

📚 Bin-packing algorithms | 装箱问题算法

In computer science, the bin-packing problem is a classic optimisation problem. Given a set of items, each with a size, and bins of a fixed capacity, the goal is to pack all the items into as few bins as possible. This problem is NP-hard, meaning efficient exact solutions are unlikely to exist for large instances. Therefore, heuristic algorithms such as first-fit and best-fit are used to find good but not necessarily optimal solutions. Bin-packing has direct applications in memory allocation, container loading, and cutting stock problems.

在计算机科学中，装箱问题是一个经典的优化问题。给定一组物品，每个物品有一个尺寸，以及固定容量的箱子，目标是用尽可能少的箱子装下所有物品。该问题是NP难问题，意味着对于大规模实例不太可能存在高效精确解。因此，首次适应、最佳适应等启发式算法被用来寻找接近最优的解。装箱问题在内存分配、集装箱装载和切割材料问题中有直接应用。

1. Introduction to Bin-Packing Problem | 装箱问题简介

In computer science, the bin-packing problem is a classic optimisation problem. Given a set of items, each with a size, and bins of a fixed capacity, the goal is to pack all the items into as few bins as possible. This problem is NP-hard, meaning efficient exact solutions are unlikely to exist for large instances. Therefore, heuristic algorithms such as first-fit and best-fit are used to find good but not necessarily optimal solutions. Bin-packing has direct applications in memory allocation, container loading, and cutting stock problems.

在计算机科学中，装箱问题是一个经典的优化问题。给定一组物品，每个物品有一个尺寸，以及固定容量的箱子，目标是用尽可能少的箱子装下所有物品。该问题是NP难问题，意味着对于大规模实例不太可能存在高效精确解。因此，首次适应、最佳适应等启发式算法被用来寻找接近最优的解。装箱问题在内存分配、集装箱装载和切割材料问题中有直接应用。

2. Formal Definition and Lower Bound | 形式化定义和下界

Formally, we have n items with sizes s₁, s₂, ..., sₙ (positive numbers) and bins of capacity C. We must partition the items into bins such that the sum of sizes in each bin does not exceed C. The objective is to minimise the number of bins used. A simple lower bound (minimum possible bins) can be computed:

形式化地，我们有n个物品，尺寸为s₁, s₂, ..., sₙ（正数），箱子容量为C。我们需要将物品分配到箱子中，使得每个箱子中物品尺寸之和不超过C。目标是最小化使用的箱子数量。一个简单的下界（最少可能箱子数）可以计算：

Lower bound = ⌈(∑ sᵢ) / C⌉

下界 = ⌈(∑ sᵢ) / C⌉

where ⌈x⌉ denotes the ceiling function. For example, if total size = 25 and C = 10, then LB = ⌈2.5⌉ = 3 bins. This lower bound provides a benchmark; no valid packing can use fewer bins.

其中⌈x⌉表示向上取整。例如，总尺寸为25，C=10，则下界 = ⌈2.5⌉ = 3个箱子。这个下界提供了一个基准；任何有效打包都不能使用更少的箱子。

3. First-Fit Algorithm | 首次适应算法

The first-fit (FF) algorithm is an online greedy approach. It processes items in the order they arrive. For each item, it places the item into the first bin (lowest index) that has enough remaining capacity. If no such bin exists, it opens a new bin. This algorithm is easy to implement and has a time complexity of O(n log n) if bins are tracked efficiently, or O(n²) with a linear scan.

首次适应（FF）算法是一种在线贪心方法。它按照物品到达顺序处理。对于每个物品，将其放入第一个具有足够剩余容量的箱子（最低索引）中。如果不存在这样的箱子，则打开一个新箱子。该算法易于实现，如果高效跟踪箱子，时间复杂度为O(n log n)；若使用线性扫描则为O(n²)。

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4. Worked Example: First-Fit | 示例：首次适应

Consider bin capacity 10 and items with sizes: 4, 8, 1, 4, 2, 1 (arriving in this order). Let’s simulate first-fit.

Result: 2 bins used. Lower bound ⌈20/10⌉ = 2, optimal packing achieved.

考虑箱子容量10，物品尺寸依次为：4, 8, 1, 4, 2, 1。模拟首次适应算法。

步骤	物品尺寸	动作	箱子（内容）
1	4	放入Bin1	Bin1: [4] (剩余6)
2	8	Bin1剩余6 < 8，打开Bin2	Bin1: [4] (6), Bin2: [8] (2)
3	1	Bin1剩余6 &gt;= 1，放入Bin1	Bin1: [4,1] (5), Bin2: [8] (2)
4	4	Bin1剩余5 &gt;= 4，放入Bin1	Bin1: [4,1,4] (9), Bin2: [8] (2)
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