Bin Packing Problem

Bin Packing Problem is a combinatorial optimization problem that involves packing objects of varying sizes into a finite number of bins with a fixed capacity while minimizing the number of bins used.

1 Definition

Given:

• A set of n items, each with a weight w_i.
• A set of bins, each with a fixed capacity C.

The objective is to pack all items into the smallest number of bins such that:

• The total weight of items in any bin does not exceed its capacity.
• Each item is placed in exactly one bin.

2 Variants

There are multiple variations of the bin packing problem:

• Online vs. Offline Bin Packing
  • In the offline version, all items are known in advance.
  • In the online version, items arrive sequentially and must be placed immediately.
• One-Dimensional vs. Multi-Dimensional
  • In 1D bin packing, items have only one attribute (e.g., weight).
  • In 2D or 3D bin packing, items also have height, width, or volume constraints.
• Bin Packing with Conflicts
  • Some items cannot be placed in the same bin due to constraints.

3 Example

Suppose we have 5 items with weights: [4, 8, 1, 4, 7] and bin capacity 10. The optimal packing might be:

Bin	Items
Bin 1	[8, 1]
Bin 2	[7]
Bin 3	[4, 4]

Total bins used: 3

4 Heuristic Algorithms

Since the bin packing problem is NP-hard, exact solutions are computationally expensive for large inputs. Common heuristic approaches include:

• First-Fit (FF)
  • Place each item in the first available bin that has enough space.
• Best-Fit (BF)
  • Place each item in the bin that will have the least remaining space after insertion.
• Worst-Fit (WF)
  • Place each item in the bin with the most remaining space.
• Next-Fit (NF)
  • Similar to First-Fit but keeps using the current bin until it is full.

5 Implementation

A simple First-Fit algorithm in Python:

def first_fit(weights, bin_capacity):
    bins = []
    for w in weights:
        placed = False
        for b in bins:
            if sum(b) + w <= bin_capacity:
                b.append(w)
                placed = True
                break
        if not placed:
            bins.append([w])
    return bins

items = [4, 8, 1, 4, 7]
capacity = 10
bins = first_fit(items, capacity)
print("Bins used:", bins)

6 Applications

• Cloud Computing
  • Allocating virtual machines to servers efficiently.
• Logistics
  • Packing cargo in containers with minimal waste.
• Memory Management
  • Allocating memory blocks in operating systems.

7 See Also

• Knapsack Problem
• Bin Covering Problem
• Packing Problem
• Combinatorial Optimization
• Approximation Algorithm